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ESAIM: M2AN 54 (2020) 1917โ1949 ESAIM: Mathematical Modelling and Numerical Analysishttps://doi.org/10.1051/m2an/2020025 www.esaim-m2an.org
SOME MULTIPLE FLOW DIRECTION ALGORITHMS FOR OVERLAND FLOWON GENERAL MESHES
Julien Coatlevenห
Abstract. After recalling the most classical multiple flow direction algorithms (MFD), we establishtheir equivalence with a well chosen discretization of ManningโStrickler models for water flow. Fromthis analogy, we derive a new MFD algorithm that remains valid on general, possibly non conformingmeshes. We also derive a convergence theory for MFD algorithms based on the ManningโStricklermodels. Numerical experiments illustrate the good behavior of the method even on distorted meshes.
Mathematics Subject Classification. 65N08, 65N12, 65N15.
Received September 17, 2019. Accepted April 9, 2020.
1. Introduction
Overland water flow plays a major role in hydrogeology at many time scales, from million of years forstratigraphic studies of interest for the oil and gas industry to several days or weeks for predicting river floodingand landslides. Intermediate time scales are also increasingly studied as they are of crucial interest for climaticforecasts, from glacier withdrawal to desert expansion. Of course, a huge literature exists in the correspondingfields, describing the physical phenomenons involved with an equally huge model diversity (NavierโStokes,Stokes, shallow water, etc...). Due to the complexity underlying flow models, approximate, phenomenologicallybased models have been developed to increase computational speed. Among those models, a common approachthat has given satisfactory results is the one based on the so-called multiple flow directions algorithms (MFD).
The idea underlying all MFD algorithms is that water routing at large space scales will be mainly governed bythe topographic slope. Using digital elevation models that provide a meshed representation of the topography,those algorithms compute water flow by distributing water from the topographically higher discrete cells tothe topographically lower ones, each distribution formula corresponding to a specific MFD algorithm. In mostcases, the MFD algorithms are developed assuming that the mesh is a uniform cartesian grid, with the samespace step in each direction. Historically, the first MFD algorithms were in fact single flow direction algorithms,where the distribution formula selected only one neighbour, generally the one with the steepest slope. Thedeficiencies of such simple algorithms are the origin of the development of MFD algorithms (see [26]). The mostclassical MFD algorithm [16, 22] uses directly the slope to distribute the flow, while models using powers ofthe slope were developed to concentrate the flow and limit diffusion effects due to the use of coarse meshes
Keywords and phrases. Multiple flow direction algorithm, overland flow, virtual element method, hybrid finite volume, generalmeshes.
IFP Energies nouvelles, 1 et 4 avenue de Bois-Preau, Rueil-Malmaison 92852, France.หCorresponding author: [email protected]
Article published by EDP Sciences c EDP Sciences, SMAI 2020
1918 J. COATLEVEN
(see [18, 21, 25]). Some attempts have been made to apply MFD algorithms on more general meshes, with anemphasis on triangular ones (see [24,27]). We refer the reader to [9] for a comparison of some of those methods,and to the references in the aforementioned papers for a more complete overview of the tremendous literatureon MFD algorithms, on which we will not try to be exhaustive.
All those algorithms proceed sequentially from the higher cells to the lower ones, however as was alreadynoticed in [23], using an MFD algorithm is in fact equivalent to solving a linear system using a specific cellordering. The key idea underlying the present paper comes from a deeper study of the linear system associatedwith one of the earliest MFD algorithm. Indeed, we will explain that this linear system is completely equivalent toa two point flux finite volume scheme (see [10]) applied to a stationary ManningโStrickler model for water flow.It was then clear that replacing the TPFA scheme that requires a strong orthogonality hypothesis on meshes toremain valid by more advanced flux approximation schemes would allow us to derive MFD algorithms adaptedto general meshes. Moreover, this equivalence will also allow us to derive a theoretical framework within whichwe will be able to study the convergence properties of MFD algorithms. In our numerical experiments, we haveconsidered two flux reconstructions inspired by two finite volume schemes: the hybrid finite volume (see [11,12])and the virtual volume method (or conservative first order virtual element method, see [5]). Of course, otherchoices could have been made (for instance VAG finite volumes [13,14] or discontinuous Galerkin methods [7]),however those two schemes will be sufficient to illustrate our approach.
The paper will be organized as follows: after describing the data and meshes, we recall the most classical mul-tiple flow direction algorithms, and reformulate them in a more algebraic fashion. Next, using this reformulationwe explain how they are linked with the TPFA scheme for a family of ManningโStrickler models. We elaborateon this basis to overcome the mesh limitations induced by the TPFA scheme, introducing a new family of MFDalgorithms that will remain valid on a huge class of meshes, including those with hanging nodes. We then studythe convergence properties of all methods and conclude by some numerical illustrations.
2. Classical multiple flow direction algorithms
2.1. Mesh and data description
Let ฮฉ be a bounded polyhedral connected domain of R2, whose boundary is denoted Bฮฉ โ ฮฉzฮฉ. We recall theusual notations describing a mesh โณ โ p๐ฏ ,โฑq of ฮฉ. ๐ฏ is a finite family of connected open disjoint polygonalsubsets of ฮฉ (the cells of the mesh), such that ฮฉ โ Y๐พP๐ฏ ๐พ. For any ๐พ P ๐ฏ , we denote by |๐พ| the measure of|๐พ|, by B๐พ โ ๐พz๐พ the boundary of ๐พ, by โ๐พ its diameter and by ๐ฅ๐พ its barycenter. โฑ is a finite family ofdisjoint subsets of hyperplanes of R2 included in ฮฉ (the faces of the mesh) such that, for all ๐ P โฑ , its measureis denoted |๐|, its diameter โ๐ and its barycenter ๐ฅ๐. For any ๐พ P ๐ฏ , there exists a subset โฑ๐พ of โฑ such thatB๐พ โ Y๐Pโฑ๐พ
๐. Then, for any ๐ P โฑ , we denote by ๐ฏ๐ โ t๐พ P ๐ฏ | ๐ P โฑ๐พu. Next, for all ๐พ P ๐ฏ and all ๐ P โฑ๐พ ,we denote by ๐๐พ,๐ the unit normal vector to ๐ outward to ๐พ, and ๐๐พ,๐ โ ||๐ฅ๐ยด๐ฅ๐พ ||. The set of boundary facesis denoted โฑext, while interior faces are denoted โฑint. Finally for any ๐ P โฑint, we denote ๐๐พ๐ฟ โ ||๐ฅ๐พ ยด ๐ฅ๐ฟ||
where ๐ฏ๐ โ t๐พ, ๐ฟu. We assume that there exists a subset โฑext,in of โฑext such that:
Bฮฉin โฤ
๐Pโฑext,in
๐ where Bฮฉin โ t๐ฅ P Bฮฉ | โ๐ ยจ ๐ ฤ 0u
and we denote of course โฑext,out โ โฑextzโฑext,in. As usual, โ โ max๐พP๐ฏ โ๐พ will denote the mesh size. In whatfollows, we will assume that our mesh satisfies:
(A1) There exists a real number ๐ ฤ 0 and a matching simplicial submesh ๐ฎ๐ฏ of โณ such that for any ๐ P ๐ฎ๐ฏ ,๐โ๐ ฤ ๐๐ where ๐๐ is the inradius of ๐ , and for any ๐ P ๐ฏ and any ๐ P ๐ฎ๐ฏ such that ๐ ฤ ๐พ, ๐โ๐พ ฤ โ๐ .
From [3, 7], we know that assumption (A1) implies that for any ๐ P N, there exists ๐ถ๐ก๐,๐ ฤ 0 independent on โsuch that for any ๐พ P ๐ฏ , any ๐ P โฑ๐พ and any ๐ P P๐p๐พq:
||๐||๐ฟ2p๐q ฤ ๐ถ๐ก๐,๐โยด12๐ ||๐||๐ฟ2p๐พq. (2.1)
SOME MULTIPLE FLOW DIRECTION ALGORITHMS 1919
Figure 1. Example of real topography.
There also exists ๐ถ๐ก๐ ฤ 0 independent on โ such that for any ๐ฃ P ๐ป1p๐พq:
||๐ฃ||๐ฟ2p๐q ฤ ๐ถ๐ก๐
ยด
โยด1๐พ ||๐ฃ||2๐ฟ2p๐พq ` โ๐พ ||โ๐ฃ||2๐ฟ2p๐พq
ยฏ12
. (2.2)
Finally, assumption (A1) implies that for any integer ๐, there exists ๐ถpoly,๐ ฤ 0 such that for any ๐พ P ๐ฏ andany ๐ฃ P ๐ป๐ p๐พq with ๐ P t1, . . . , ๐ ` 1u, there exists ๐ P P๐p๐พq such that
|๐ฃ ยด ๐|๐ป๐p๐พq ` โ12๐พ |๐ฃ ยด ๐|๐ป๐pB๐พq ฤ ๐ถpoly,๐โ๐ ยด๐
๐พ |๐ฃ|๐ป๐ p๐พq for ๐ P t0, . . . , ๐ ยด 1u . (2.3)
In the estimates that will follow, the constant ๐ถ ฤ 0 will always denote by convention a quantity independenton the mesh size โ, whose value can change from line to line. Also by convention for any ๐ P โฑint we willdenote ๐ฏ๐ โ t๐พ, ๐ฟu. In the same way, when considering a cell ๐พ and one of its interior faces ๐ P โฑ๐พ Xโฑint, byconvention the cell ๐ฟ will denote the other element of ๐ฏ๐.
In order to be able to establish error estimates, we assume that the topography, often called the basement inthe geological community and thus usually denoted ๐, satisfies ๐ P ๐ 2,8pฮฉq and that there exists ๐ผ ฤ 0 suchthat ยดโ๐ ฤ ๐ผ for almost every ๐ฅ P ฮฉ. However, from the practical point of view ๐ is only known through somepointwise values, very often given as a digitized surface elevation, on which some interpolation and upscaling ordownscaling processes have been applied (see Fig. 1). Thus, its Laplacian cannot be expected to be practicallycomputable, and the above assumption should really be considered as an abstract technical requirement forestablishing convergence estimates. Thus, our data will more realistically consists in some discrete mesh-basedp๐ต๐พq๐พP๐ฏ representation of ๐, where each ๐ต๐พ can represent several pointwise values of ๐, complemented by awater source term ๐ P ๐ฟ8pฮฉq, possibly also only known through a mesh-based representation.
2.2. The classical multiple flow direction algorithm
In the geological literature, multiple flow direction algorithms are considered as purely algorithmic ways ofdistributing water from one cell to another. Thus, they are generally described in a purely algorithmic fashion.Consequently, in this section to introduce the most classical MFD algorithms we will follow the presentation
1920 J. COATLEVEN
generally found in the geological community, that is to say a purely algorithmic point of view. One of our firsttasks will precisely consists in abstracting ourselves from this algorithmic setting. For any ๐พ P ๐ฏ , let ๐๐พ be avalue for the topography associated with cell ๐พ. To fix notations, consider for the moment that
๐๐พ โ1|๐พ|
ลผ
๐พ
๐ @๐พ P ๐ฏ
but other approximations can be considered, for instance ๐๐พ โ ๐ p๐ฅ๐พq for any ๐พ P ๐ฏ if ๐ is regular enough.Multiple flow direction algorithms are based on formulae to distribute the water flow from a cell to its neighboursthat are topographically lower. The most classical distribution formula consists simply in distributing the flowproportionally to the ratio ๐ ๐พ๐ฟ๐ ๐พ of the discrete slope ๐ ๐พ๐ฟ between the high cell ๐พ and the low cell ๐ฟ regardingthe total positive slope ๐ ๐พ of the high cell ๐พ, where the discrete slope ๐ ๐พ๐ฟ is given by
๐ ๐พ๐ฟ โ|๐|
๐๐พ๐ฟp๐๐พ ยด ๐๐ฟq
and the total positive slope ๐ ๐พ of cell ๐พ is given by
๐ ๐พ โรฟ
๐Pโฑ๐พXโฑint,๐๐พฤ๐๐ฟ
|๐|
๐๐พ๐ฟp๐๐พ ยด ๐๐ฟq .
Notice that in many cases of the literature, as the MFD algorithm is applied on a uniform cartesian mesh withthe same space step in every direction, the face measure |๐| is simply omitted (see for instance [16, 22]), withno impact on the ratio ๐ ๐พ๐ฟ๐ ๐พ . To give a detailed description of the most classical multiple flow directionalgorithms, we need to introduce some notations. Let ๐0 โ max t๐๐พ | ๐พ P ๐ฏ u, ๐ฏ0 โ t๐พ P ๐ฏ | ๐๐พ โ ๐0u and๐ฏยด1 โ H. The set ๐ฏ0 thus denotes the set of cells with maximum topographic height. Then, for any ๐ P N wedefine the set ๐ฏ๐ of elements of ๐ฏ by setting:
๐ฏ๐ โฤ
0ฤ๐ฤ๐
๐ฏ๐ where ๐ฏ๐ โ t๐พ P ๐ฏ | ๐๐พ โ ๐๐u and ๐๐ โ max!
๐๐พ | ๐พ P ๐ฏ z๐ฏ๐ยด1
)
.
By construction, as ๐ฏ is a finite set there exists ๐๐ ฤ 0 such that ๐ฏ๐๐ยด1 โฐ H and ๐ฏ๐๐ยด1 โ ๐ฏ . Thus, for any๐ ฤ ๐๐, we have ๐ฏ๐ โ H and ๐ฏ๐ โ ๐ฏ๐๐ยด1.
To model water sources, we define a family p๐๐พq๐พP๐ฏ by setting:
๐๐พ โ1|๐พ|
ลผ
๐พ
๐ @๐พ P ๐ฏ .
The classical MFD algorithm (reformulated from [16,22]) then reads as follows (see Fig. 2 for a visual illustration,where the width of the arrows is roughly following the amount of distributed water):
(i) For any ๐พ P ๐ฏ , the water influx r๐๐พ is initialized at 0.
(ii) For any ๐พ P ๐ฏ0, the water influx r๐๐พ is given by r๐๐พ โ |๐พ|๐๐พ .
(iii) Loop for ๐ โ 1 . . . ๐๐ ยด 1.
(iii-1) For any ๐ฟ P ๐ฏ๐ยด1, we distribute the entire water influx r๐๐ฟ to the neighbours that belong to ๐ฏ๐ propor-tionally to the slope between the cell ๐ฟ and its neighbour, i.e.:
r๐๐พ รร r๐๐พ `รฟ
๐Pโฑ๐พXโฑint,๐๐พฤ๐๐ฟ,๐ฟP๐ฏ๐ยด1
|๐|r๐๐ฟ
๐๐พ๐ฟ๐ ๐ฟp๐๐ฟ ยด ๐๐พq for all ๐พ P ๐ฏ๐
SOME MULTIPLE FLOW DIRECTION ALGORITHMS 1921
Figure 2. Basic principle of MFD algorithms: water is distributed to lower neighbouring cellsproportionally to the slope.
where ๐ ๐ฟ is the total positive slope in cell ๐ฟ, and รร denotes the action of updating r๐๐พ .
(iii-2) For any ๐พ P ๐ฏ๐, the water influx is complemented by the local sources by setting
r๐๐พ รร r๐๐พ ` |๐พ|๐๐พ .
(iv) End loop for ๐ โ 1 . . . ๐๐ ยด 1.The MFD algorithm admits a reformulation as a linear system that will play a key role in the remaining of
the paper. To our knowledge, only [23] mentioned this reformulation, although without exhibiting an explicitformula. This link seemed to have been most of the time simply overlooked by the geological community:
Theorem 2.1. The MFD algorithm is equivalent to solving the following linear system for the unknownpr๐๐พq๐พP๐ฏ
r๐๐พ ยดรฟ
๐Pโฑ๐พXโฑint,๐๐พฤ๐๐ฟ
|๐|r๐๐ฟ
๐๐พ๐ฟ๐ ๐ฟp๐๐ฟ ยด ๐๐พq โ |๐พ|๐๐พ @๐พ P ๐ฏ (2.4)
where
๐ ๐พ โรฟ
๐Pโฑ๐พXโฑint,๐๐พฤ๐๐ฟ
|๐|
๐๐พ๐ฟp๐๐พ ยด ๐๐ฟq
using an ordering for the cells of ๐ฏ based on decreasing topography ๐๐พ and a lower triangular solver.
Proof. First, notice that as cells ๐พ P ๐ฏ๐ are updated only at step ๐, their expression is complete past this stepand given by:
r๐๐พ โรฟ
๐Pโฑ๐พXโฑint,๐๐พฤ๐๐ฟ,๐ฟP๐ฏ๐ยด1
|๐|r๐๐ฟ
๐๐พ๐ฟ๐ ๐ฟp๐๐ฟ ยด ๐๐พq ` |๐พ|๐๐พ for all ๐พ P ๐ฏ z๐ฏ๐.
1922 J. COATLEVEN
However, by construction of the sets ๐ฏ๐, any ๐ฟ P ๐ฏ such that ๐๐ฟ ฤ ๐๐พ for ๐พ P ๐ฏ๐, will satisfy ๐ฟ P ๐ฏ๐ยด1. Thus,the above expression can be simplified in:
r๐๐พ โรฟ
๐Pโฑ๐พXโฑint,๐๐พฤ๐๐ฟ
|๐|r๐๐ฟ
๐๐พ๐ฟ๐ ๐ฟp๐๐ฟ ยด ๐๐พq ` |๐พ|๐๐พ for all ๐พ P ๐ฏ z๐ฏ๐
where the set ๐ฏ๐ยด1 no longer appears. As for any ๐พ P ๐ฏ , there exists 0 ฤ ๐ ฤ ๐๐ ยด 1 such that ๐พ P ๐ฏ๐,and noticing that ๐ ๐ฟ ฤ 0 as soon as there exists ๐ P โฑ๐ฟ X โฑint, ๐ฏ๐ โ t๐พ, ๐ฟu such that ๐๐ฟ ฤ ๐๐พ , (2.4) followsby induction. Finally, starting from (2.4), if one chooses an ordering for the cells of ๐ฏ based on decreasingtopography ๐๐พ , it is clear that the above system becomes a lower triangular one for r๐๐พ . It should be obvi-ous at this point that the associated lower triangular solver then coincides exactly with the classical MFDalgorithm.
A great advantage of considering the above system rather than its algorithmic counterpart is that it makesclear that triangular solvers are not the only possible linear solvers, which was indeed the main point of [23].A wide range of linear solvers can be used to solve (2.4), and most importantly parallel solvers than canconsiderably speed up the solving process on meshes with a huge cell number.
Remark 2.2. Many variants of this classical MFD algorithm exist, which at least to the authors knowledgemainly consist in modifying the way the influx is distributed from an upper cell to its lower neighbours: powersof the slope instead of the slope, probabilistic repartitions, repartitions using a specified number of neighboursfor a given mesh structure, etc... With the exception of distribution formulae that use powers of the slope insteadof the slope itself, proceeding as we have done for the most classical MFD algorithm it is not difficult to showthat all those variants can be rewritten under the form:
r๐๐พ ยดรฟ
๐Pโฑ๐พXโฑint,๐๐พฤ๐๐ฟ
๐๐|๐|r๐๐ฟ
๐๐พ๐ฟ๐ ๐ฟp๐๐ฟ ยด ๐๐พq โ |๐พ|๐๐พ @๐พ P ๐ฏ
with this time
๐ ๐พ โรฟ
๐Pโฑ๐พXโฑint,๐๐พฤ๐๐ฟ
๐๐|๐|
๐๐พ๐ฟp๐๐พ ยด ๐๐ฟq .
The coefficient ๐๐ associated with each face will take different values depending on the way one want to modifythe influx repartition. Notice that we have chosen to not consider distribution formulae based on powers of theslope as they only had some technical difficulties with no fundamental differences in the analysis.
3. On the link with ManningโStrickler models
Consider the following classical transient ManningโStrickler model:ห
ห
ห
ห
ห
ห
B๐ข
B๐กยด div p๐๐ข๐โ๐q โ ๐ in ฮฉ
ยด๐ข๐โ๐ ยจ ๐ โ 0 on Bฮฉin
where ๐ข is the water height, ๐ a parameter and ๐ denotes the outward normal to ฮฉ. The coefficient ๐ P ๐ฟ8pฮฉqis the inverse of the ManningโStrickler coefficient expressing soil roughness, and is assumed to satisfy 0 ฤ ๐ยด ฤ๐ ฤ ๐` ฤ `8 almost everywhere in ฮฉ. Formally, the steady state associated with the above system is thefollowing stationary ManningโStrickler model for overland flow:
ห
ห
ห
ห
ห
ยดdiv p๐๐ข๐โ๐q โ ๐ in ฮฉ
ยด๐๐ข๐โ๐ ยจ ๐ โ 0 on Bฮฉin.(3.1)
SOME MULTIPLE FLOW DIRECTION ALGORITHMS 1923
Denoting ๐ข๐,๐ โ ๐๐ข๐ , ๐ฝ โ ยดโ๐ P ๐ 1,8pฮฉq and ๐ โ ยดโ๐ ฤ ๐ผ ฤ 0 P ๐ฟ8pฮฉq, remark that (3.1) can berewritten:
ห
ห
ห
ห
ห
๐ฝ ยจโ๐ข๐,๐ ` ๐๐ข๐,๐ โ ๐ in ฮฉ
๐ข๐,๐ โ 0 on Bฮฉin.(3.2)
Stationary transport problems of the form (3.2) have of course received a lot of attention in the existingliterature, in particular as they correspond to a popular model for neutron transport. Their well-posedness hasbeen considered for instance in [1, 2], or more recently in [8, 15, 17]. Thus, from the results of [8, 15] and theregularity hypotheses on ๐, ๐ , and ๐, we know that there exists a unique ๐ข P ๐ฟ8pฮฉq solution of (3.2). As thewater height ๐ข only appears through its ๐-th power ๐ข๐, in the remaining of the paper we will with a slightabuse of notations directly use ๐ข instead of ๐ข๐.
We are now going to explain that the classical MFD algorithm coincides with a well chosen discretizationof the stationary ManningโStrickler model. Assume that the mesh is orthogonal, i.e. there exists a family ofcentroids p๐ฅ๐พq๐พP๐ฏ such that:
๐ฅ๐พ P ๐พ @๐พ P ๐ฏ and๐ฅ๐ฟ ยด ๐ฅ๐พ
||๐ฅ๐ฟ ยด ๐ฅ๐พ ||โ ๐๐พ,๐ for ๐ P โฑint, ๐ โ t๐พ, ๐ฟu
and let us denote ๐๐พ,๐ the distance of ๐ฅ๐พ to the hyperplane containing ๐ for any ๐ P โฑ๐พ and any ๐พ P ๐ฏ .Then, one can use a two-point finite volume scheme to discretize the steady-state Manning model. Assume that๐๐พ โ ๐ p๐ฅ๐พq or at least a second order approximation of it. Denoting ๐ข๐พ for any ๐พ P ๐ฏ the discrete waterheight unknown, if one further assumes that ๐๐ โ ๐๐พ for any ๐ P โฑext and ๐พ P ๐ฏ๐ which is generally what isdone in practical applications of the MFD algorithm, for any ๐พ P ๐ฏ we get:
รฟ
๐Pโฑ๐พXโฑint
๐๐พ๐ฟr๐ขup๐ p๐๐พ ยด ๐๐ฟq โ |๐พ|๐๐พ
where r๐ขup๐ โ ๐ข๐พ if ๐๐พ ฤ ๐๐ฟ and r๐ขup
๐ โ ๐ข๐ฟ if ๐๐พ ฤ ๐๐ฟ and the transmissivity ๐๐พ๐ฟ is given for instance by theharmonic mean:
๐๐พ๐ฟ โ|๐|๐๐พ๐๐ฟ
๐๐พ๐๐ฟ,๐ ` ๐๐ฟ๐๐พ,๐
where ๐๐พ โ1|๐พ|
ลผ
๐พ
๐.
Immediately we deduce the following equivalence result, which despite its simplicity is the cornerstone of thepresent paper:
Theorem 3.1. The TPFA scheme for Manning Stricklerโs model is equivalent to the MFD algorithm.
Proof. Gathering the faces by upwinding kind, we get:รฟ
๐Pโฑ๐พXโฑint,๐๐พฤ๐๐ฟ
๐๐พ๐ฟ๐ข๐พ p๐๐พ ยด ๐๐ฟq ยดรฟ
๐Pโฑ๐พXโฑint,๐๐พฤ๐๐ฟ
๐๐พ๐ฟ๐ข๐ฟ p๐๐ฟ ยด ๐๐พq โ |๐พ|๐๐พ . (3.3)
Setting๐ ๐พ โ
รฟ
๐Pโฑ๐พXโฑint,๐๐พฤ๐๐ฟ
๐๐พ๐ฟ p๐๐พ ยด ๐๐ฟq
and noticing that ๐ ๐ฟ ฤ 0 as soon as there exists ๐ P โฑ๐ฟ X โฑint such that ๐๐ฟ ฤ ๐๐พ , we see that equation (3.3)can be rewritten:
๐ ๐พ๐ข๐พ ยดรฟ
๐Pโฑ๐พXโฑint,๐๐พฤ๐๐ฟ
๐๐พ๐ฟ๐ข๐ฟ p๐๐ฟ ยด ๐๐พq โ |๐พ|๐๐พ .
Defining the water influx by r๐๐พ โ ๐ ๐พ๐ข๐พ , we thus obtain:
r๐๐พ ยดรฟ
๐Pโฑ๐พXโฑint,๐๐พฤ๐๐ฟ
๐๐พ๐ฟr๐๐ฟ
๐ ๐ฟp๐๐ฟ ยด ๐๐พq โ |๐พ|๐๐พ . (3.4)
If we take ๐ โ 1, immediately we see that (3.4) is exactly the MFD equation (2.4).
1924 J. COATLEVEN
Surprisingly, this result seems to be absent of the MFD literature. The main reason is probably that formu-lations of MFD algorithms as linear systems are equally difficult to find. From this equivalence between theclassical MFD and the two-point flux approximation (TPFA) of the classical ManningโStrickler model, someuseful observations can be made. Probably the most surprising one is that the MFD unknown pr๐๐พq๐พP๐ฏ can beused instead of ๐ข๐พ to solve the two-point approximation of the ManningโStrickler model. Moreover, existenceand uniqueness of solutions of the MFD problem (3.4) are now an immediate consequence of its equivalencewith the MFD algorithm without requiring any hypothesis on the topography. Next, let us mention that whenmodeling overland flow, the quantity of interest is not the water height but the water discharge, i.e. the norm||๐โ๐โ๐|| of the water flux vector. This is the reason why no water height explicitly appears in MFD algorithms.However, a crucial consequence of our equivalence result is that the usual unknown r๐๐พ of the MFD algorithm,while an excellent choice from the algebraic perspective, is probably not the good quantity to represent the waterdischarge. Indeed, using r๐๐พ โ ๐ ๐พ๐ข๐พ and the consistency of the two-point formula we see that it approximates:
r๐๐พ ยซรฟ
๐Pโฑ๐พ
ลผ
๐
๐ข pยด๐โ๐ ยจ ๐๐พ,๐q`
where ๐ฃ` โ maxp0, ๐ฃq. We cannot expect such a r๐๐พ to be an approximation of the norm of ๐๐ขโ๐, as itapproximates the accumulated influx in a cell which is a mesh dependent quantity. Thus, no kind of convergencelet alone approximation properties can be expected for such a quantity in general. To effectively compute adiscrete water discharge ๐๐พ for each cell ๐พ P ๐ฏ , we reconstruct cellwise the water flux vector by setting:
๐๐พ โรฟ
๐Pโฑ๐พXโฑint,๐๐พฤ ๐๐ฟ
๐๐พ๐ฟr๐๐พ
|๐พ|๐ ๐พp๐๐พ ยด ๐๐ฟq p๐ฅ๐ ยด ๐ฅ๐พq ยด
รฟ
๐Pโฑ๐พXโฑint,๐๐พฤ๐๐ฟ
๐๐พ๐ฟr๐๐ฟ
|๐พ|๐ ๐ฟp๐๐ฟ ยด ๐๐พq p๐ฅ๐ ยด ๐ฅ๐พq .
The consistent water discharge is immediately given by ๐๐พ โ ||๐๐พ ||. We will illustrate in the numerical sectionthe convergence deficiency of r๐๐พ , and how ๐๐พ has a much better behavior. Obviously, for any ๐พ P ๐ฏ such that๐ ๐พ โฐ 0, we can compute an equivalent positive water height by setting ๐ข๐พ โ r๐๐พ๐ ๐พ . Cells where ๐ ๐พ โ 0 arecells where all discrete fluxes are ingoing fluxes, thus it is clear that one should either set ๐ข๐พ โ `8 or ๐ข๐พ โ 0depending if water effectively reaches the cell or not. From the definition of ๐๐พ , we see that the value of ๐ข๐พ
for such cells will have no influence on the water discharge and its asymptotic convergence. It is thus clear thatone can always define ๐ข๐พ by setting:
๐ข๐พ โ
ห
ห
ห
ห
ห
ห
r๐๐พ
๐ ๐พif ๐ ๐พ ฤ 0
0 otherwise.
Remark 3.2. Notice that using the harmonic mean is not mandatory if one can use an approximation of ๐directly on the faces, leading to the transmissivity ๐๐พ๐ฟ โ |๐|๐๐๐๐พ๐ฟ in which case (3.4) will correspond to oneof the many variants of the classical MFD.
4. Some conservative and consistent flux reconstructions on general meshes
From the equivalence between the MFD algorithms and the TPFA scheme for the stationary ManningโStrickler model, an obvious way of generalizing MFD algorithms to general meshes consists in replacing theTPFA flux reconstruction formula by more advanced flux reconstruction techniques that are still valid ongeneral meshes. We follow the idea of [4]: we consider a gradient reconstruction in each cell that will consist ina consistent part plus a stabilization part. However, contrary to [4] where the stabilization was used to obtaincoercivity, here we will use this stabilization to enforce conservativity.
SOME MULTIPLE FLOW DIRECTION ALGORITHMS 1925
4.1. A conservative flux reconstruction formula
For any cell ๐พ P ๐ฏ , let us denote ๐ต๐พ โ ๐๐พp๐q the local values associated with the topography ๐, and๐๐พ โ R๐ฉ๐พ the set of local values, with ๐ฉ๐พ the number of local values. The operator ๐๐พ : ๐ป1p๐พq รรร ๐๐พ
will of course depend on the considered reconstruction formula, however to simplify notations we consider thatthe value ๐๐พ at ๐ฅ๐พ always belongs to the set of local values. Those local values represent all the knowledge wehave in practice of the field ๐, and once again its Laplacian cannot be expected to be computable. For typicalapplications such as stratigraphic modelling it consists in cell values complemented by vertex or face values, thusconditioning the schemes we can effectively use. We assume that we are given a gradient reconstruction operatorโ๐พ : ๐๐พ รรร R2, and we define a reconstruction formula in each cell through the operator ฮ ๐พ : ๐๐พ รรร P1p๐พq,where P1p๐พq is the set of first order polynomials on ๐พ and:
ฮ ๐พ p๐ต๐พq p๐ฅq โ ๐๐พ `โ๐พ p๐ต๐พq ยจ p๐ฅยด ๐ฅ๐พq
and thus โฮ ๐พ p๐ต๐พq โ โ๐พ p๐ต๐พq. Such gradient reconstructions are the usual basis of modern finite volumemethods (see [4, 5, 12]), and many formulae can be found in the literature. Using those reconstructions, it istempting to define the flux operator ๐น๐พ,๐ p๐ต๐พq by setting:
๐น๐พ,๐ p๐ต๐พq โ ยด|๐|๐๐พโ๐พ p๐ต๐พq ยจ ๐๐พ,๐.
As consistency of the flux will of course rely on the consistency of the gradient reconstruction operators, theabove definition will indeed lead to consistent fluxes, however to construct our generalized MFD algorithmsconservativity will play a crucial role. This means that we will require the conservativity of the flux:
รฟ
๐พP๐ฏ๐
๐น๐พ,๐ p๐ต๐พq โ 0
which cannot be satisfied in general (except by the two-point fluxes) with such a simple formula. This is thereason why, following the ideas of [4], we introduce a stabilized gradient operator โ๐พ,๐ : ๐๐พ ห R รรร R bysetting:
โ๐พ,๐ p๐ต๐พ , ๐๐q โ โ๐พ p๐ต๐พq `๐
๐๐พ,๐pp๐๐ ยด ๐๐พq ยดโ๐พ p๐ต๐พq ยจ p๐ฅ๐ ยด ๐ฅ๐พqq๐๐พ,๐
where ๐ ฤ 0 is a stabilization parameter. Then, for any ๐พ P ๐ฏ and any ๐ P โฑ๐พ , we define an intermediate fluxoperator ๐น๐พ,๐ : ๐๐พ ห R รรร R by setting:
๐น๐พ,๐ p๐ต๐พ , ๐๐q โ ยด|๐|๐๐พโ๐พ,๐ p๐ต๐พ , ๐๐q ยจ ๐๐พ,๐.
Obviously, for any ๐ P โฑint, we would like to define ๐ as the solution ofรฟ
๐พP๐ฏ๐
๐น๐พ,๐
ยด
๐ต๐พ , ๐
ยฏ
โ 0. (4.1)
Fortunately, as ๐ is positive almost everywhere we immediately deduce that such a ๐ is always defined and isgiven by:
๐ โ1
ห
รฟ
๐P๐ฏ๐
๐๐๐
๐๐,๐
ยธ
ห
รฟ
๐พP๐ฏ๐
๐๐๐พ
๐๐พ,๐๐๐พ `
รฟ
๐พP๐ฏ๐
โ๐พ p๐ต๐พq ยจ
ห
๐๐๐พ
๐๐พ,๐p๐ฅ๐ ยด ๐ฅ๐พq ยด ๐๐พ๐๐พ,๐
ห
ยธ
. (4.2)
Thus, for any ๐ P โฑint, it is legitimate to define the conservative flux operator ๐น๐พ,๐ :ล
๐ฟP๐ฏ๐๐๐ฟ รรร R by
setting:๐น๐พ,๐ p๐ต๐q โ ๐น๐พ,๐
ยด
๐ต๐พ , ๐
ยฏ
(4.3)
1926 J. COATLEVEN
where ๐ is the unique solution of (4.1), while for any ๐ P โฑext, we simply set:
๐น๐พ,๐ p๐ต๐q โ ยด|๐|๐๐พโ๐พ p๐ต๐พq ยจ ๐๐พ,๐. (4.4)
Remark 4.1. It is tempting to recover conservativity using the following simpler formula for ๐น๐พ,๐ p๐ต๐q:
๐น๐พ,๐ p๐ต๐q โ12p๐น๐พ,๐ p๐ต๐พq ยด ๐น๐ฟ,๐ p๐ต๐ฟqq .
The main drawback of this alternative formula can be understood in cases where ๐ is discontinuous, where thiswould obviously lead to a very coarse approximation of the flux along the lines of discontinuity of ๐. Providedthose discontinuities are resolved by the mesh, our formula for ๐น๐พ,๐ p๐ต๐q will instead behave as the usualharmonic mean. It is thus a more robust and versatile choice.
4.2. Consistency properties of the flux
Denoting ๐ตp๐ฅ, ๐q the ball centered at ๐ฅ of radius ๐ and ๐ตฮฉp๐ฅ, ๐q โ ๐ตp๐ฅ, ๐qXฮฉ, we recall the usual definitionof strong consistency for gradient reconstruction operators:
Definition 4.2 (Strong consistency). The gradient reconstruction operator โ๐พ is strongly consistent if andonly if there exists ๐ถ ฤ 0 independent on โ such that for any ๐ P ๐ถ2
ยด
๐ตฮฉ p๐ฅ๐พ , ๐qยฏ
, every 0 ฤ ๐ ฤ 2โ๐พ andevery ๐ฅ P ๐ตฮฉ p๐ฅ๐พ , ๐q:
|โ๐p๐ฅq ยดโ๐พ p๐๐พp๐qq| ฤ ๐ถ๐||๐||๐ 2,8p๐ตฮฉp๐ฅ๐พ ,๐qXฮฉq.
As an immediate consequence of the above definition, we have, using Taylorโs expansion and the density of๐ถ8
ยด
๐ตฮฉ p๐ฅ๐พ , ๐qยฏ
in ๐ 2,8 p๐ตฮฉp๐ฅ, ๐qq that for any ๐ P ๐ 2,8pฮฉq there exists ๐ถ ฤ 0 depending on ๐ but not onโ such that for every โ๐พ ฤ ๐ ฤ 2โ๐พ , any ๐พ P ๐ฏ and almost every ๐ฅ P ๐ตฮฉ p๐ฅ๐พ , ๐q:
|โ๐p๐ฅq ยดโ๐พ p๐๐พp๐qq| ฤ ๐ถ๐||๐||๐ 2,8p๐ตฮฉp๐ฅ๐พ ,๐qq
and|๐p๐ฅq ยดฮ ๐พ p๐๐พp๐qq p๐ฅq| ฤ ๐ถ๐2||๐||๐ 2,8p๐ตฮฉp๐ฅ๐พ ,๐qq.
Another immediate consequence is the consistency of our discrete fluxes:
Proposition 4.3. Assume that assumption (A1) is satisfied and that the gradient reconstruction operator isstrongly consistent. Then, there exists ๐ถ ฤ 0 independent on โ such that for any ๐พ P ๐ฏ , any ๐ P โฑ๐พ , any๐ P ๐ 2,8pฮฉq and any ๐ P ๐ 1,8pฮฉq, we have:
ห
ห
ห
ห
ห
ห
ห
ห
1|๐|
๐น๐พ,๐ p๐ต๐q ` ๐โ๐ ยจ ๐๐พ,๐
ห
ห
ห
ห
ห
ห
ห
ห
๐ฟ8p๐ตฮฉp๐ฅ๐พ ,2โ๐พqq
ฤ ๐ถโ๐พ ||๐||๐ 1,8pฮฉq||๐||๐ 2,8pฮฉq. (4.5)
Proof. By density of ๐ถ8`
ฮฉห
in ๐ 1,8pฮฉq and ๐ 2,8pฮฉq, it is clear that it suffices to establish the result for๐ P ๐ถ8
`
ฮฉห
and ๐ P ๐ถ8`
ฮฉห
. Assuming such regularity, we start by establishing that there exists ๐ถ ฤ 0independent on โ such that for any ๐ P โฑint:
|๐ ยด ๐ p๐ฅ๐q | ฤ ๐ถโ2๐พ ||๐||๐ 2,8pฮฉq and |๐ ยดฮ ๐พ p๐ต๐พq p๐ฅ๐q | ฤ ๐ถโ2
๐พ ||๐||๐ 2,8pฮฉq @๐พ P ๐ฏ๐. (4.6)
As by construction, we have ฮ ๐พ p๐ต๐พq p๐ฅ๐q โ ๐๐พ `โ๐พ p๐ต๐พq ยจ p๐ฅ๐ ยด ๐ฅ๐พq, slightly rewriting (4.2) we get:
๐ ยด ๐ p๐ฅ๐q โ1
ห
รฟ
๐P๐ฏ๐
๐๐๐
๐๐,๐
ยธ
ห
รฟ
๐ฟP๐ฏ๐
๐๐๐ฟ
๐๐ฟ,๐pฮ ๐ฟ p๐ต๐ฟq p๐ฅ๐q ยด ๐ p๐ฅ๐qq ยด
รฟ
๐ฟP๐ฏ๐
๐๐ฟโ๐ฟ p๐ต๐ฟq ยจ ๐๐ฟ,๐
ยธ
.
SOME MULTIPLE FLOW DIRECTION ALGORITHMS 1927
Immediately, using the strong consistency of the gradient operator we get that for all ๐พ P ๐ฏ๐:
|ฮ ๐พ p๐ต๐พq p๐ฅ๐q ยด ๐ p๐ฅ๐q | ฤ ๐ถโ2๐พ ||๐||๐ 2,8p๐ตฮฉp๐ฅ๐พ ,โ๐พqq
and thus for all ๐พ P ๐ฏ๐:
1ห
รฟ
๐P๐ฏ๐
๐๐๐
๐๐,๐
ยธ
รฟ
๐ฟP๐ฏ๐
๐๐๐ฟ
๐๐ฟ,๐|ฮ ๐ฟ p๐ต๐ฟq p๐ฅ๐q ยด ๐ p๐ฅ๐q | ฤ ๐ถ
รฟ
๐ฟP๐ฏ๐
๐๐๐ฟ
๐๐ฟ,๐โ2
๐ฟ
ห
รฟ
๐P๐ฏ๐
๐๐๐
๐๐,๐
ยธ ||๐||๐ 2,8p๐ตฮฉp๐ฅ๐พ ,โ๐พqq ฤ ๐ถโ2๐พ ||๐||๐ 2,8pฮฉq
as under assumption (A1), we know (see [3,7]) that there exists ๐ถ ฤ 0 such that for any p๐พ, ๐q P ๐ฏ 2 such thatโฑ๐พ Xโฑ๐ โฐ H, we have โ๐โ๐พ ฤ ๐ถ. Moreover, as the gradient operator is strongly consistent, we know that:
|๐๐พโ๐พ p๐ต๐พq ยด ๐ p๐ฅ๐qโ๐ p๐ฅ๐q | ฤ ๐ถโ`
||๐||๐ 1,8pฮฉq||๐||๐ 1,8pฮฉq ` ||๐||๐ฟ8pฮฉq||๐||๐ 2,8pฮฉq
ห
.
Using:รฟ
๐พP๐ฏ๐
๐ p๐ฅ๐qโ๐ p๐ฅ๐q ยจ ๐๐พ,๐ โ 0
we get:ห
ห
ห
ห
ห
รฟ
๐พP๐ฏ๐
๐๐พโ๐พ p๐ต๐พq ยจ ๐๐พ,๐
ห
ห
ห
ห
ห
ฤ ๐ถโ`
||๐||๐ 1,8pฮฉq||๐||๐ 1,8pฮฉq ` ||๐||๐ฟ8pฮฉq||๐||๐ 2,8pฮฉq
ห
.
Meanwhile, we clearly get using again assumption (A1):
1ห
รฟ
๐P๐ฏ๐
๐๐๐
๐๐,๐
ยธ ฤ1
๐๐ยด
1ห
รฟ
๐P๐ฏ๐
1๐๐,๐
ยธ ฤ1
๐๐ยดcard p๐ฏ๐qmax๐P๐ฏ๐
๐๐,๐ ฤ1
๐๐ยดmax๐P๐ฏ๐
โ๐ ฤ๐ถโ๐พ
๐๐ยด
which finally establishes that:|๐ ยด ๐ p๐ฅ๐q | ฤ ๐ถโ2
๐พ ||๐||๐ 2,8pฮฉq
and (4.6) immediately follows using the triangular inequality and the strong consistency of the gradient operator.Then, for any ๐ฅ P ๐ตฮฉ p๐ฅ๐พ , 2โ๐พq and any ๐ P โฑint:
1|๐|
๐น๐พ,๐ p๐ต๐q ` ๐p๐ฅqโ๐p๐ฅq ยจ ๐๐พ,๐ โ ๐๐พpโ๐p๐ฅq ยดโ๐พ p๐ต๐พqq ยจ ๐๐พ,๐ ` p๐p๐ฅq ยด ๐๐พqโ๐p๐ฅq ยจ ๐๐พ,๐
ยด๐๐๐พ
๐๐พ,๐
ยด
๐ ยดฮ ๐พ p๐ต๐พq p๐ฅ๐q
ยฏ
.
Immediately, as the gradient reconstruction operator is strongly consistent we get:
|๐๐พ pโ๐p๐ฅq ยดโ๐พ p๐ต๐พqq ยจ ๐๐พ,๐| ฤ ๐ถ๐`โ๐พ ||๐||๐ 2,8pฮฉq
while Taylorโs expansion immediately gives:
|p๐p๐ฅq ยด ๐๐พqโ๐p๐ฅq ยจ ๐๐พ,๐| ฤ ๐ถโ๐พ ||๐||๐ 1,8pฮฉq||๐||๐ 1,8 .
Under assumption (A1), we have โ๐พ๐๐พ,๐ ฤ โ๐พ๐๐พ ฤ 1๐ which combined with (4.6) gives the desired resultfor ๐ P โฑint. For ๐ P โฑext, we directly get:
1|๐|
๐น๐พ,๐ p๐ต๐q ` ๐p๐ฅqโ๐p๐ฅq ยจ ๐๐พ,๐ โ ๐๐พ pโ๐p๐ฅq ยดโ๐พ p๐ต๐พqq ยจ ๐๐พ,๐ ` p๐p๐ฅq ยด ๐๐พqโ๐p๐ฅq ยจ ๐๐พ,๐
and thus applying the same estimates than in the case of interior faces concludes the proof of (4.5).
1928 J. COATLEVEN
5. A multiple flow direction algorithm on general meshes
5.1. The generalized MFD algorithm
Given an approximation p๐ต๐พq๐พP๐ฏ of the topography and p๐๐พq๐พP๐ฏ of the source term as data, and usingboth upwinding for the water height and our discrete fluxes, the most natural discrete formulation of theManningโStrickler model consists in finding
ยด
p๐ข๐พq๐พP๐ฏ , p๐ข๐q๐Pโฑ๐ext,in
ยฏ
such that:
ห
ห
ห
ห
ห
ห
ห
รฟ
๐Pโฑ๐พ
๐ขup๐ ๐น๐พ,๐ p๐ต๐q โ |๐พ|๐๐พ @๐พ P ๐ฏ
๐ข๐ โ 0 @๐ P โฑ๐ext,in
(5.1)
where the upwinding formula ๐ฃup๐ for any set of cell values p๐ฃ๐พq๐พP๐ฏ is defined by:
๐ฃup๐ โ
ห
ห
ห
ห
ห
๐ฃ๐พ if ๐น๐พ,๐ p๐ต๐q ฤ 0
๐ฃ๐ฟ if ๐น๐พ,๐ p๐ต๐q ฤ 0@๐ P โฑint, ๐ฏ๐ โ t๐พ, ๐ฟu
and
๐ฃup๐ โ
ห
ห
ห
ห
ห
๐ฃ๐พ if ๐น๐พ,๐ p๐ต๐q ฤ 0
0 if ๐น๐พ,๐ p๐ต๐q ฤ 0@๐ P โฑext, ๐ฏ๐ โ t๐พu
and where we have defined the influx boundary associated with the discrete fluxes by setting
โฑ๐ext,in โ t๐ P โฑext | ๐น๐พ,๐ p๐ต๐q ฤ 0, ๐ฏ๐ โ t๐พu u .
By construction of the fluxes, for any ๐ P โฑint we have that ๐น๐ฟ,๐ p๐ต๐q โ ยด๐น๐พ,๐ p๐ต๐q. Mimicking what we havedone for the TPFA scheme, we gather faces by upwinding kind and use the fact that ๐ขup
๐ โ 0 for all ๐ P โฑ๐ext,in,leading to:
ยจ
ห
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
๐น๐พ,๐ p๐ต๐q
ห
โ๐ข๐พ ยดรฟ
๐Pโฑ๐พXโฑint,๐น๐ฟ,๐p๐ต๐qฤ 0
๐ข๐ฟ๐น๐ฟ,๐ p๐ต๐q โ |๐พ|๐๐พ .
Defining:๐ ๐พ โ
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
๐น๐พ,๐ p๐ต๐q and r๐๐พ โ ๐ ๐พ๐ข๐พ
and noticing that ๐ ๐ฟ ฤ 0 as soon as there exists ๐ P โฑ๐ฟ such that ๐น๐ฟ,๐ p๐ต๐q ฤ 0, we see that the above equationcan be rewritten:
r๐๐พ ยดรฟ
๐Pโฑ๐พXโฑint,๐น๐ฟ,๐p๐ต๐qฤ 0
r๐๐ฟ
๐ ๐ฟ๐น๐ฟ,๐ p๐ต๐q โ |๐พ|๐๐พ .
For any ๐ P โฑ๐ext,in, we set:
๐ ๐ โ ยดรฟ
๐พP๐ฏ๐,๐น๐พ,๐p๐ต๐qฤ0
๐น๐พ,๐ p๐ต๐q and r๐๐ โ ๐ ๐๐ข๐.
Gathering all those results, we obtain:ห
ห
ห
ห
ห
ห
ห
ห
r๐๐พ ยดรฟ
๐Pโฑ๐พXโฑint,๐น๐ฟ,๐p๐ต๐qฤ 0
r๐๐ฟ
๐ ๐ฟ๐น๐ฟ,๐ p๐ต๐q โ |๐พ|๐๐พ @๐พ P ๐ฏ
r๐๐ โ 0 @๐ P โฑ๐ext,in.
(5.2)
SOME MULTIPLE FLOW DIRECTION ALGORITHMS 1929
Notice that the above system is set for the unknownsยด
pr๐๐พq๐พP๐ฏ , pr๐๐q๐Pโฑ๐ext,in
ยฏ
, thus it has the same number ofunknowns than the original system (5.1). We still reconstruct a water flux vector cellwise by setting:
๐๐พ โรฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ 0
r๐๐พ
|๐พ|๐ ๐พ๐น๐พ,๐ p๐ต๐q p๐ฅ๐ ยด ๐ฅ๐พq `
รฟ
๐Pโฑ๐พXโฑint,๐น๐พ,๐p๐ต๐qฤ0
r๐๐ฟ
|๐พ|๐ ๐ฟ๐น๐พ,๐ p๐ต๐q p๐ฅ๐ ยด ๐ฅ๐พq (5.3)
the consistent water discharge being given by ๐๐พ โ ||๐๐พ ||. Finally, for any cell ๐พ P ๐ฏ , we set ๐ข๐พ as we havedone for the TPFA case:
๐ข๐พ โ
ห
ห
ห
ห
ห
ห
r๐๐พ
๐ ๐พif ๐ ๐พ โฐ 0
0 otherwise(5.4)
while for any ๐ P โฑ๐ext,in, ๐ข๐ โ 0. Notice that this new system allows to correctly define the numerical method,while the system (5.1) does not uniquely define the water height on cells where ๐ ๐พ โ 0. Contrary to system (3.4),system (5.2) admits no obvious renumbering of mesh elements that makes the system triangular. However, asit is anyway necessary to enable parallelism, we say that the generalized MFD algorithm will consists in solvingthe linear system (5.2) using any linear solver of the literature. Notice that as we are using the unknown r๐ ratherthan ๐ข, the discrete system (5.2) takes the form of a perturbation of the identity matrix. Thus, we expect itscondition number to be relatively good (depending of course of the coefficient ๐) and in any case much betterthan if had tried to solve (5.2) directly.
5.2. On existence and uniqueness for the generalized MFD algorithm
Existence and uniqueness of a solution to the above system are much less obvious than in the case of theTPFA scheme. To explain this fact, let us denote
๐ฏ ห โ t๐พ P ๐ฏ | ๐ ๐พ ฤ 0u .
Using the definition of ๐ฏ ห, summing (5.2) over ๐พ P ๐ฏ , after a straightforward manipulation of the last sum weget:
รฟ
๐พP๐ฏ z๐ฏ หr๐๐พ `
รฟ
๐พP๐ฏ ห
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ 0
r๐๐พ
๐ ๐พ๐น๐พ,๐ p๐ต๐q ยด
รฟ
๐ฟP๐ฏ ห
รฟ
๐Pโฑ๐ฟXโฑint,๐น๐ฟ,๐p๐ต๐qฤ 0
r๐๐ฟ
๐ ๐ฟ๐น๐ฟ,๐ p๐ต๐q โ
รฟ
๐พP๐ฏ|๐พ|๐๐พ
and consequently:
รฟ
๐พP๐ฏ z๐ฏ หr๐๐พ `
รฟ
๐พP๐ฏ ห
รฟ
๐Pโฑ๐พXโฑext,๐น๐พ,๐p๐ต๐qฤ 0
r๐๐พ
๐ ๐พ๐น๐พ,๐ p๐ต๐q โ
รฟ
๐พP๐ฏ|๐พ|๐๐พ . (5.5)
Thus, the existence of a solution for any second member p๐๐พq๐พP๐ฏ requires that:
๐ โ p๐ฏ z๐ฏ หq Y ๐ฏ หห โฐ H where ๐ฏ หห โ t๐พ P ๐ฏ ห | t๐ P โฑ๐พ X โฑext | ๐น๐พ,๐ p๐ต๐q ฤ 0u โฐ Hu . (5.6)
In the case of the TPFA scheme, condition (5.6) is necessarily satisfied. If condition (5.6) is not satisfied,then (5.2) can have a solution only if
รฟ
๐พP๐ฏ|๐พ|๐๐พ โ 0.
From this observation, it seems clear that establishing an exhaustive existence and uniqueness theory for system(5.2) is a more complex task than what one could expect. For the continuous problem, well-posedness is directlylinked to the properties of the topography ๐ and in particular to its Laplacian. Unsurprisingly, well-posednessof (5.2) will be directly linked to the properties of the quantity that plays the role of the topography ๐ at the
1930 J. COATLEVEN
discrete level, that is to say the flux family p๐น๐พ,๐ p๐ต๐qq๐พP๐ฏ ,๐Pโฑ๐พ. Without any further assumption on the mesh,
far from the asymptotic regime consistency alone cannot be expected to ensure well-posedness for coarse meshes.This means that well-posedness will in general be controlled by the structure of the discrete flux family ratherthan by the properties of ๐. As in practice the Laplacian of ๐ cannot be computed in general, it is anyway betterto have an existence result that uses only computable quantities. In fact, not only the necessary condition (5.6)must be satisfied, but it also requires that there does not exist what we will call a flux cycle. For any subset ๐of ๐ฏ , we denote:
โฑ๐int โ t๐ P โฑint | ๐ฏ๐ ฤ ๐u .
We say that a set of cells ๐ ฤ ๐ฏ form a flux cycle if and only if for any ๐พ P ๐
๐น๐พ,๐ p๐ต๐q ฤ 0 @๐ P โฑ๐พz`
โฑ๐พ X โฑ๐int
ห
andรฟ
๐Pโฑ๐พXโฑ๐int,๐น๐พ,๐p๐ต๐qฤ 0
๐น๐พ,๐ p๐ต๐q ฤ 0 andรฟ
๐Pโฑ๐พXโฑ๐int,๐น๐พ,๐p๐ต๐qฤ0
๐น๐พ,๐ p๐ต๐q ฤ 0.
The first condition implies that water can enter the cycle but not leave it, while the second condition implies thatwater will run through the cycle without stopping anywhere. Whether such configurations can exist on coarsemeshes for consistent flux families is a difficult question. However, from the physical point of view flux cyclesare completely unrealistic as they represent regions where water will at some point go from a topographicallylow region to a topographically higher one. Moreover, under the positivity hypothesis on the Laplacian of thetopography, we know that for the exact fluxes we have:
รฟ
๐Pโฑ๐พ
ลผ
๐
ยด๐โ๐ ยจ ๐๐พ,๐ ฤ 0
and thus no flux cycle can exist. Thus, the presence of flux cycles should be considered as anomalous and asign that a too coarse mesh has been used. A natural sufficient condition of existence and uniqueness for (5.2)should then be one that is satisfied by the continuous fluxes and immediately implies that no flux cycle exist.
Proposition 5.1. We say that there exists a flowing path from ๐พ P ๐ฏ z๐ to ๐ if there exists r๐พ P ๐ such that
D ๐๐พ P N, ๐๐พ ฤ 0 and p๐๐q0ฤ๐ฤ๐๐พยด1 such that ๐๐ P โฑint @0 ฤ ๐ ฤ ๐๐พ ยด 1
and ๐ฏ๐๐ โ t๐พ๐, ๐พ๐`1u @0 ฤ ๐ ฤ ๐๐พ ยด 1 where ๐พ โ ๐พ0r๐พ โ ๐พ๐๐พ
and ๐น๐พ๐,๐๐p๐ต๐๐
q ฤ 0.
If ๐ โฐ H and for any cell ๐พ P ๐ฏ z๐ there exists a flowing path to ๐, then (5.2) is well-posed.
Proof. As system (5.2) is linear, it suffices to establish uniqueness to ensure the existence of solutions. Thus,assume that r๐ โ
ยด
pr๐๐พq๐พP๐ฏ , pr๐๐q๐Pโฑ๐ext,in
ยฏ
is solution of (5.2) with zero right-hand side. Then, taking themodulus of (5.2) and summing over ๐พ P ๐ฏ we get:
รฟ
๐พP๐ฏ|r๐๐พ | ฤ
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑint,๐น๐ฟ,๐p๐ต๐qฤ 0
|r๐๐ฟ|
๐ ๐ฟ๐น๐ฟ,๐ p๐ต๐q . (5.7)
Again, we have:
รฟ
๐พP๐ฏ|r๐๐พ | โ
รฟ
๐พP๐ฏ z๐ฏ ห|r๐๐พ |`
รฟ
๐พP๐ฏ ห
รฟ
๐Pโฑ๐พXโฑext,๐น๐พ,๐p๐ต๐qฤ 0
|r๐๐พ |
๐ ๐พ๐น๐พ,๐ p๐ต๐q`
รฟ
๐พP๐ฏ ห
รฟ
๐Pโฑ๐พXโฑint,๐น๐พ,๐p๐ต๐qฤ 0
|r๐๐พ |
๐ ๐พ๐น๐พ,๐ p๐ต๐q
while:รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑint,๐น๐ฟ,๐p๐ต๐qฤ 0
|r๐๐ฟ|
๐ ๐ฟ๐น๐ฟ,๐ p๐ต๐q โ
รฟ
๐ฟP๐ฏ ห
รฟ
๐Pโฑ๐ฟXโฑint,๐น๐ฟ,๐p๐ต๐qฤ 0
|r๐๐ฟ|
๐ ๐ฟ๐น๐ฟ,๐ p๐ต๐q .
SOME MULTIPLE FLOW DIRECTION ALGORITHMS 1931
Injecting this into (5.7), we obtain:
รฟ
๐พP๐ฏ z๐ฏ ห|r๐๐พ | `
รฟ
๐พP๐ฏ ห
รฟ
๐Pโฑ๐พXโฑext,๐น๐พ,๐p๐ต๐qฤ 0
|r๐๐พ |
๐ ๐พ๐น๐พ,๐ p๐ต๐q ฤ 0
which immediately implies that r๐๐พ โ 0 for any ๐พ P ๐. Next, let us denote ๐ฏ0 โ ๐ฏ and ๐0 โ ๐, and define ๐ฏ๐
for any ๐ ฤ 1 by setting:๐ฏ๐ โ ๐ฏ z
ฤ
0ฤ๐ฤ๐ยด1
๐๐ where ๐๐ โ p๐ฏ๐z๐ฏ ห๐ q Y ๐ฏ หห๐
with๐ฏ ห๐ โ t๐พ P ๐ฏ๐ | ๐ ๐พ ฤ 0u and ๐ฏ หห๐ โ
๐พ P ๐ฏ ห๐ |
๐ P โฑ๐พ X โฑ ๐ext | ๐น๐พ,๐ p๐ต๐q ฤ 0
(
โฐ H(
and for any ๐ ฤ 1:
โฑ๐ โ t๐ P โฑ | ๐ฏ๐ X ๐ฏ๐ โฐ Hu and โฑ ๐int โ t๐ P โฑ๐ X โฑint | ๐ฏ๐ ฤ ๐ฏ๐u and โฑ ๐
ext โ โฑ๐zโฑ ๐int.
Clearly, as ๐ โฐ H we have ๐ฏ1 ฤน ๐ฏ0 or ๐ฏ1 โ H. We now proceed by induction. Assume that for ๐ ฤ 1, we haveestablished that
r๐๐พ โ 0 for all ๐พ Pฤ
0ฤ๐ฤ๐ยด1
๐๐ and ๐ฏ๐ ฤน ๐ฏ๐ยด1 or ๐ฏ๐ โ H @1 ฤ ๐ ฤ ๐ยด 1.
If ๐ฏ๐ โ H, then there is nothing to prove. Otherwise, as for any ๐พ P ๐ฏ๐, there exists a flow path to ๐, then๐ฏ หห๐ โฐ H. Thus, summing over ๐ฏ๐ and proceeding as above we obtain that:
รฟ
๐พP๐ฏ๐z๐ฏ ห๐
|r๐๐พ | `รฟ
๐พP๐ฏ ห๐
รฟ
๐Pโฑ๐พXโฑ๐ext,๐น๐พ,๐p๐ต๐qฤ 0
|r๐๐พ |
๐ ๐พ๐น๐พ,๐ p๐ต๐q ฤ 0.
Thus, we obtain that r๐๐พ โ 0 for any ๐พ P ๐๐ and ๐๐ โฐ H which implies ๐ฏ๐`1 ฤน ๐ฏ๐. Then, it is clear that thesequence p๐ฏ๐q๐ฤ0 is strictly decreasing in the sense that it satisfies ๐ฏ๐ ฤน ๐ฏ๐ยด1 or ๐ฏ๐ โ H because of the flow pathassumption. Thus, there exists ๐๐ฏ ฤ 0 such that ๐ฏ๐๐ฏ โ H and thus r๐ โ 0, which concludes the proof.
6. Convergence analysis
The main idea underlying our convergence analysis is that for all consistent fluxes, if the topography ๐ isregular enough numerical fluxes will converge to the continuous ones. The major originality of the presentanalysis regarding for instance finite volumes or discontinuous Galerkin methods for steady-advection diffusionproblems of the form (3.2) is that here the coefficients ๐ฝ and ๐ are approximated through discrete differentialoperators applied to the topography ๐. Thus, this additional approximation has to be handled carefully torecover the correct convergence estimates.
To establish precise error estimates we first need to choose a discrete norm. To this end, we define
๐๐พ,๐ โ12`
|๐น๐พ,๐ p๐ต๐q | ` |๐| ||๐โ๐ ยจ ๐๐พ,๐||๐ฟ8p๐ตp๐ฅ๐พ ,2โ๐พqq
ห
and set for ๐ฃ โ p๐ฃ๐พq๐พP๐ฏ :
||๐ฃ||2โ โ๐ผ๐ยด
2
รฟ
๐พP๐ฏ|๐พ|๐ฃ2
๐พ `12
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑext,๐น๐พ,๐p๐ต๐qฤ 0
๐๐พ,๐|๐ฃ๐พ |2 `
12
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
๐๐พ,๐p๐ฃ๐พ ยด ๐ฃup๐ q
2.
In the above expression, one would more naturally expect to find ๐น๐พ,๐ p๐ต๐q instead of ๐๐พ,๐. However, theirbehavior is difficult to control and in particular, it is complex to estimate the way this fluxes stay away from zero
1932 J. COATLEVEN
when the continuous fluxesล
๐๐โ๐ ยจ๐๐พ,๐ cancel creating technical difficulties, which ๐๐พ,๐ avoids. Immediately,
we get that:
|๐๐พ,๐ ยด |๐น๐พ,๐ p๐ต๐q | | โ12
ห
ห |๐น๐พ,๐ p๐ต๐q ยด |๐| ||๐โ๐ ยจ ๐๐พ,๐||๐ฟ8p๐ตp๐ฅ๐พ ,2โ๐พqq
ห
ห
and thus:|๐๐พ,๐ ยด |๐น๐พ,๐ p๐ต๐q | | ฤ
12๐ ๐พ,๐ (6.1)
where for any ๐พ P ๐ฏ and any ๐ P โฑ๐พ we have denoted:
๐ ๐พ,๐ โ maxหห
ห
ห
ห
๐น๐พ,๐ p๐ต๐q `
ลผ
๐
๐โ๐ ยจ ๐๐พ,๐
ห
ห
ห
ห
,ห
ห|๐น๐พ,๐ p๐ต๐q | ยด |๐| ||๐โ๐ ยจ ๐๐พ,๐||๐ฟ8p๐ตp๐ฅ๐พ ,2โ๐พqq
ห
ห
ห
. (6.2)
Notice that from (4.5), we get that there exists ๐ถ ฤ 0 such that for any ๐พ P ๐ฏ and any ๐ P โฑ๐พ , we have:
๐ ๐พ,๐ ฤ ๐ถ|๐|โ๐พ ||๐||๐ 1,8pฮฉq||๐||๐ 2,8pฮฉq. (6.3)
The main objective of this section is to establish the following explicit ๐ฟ2 error estimate:
Theorem 6.1. Assume that the gradient operator underlying the fluxes is consistent. Further assume that themesh satisfies assumption (A1), that ๐ P ๐ 1,8pฮฉq and that the solution ๐ข of (3.1) belongs to ๐ป1pฮฉq. Let
๐พpโq โ max
ห
sup๐พP๐ฏ ,๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
๐ ๐พ,๐
๐๐พ,๐, sup๐พP๐ฏ ,๐Pโฑ๐พXโฑext,๐น๐พ,๐p๐ต๐qโฐ0
๐ ๐พ,๐
๐๐พ,๐
ยธ
with ๐ ๐พ,๐ defined by (6.2). Then, there exists โโp๐ขq P ๐ฟ2pฮฉq such that for any ๐พ P ๐ฏ , โโp๐ขq|๐พ โ โ๐พp๐ขq isconstant on ๐พ and:
|๐ขยด โ๐พp๐ขq|2๐ฟ2p๐พq ` โ
12๐พ |๐ขยด โ๐พp๐ขq|
2๐ฟ2pB๐พq ฤ ๐ถpolyโ๐พ |๐ข|๐ป1p๐พq.
Moreover if ๐พpโq ร 0 when โ ร 0, then there exists โ0 ฤ 0 and ๐ถ ฤ 0 depending on ๐ข, ๐, ๐ and the meshparameters such that for any โ ฤ โ0:
||๐ขยด โโp๐ขq||โ ฤ ๐ถ maxpโ, ๐พpโqq12||๐ข||๐ป1pฮฉq (6.4)
where ๐ข is reconstructed by (5.4) from the solution of (5.2).
Notice that the existence of โโp๐ขq in the above result is an immediate consequence of assumption (A1) and(2.3). As a corollary, we will then be able to establish an explicit error estimate for the water discharge:
Corollary 6.2. Under the assumptions of Proposition 6.1, there exists โ0 ฤ 0 and ๐ถ ฤ 0 depending on ๐ข, ๐,๐ and the mesh parameters such that for any โ ฤ โ0:
ห
รฟ
๐พP๐ฏ|| ยด ๐๐ขโ๐ยด๐๐พ ||
2๐ฟ2p๐พq2
ยธ12
ฤ ๐ถ maxpโ, ๐พpโqq12p1`maxpโ, ๐พpโqq12q||๐ข||๐ป1pฮฉq. (6.5)
The error estimates use the surprising factor ๐พpโq instead of โ as one could legitimately expect. This comesfrom the fact that we use approximate fluxes, which is equivalent in some sense to using a quadrature formulafor the coefficients of (3.2). As the natural norm for the discrete problem uses those approximate fluxes, itseems unfortunately unavoidable that this impacts the error estimate. In the case where for any ๐พ P ๐ฏ and any๐ P โฑ๐พ that is indeed present in the norm || ยจ ||โ, the ๐๐พ,๐โs are bounded by below by a constant independent onโ, which will happen most of the time in practice, then ๐พpโq will behave as โ and we retrieve a convergence atrate โ12. Controlling the lower bound for ๐๐พ,๐ is the reason why we have used a supremum on a set containing
SOME MULTIPLE FLOW DIRECTION ALGORITHMS 1933
more than ๐พ in the definition of ๐๐พ,๐. If this supremum is zero, most gradient reconstruction operators andin particular those described in our numerical experiments will provide a discrete gradient that aligns with thecontinuous one and the discrete flux will be zero too, avoiding most of the problematic cases for the asymptoticbehavior of ๐๐พ,๐. In the case of exact fluxes it is known that for the water height ๐ข the optimal convergencerate to ๐ข is โ12, even if superconvergence at rate โ is very often observed in practice, as revealed by [6,19] (seealso [7]). Thus, as additional flux consistency terms could only decrease the convergence rate, it seems clearthat our estimate for the water height cannot be expected to be improved in terms of order of convergence.
The proof of Theorem 6.1 being rather lengthy, we will try to decompose it as much as possible. Let us noticethat in the special case where โ๐ is constant, the discrete flux are exact. Then our problem corresponds toa piecewise constant version of the discontinuous Galerkin method described in [20], and we can in principlefollow the steps of their proof. Denoting ||๐ฃ||2โ โ
ล
๐พP๐ฏ ||๐ฃ||2๐พ,โ with obvious notations, the first step of their
proof is to establish a local error identity for ||๐ข๐พ ยด ๐ฃ๐พ ||๐พ,โ for any ๐ฃ๐พ in R, using the exactness of the flux.Then, taking ๐ฃ๐พ โ โ๐พp๐ขq and summing over ๐พ P ๐ฏ , they obtain a global error estimate where the residualterms only involve ๐ขยด โp๐ขq, which can be straightforwardly estimated using polynomial approximation results.
The general guideline of our proof is the same, however we have to endure some additional technicalities dueto the non exactness of the flux. We first start by establishing the equivalent of the local error identity of [20],but keeping our approximate flux in the result. Thus, an additional residual term standing for the consistencyerror of the flux will appear on the right hand side of our identity. Then, to match the definition of the || ยจ ||โnorm, each time our estimates will involve the sum of the flux over the full set โฑ๐พ , we will replace it by theLaplacian of ๐, and put the difference in a residual term. For an isolated flux term, we will do the same usingthis time either ๐๐พ,๐ if the term is involved in the || ยจ ||โ norm or by the continuous flux if it is directly a residualterm. Thus, when summing the local error identities over ๐พ P ๐ฏ to obtain the global error estimate, residualterms involving the discrete counterpart of the Laplacian will be handled through a technical result describedin the following subsection, while other residual terms will be estimated directly using the strong consistency ofthe flux. The reason for doing so is to obtain an error estimate involving only the || ยจ ||โ norm on the left-handside and residual terms involving either ๐ขยด โp๐ขq or flux consistency error terms on the right-hand side. Finally,each residual term will be estimated using polynomial approximation results and the consistency of the flux.
6.1. Local error identity and approximation results for the Laplacian
Let us now establish an abstract local error identity, following the lines of [20].
Lemma 6.3. For any ๐พ P ๐ฏ , any ๐ฃ๐พ P R and any ๐ โ p๐๐q๐Pโฑ๐พP ๐ฟ2pB๐พq constant on each ๐ P โฑ๐พ , we have:
12
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ 0
๐น๐พ,๐ p๐ต๐q p๐ข๐พ ยด ๐ฃ๐พq2`
12
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
๐น๐พ,๐ p๐ต๐q p๐ขup๐ ยด ๐๐q
2
ยด12
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
๐น๐พ,๐ p๐ต๐q pp๐ข๐พ ยด ๐ฃ๐พq ยด p๐ขup๐ ยด ๐๐qq
2`
12
รฟ
๐Pโฑ๐พ
๐น๐พ,๐ p๐ต๐q p๐ข๐พ ยด ๐ฃ๐พq2
โรฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ 0
1|๐|
ลผ
๐
๐น๐พ,๐ p๐ต๐q p๐ขยด ๐ฃ๐พq p๐ข๐พ ยด ๐ฃ๐พq
`รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
1|๐|
ลผ
๐
๐น๐พ,๐ p๐ต๐q p๐ขยด ๐๐q p๐ข๐พ ยด ๐ฃ๐พq
ยดรฟ
๐Pโฑ๐พ
ลผ
๐
ห
1|๐|
๐น๐พ,๐ p๐ต๐q ` ๐โ๐ ยจ ๐๐พ,๐
ห
๐ข p๐ข๐พ ยด ๐ฃ๐พq (6.6)
where ๐ข is the solution of the continuous problem (3.1).
1934 J. COATLEVEN
Proof. This proof is a direct adaptation of its counterpart in [20]. For any ๐ค๐พ P R and any ๐ โ p๐๐q๐Pโฑ๐พP
๐ฟ2pB๐พq constant on each ๐ P โฑ๐พ , consider:
โ๐พ โ ยดรฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
๐น๐พ,๐ p๐ต๐q p๐ค๐พ ยด ๐๐q๐ค๐พ `รฟ
๐Pโฑ๐พ
๐น๐พ,๐ p๐ต๐q๐ค2๐พ .
Using the identity ๐p๐ยด ๐q โ 12
`
๐2 ยด ๐2 ` p๐ยด ๐q2ห
, we obtain:
โ๐พ โรฟ
๐Pโฑ๐พ
๐น๐พ,๐ p๐ต๐q๐ค2๐พ ยด
12
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
๐น๐พ,๐ p๐ต๐q๐ค2๐พ `
12
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
๐น๐พ,๐ p๐ต๐q ๐2๐
ยด12
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
๐น๐พ,๐ p๐ต๐q p๐ค๐พ ยด ๐๐q2
and thus:
โ๐พ โ12
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ 0
๐น๐พ,๐ p๐ต๐q๐ค2๐พ `
12
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
๐น๐พ,๐ p๐ต๐q ๐2๐
ยด12
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
๐น๐พ,๐ p๐ต๐q p๐ค๐พ ยด ๐๐q2`
12
รฟ
๐Pโฑ๐พ
๐น๐พ,๐ p๐ต๐q๐ค2๐พ .
On the other hand, from (5.1) multiplying by ๐ค๐พ and using the definition of ๐ขup๐ we have by construction:
ยดรฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
๐น๐พ,๐ p๐ต๐q p๐ข๐พ ยด ๐ขup๐ q๐ค๐พ `
รฟ
๐Pโฑ๐พ
๐น๐พ,๐ p๐ต๐q๐ข๐พ๐ค๐พ โ
ลผ
๐พ
๐๐ค๐พ .
Let us now set ๐ค๐พ โ ๐ข๐พ ยด ๐ฃ๐พ and ๐๐ โ ๐ขup๐ ยด ๐๐ for all ๐ P โฑ๐พ . We obtain:
โ๐พ โ ยดรฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
๐น๐พ,๐ p๐ต๐q p๐๐ ยด ๐ฃ๐พq๐ค๐พ `
ลผ
๐พ
๐๐ค๐พ ยดรฟ
๐Pโฑ๐พ
๐น๐พ,๐ p๐ต๐q ๐ฃ๐พ๐ค๐พ .
Using ยดdiv p๐ข๐โ๐q โ ๐ , this leads to:
โ๐พ โ ยดรฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
๐น๐พ,๐ p๐ต๐q p๐๐ ยด ๐ฃ๐พq๐ค๐พ `
ลผ
๐พ
ยดdiv p๐ข๐โ๐q๐ค๐พ ยดรฟ
๐Pโฑ๐พ
๐น๐พ,๐ p๐ต๐q ๐ฃ๐พ๐ค๐พ .
Integrating by parts, we get:ลผ
๐พ
ยดdiv p๐ข๐โ๐q๐ค๐พ โ ยดรฟ
๐Pโฑ๐พ
ลผ
๐
๐โ๐ ยจ ๐๐พ,๐๐ข๐ค๐พ
โรฟ
๐Pโฑ๐พ
1
|๐|
ลผ
๐
๐น๐พ,๐ p๐ต๐q๐ข๐ค๐พ ยดรฟ
๐Pโฑ๐พ
ลผ
๐
ห
1
|๐|๐น๐พ,๐ p๐ต๐q ` ๐โ๐ ยจ ๐๐พ,๐
ห
๐ข๐ค๐พ .
Then:
โ๐พ โ ยดรฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
1
|๐|
ลผ
๐
๐น๐พ,๐ p๐ต๐q p๐๐ ยด ๐ฃ๐พq๐ค๐พ `รฟ
๐Pโฑ๐พ
1
|๐|
ลผ
๐
๐น๐พ,๐ p๐ต๐q p๐ขยด ๐ฃ๐พq๐ค๐พ
ยดรฟ
๐Pโฑ๐พ
ลผ
๐
ห
1
|๐|๐น๐พ,๐ p๐ต๐q ` ๐โ๐ ยจ ๐๐พ,๐
ห
๐ข๐ค๐พ
โรฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
1
|๐|
ลผ
๐
๐น๐พ,๐ p๐ต๐q p๐ขยด ๐๐q๐ค๐พ `รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ 0
1
|๐|
ลผ
๐
๐น๐พ,๐ p๐ต๐q p๐ขยด ๐ฃ๐พq๐ค๐พ
ยดรฟ
๐Pโฑ๐พ
ลผ
๐
ห
1
|๐|๐น๐พ,๐ p๐ต๐q ` ๐โ๐ ยจ ๐๐พ,๐
ห
๐ข๐ค๐พ
which finally establishes the desired identity.
SOME MULTIPLE FLOW DIRECTION ALGORITHMS 1935
In the above local error identity, one can see that the left hand side strongly looks like the ๐พ-term of the || ยจ ||โnorm. However, to match the || ยจ ||โ norm, we need to replace the sum of the fluxes over โฑ๐พ by the Laplacianof ๐, as well as isolated flux by ๐๐พ,๐. If this last operation straightforwardly leads to a residual term because of(6.1), however the following lemma precise what kind of link we can expect between the discrete Laplacian andthe true Laplacian:
Lemma 6.4. For any p๐ค๐พq๐พP๐ฏ , one has:
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ
๐น๐พ,๐ p๐ต๐q๐ค๐พ โ ยดรฟ
๐พP๐ฏ
ลผ
๐พ
๐โ๐๐ค๐พ `รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
ลผ
๐
ห
1|๐|
๐น๐พ,๐ p๐ต๐q ` ๐โ๐ ยจ ๐๐พ,๐
ห
ห p๐ค๐พ ยด ๐คup๐ q `
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑext,๐น๐พ,๐p๐ต๐qฤ 0
ลผ
๐
ห
1|๐|
๐น๐พ,๐ p๐ต๐q ` ๐โ๐ ยจ ๐๐พ,๐
ห
๐ค๐พ .
Proof. Let us first notice that:รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ
๐น๐พ,๐ p๐ต๐q๐คup๐ โ
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑext,๐น๐พ,๐p๐ต๐qฤ 0
๐น๐พ,๐ p๐ต๐q๐คup๐ .
Indeed, using ๐น๐พ,๐ p๐ต๐q ` ๐น๐ฟ,๐ p๐ต๐q โ 0 for any ๐ P โฑint, ๐ฏ๐ โ t๐พ, ๐ฟu we haveรฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑint
๐น๐พ,๐ p๐ต๐q๐คup๐ โ 0
while for ๐ P โฑext such that ๐น๐พ,๐ p๐ต๐q ฤ 0, ๐ฏ๐ โ t๐พu, ๐คup๐ โ 0 immediately implies that:
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑext,๐น๐พ,๐p๐ต๐qฤ0
๐น๐พ,๐ p๐ต๐q๐คup๐ โ 0.
Thus, we have:รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ
๐น๐พ,๐ p๐ต๐q๐ค๐พ โรฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
๐น๐พ,๐ p๐ต๐q p๐ค๐พ ยด ๐คup๐ q
`รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑext,๐น๐พ,๐p๐ต๐qฤ 0
๐น๐พ,๐ p๐ต๐q๐ค๐พ
using the definition of ๐คup๐ . In the same way:
รฟ
๐พP๐ฏ
ลผ
๐พ
ยด๐โ๐๐ค๐พ โรฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ
ลผ
๐
ยด๐โ๐ ยจ ๐๐พ,๐๐ค๐พ โรฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
ลผ
๐
ยด๐โ๐ ยจ ๐๐พ,๐ p๐ค๐พ ยด ๐คup๐ q
`รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑext,๐น๐พ,๐p๐ต๐qฤ 0
ลผ
๐
ยด๐โ๐ ยจ ๐๐พ,๐๐ค๐พ .
Combining those expressions gives the expected result.
6.2. Global error estimate
From the local error identity, and the link between the discrete and continuous Laplacians, we deduce a globalestimate for the discrete norm of the error:
1936 J. COATLEVEN
Lemma 6.5. Using the hypothesis and notations of Proposition 6.1, we have:
||๐ขยด โโp๐ขq||2โ ฤ
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑint,๐น๐พ,๐p๐ต๐qฤ0
1|๐|
ลผ
๐
๐น๐พ,๐ p๐ต๐q p๐ขยด โup๐ p๐ขqq pp๐ข
up๐ ยด โup
๐ p๐ขqq ยด p๐ข๐พ ยด โ๐พp๐ขqqq
`รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑext,๐น๐พ,๐p๐ต๐qฤ 0
1|๐|
ลผ
๐
๐น๐พ,๐ p๐ต๐q p๐ขยด โ๐พp๐ขqq p๐ข๐พ ยด โ๐พp๐ขqq
`รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑext,๐น๐พ,๐p๐ต๐qฤ0
1|๐|
ลผ
๐
๐น๐พ,๐ p๐ต๐q p๐ขยด โup๐ p๐ขqq pp๐ข๐พ ยด โ๐พp๐ขqq ยด p๐ข
up๐ ยด โup
๐ p๐ขqqq
ยดรฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ
ลผ
๐
ห
1|๐|
๐น๐พ,๐ p๐ต๐q ` ๐โ๐ ยจ ๐๐พ,๐
ห
๐ข p๐ข๐พ ยด โ๐พp๐ขqq
ยด12
ห
รฟ
๐พP๐ฏ
ลผ
๐พ
๐โ๐|๐ข๐พ ยด โ๐พp๐ขq|2 `
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ
๐น๐พ,๐ p๐ต๐q |๐ข๐พ ยด โ๐พp๐ขq|2
ยธ
ยด12
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
p|๐น๐พ,๐ p๐ต๐q | ยด ๐๐พ,๐q pp๐ข๐พ ยด โ๐พp๐ขqq ยด p๐ขup๐ ยด โup
๐ p๐ขqqq2
ยด12
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑext,๐น๐พ,๐p๐ต๐qฤ 0
p|๐น๐พ,๐ p๐ต๐q | ยด ๐๐พ,๐q |๐ข๐พ ยด โ๐พp๐ขq|2.
Proof. We start from (6.6) with ๐ฃ๐พ โ โ๐พp๐ขq and ๐๐ โ โup๐ p๐ขq. Summing over ๐พ P ๐ฏ the second term of the
left hand side of (6.6) leads to:
12
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
๐น๐พ,๐ p๐ต๐q p๐ขup๐ ยด โup
๐ p๐ขqq2
โ ยด12
รฟ
๐ฟP๐ฏ
รฟ
๐Pโฑ๐ฟXโฑint,๐น๐ฟ,๐p๐ต๐qฤ 0
๐น๐ฟ,๐ p๐ต๐q p๐ข๐ฟ ยด โ๐ฟp๐ขqq2
`12
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑext,๐น๐พ,๐p๐ต๐qฤ0
๐น๐พ,๐ p๐ต๐q p๐ข๐ ยด โ๐p๐ขqq2
โ ยด12
รฟ
๐ฟP๐ฏ
รฟ
๐Pโฑ๐ฟXโฑint,๐น๐ฟ,๐p๐ต๐qฤ 0
๐น๐ฟ,๐ p๐ต๐q p๐ข๐ฟ ยด โ๐ฟp๐ขqq2
as both ๐ข๐ and โ๐p๐ขq cancels for ๐ P โฑ๐ext,in. Thus, summing over ๐พ P ๐ฏ the first two terms of the left handside of (6.6), we obtain:
12
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ 0
๐น๐พ,๐ p๐ต๐q pโ๐,๐พ ยด โ๐พp๐ขqq2`
12
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
๐น๐พ,๐ p๐ต๐q p๐ขup๐ ยด โup
๐ p๐ขqq2
โ12
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑext,๐น๐พ,๐p๐ต๐qฤ 0
๐น๐พ,๐ p๐ต๐q pโ๐,๐พ ยด โ๐พp๐ขqq2.
SOME MULTIPLE FLOW DIRECTION ALGORITHMS 1937
Consequently, summing over ๐พ P ๐ฏ the entire left hand side of (6.6) leads to:
รฟ
๐พP๐ฏโ๐พ โ
12
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ
๐น๐พ,๐ p๐ต๐q |๐ข๐พ ยด โ๐พp๐ขq|2
ยด12
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
๐น๐พ,๐ p๐ต๐q pp๐ข๐พ ยด โ๐พp๐ขqq ยด p๐ขup๐ ยด โup
๐ p๐ขqqq2
`12
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑext,๐น๐พ,๐p๐ต๐qฤ 0
๐น๐พ,๐ p๐ต๐q |๐ข๐พ ยด โ๐พp๐ขq|2 โ ๐น1 ` ๐น2 ` ๐น3.
The first term of the above expression rewrites:
๐น1 โ12
รฟ
๐พP๐ฏ
ลผ
๐พ
ยด๐โ๐|๐ข๐พ ยด โ๐พp๐ขq|2
`12
ห
รฟ
๐พP๐ฏ
ลผ
๐พ
๐โ๐|๐ข๐พ ยด โ๐พp๐ขq|2 `
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ
๐น๐พ,๐ p๐ต๐q |๐ข๐พ ยด โ๐พp๐ขq|2
ยธ
โ ๐ฟ0 ` ๐ฟ1
while the second term and third term give:
๐น2 ` ๐น3 โ12
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
๐๐พ,๐ pp๐ข๐พ ยด โ๐พp๐ขqq ยด p๐ขup๐ ยด โup
๐ p๐ขqqq2
`12
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑext,๐น๐พ,๐p๐ต๐qฤ 0
๐๐พ,๐|๐ข๐พ ยด โ๐พp๐ขq|2
`12
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
p|๐น๐พ,๐ p๐ต๐q | ยด ๐๐พ,๐q pp๐ข๐พ ยด โ๐พp๐ขqq ยด p๐ขup๐ ยด โup
๐ p๐ขqqq2
`12
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑext,๐น๐พ,๐p๐ต๐qฤ 0
p|๐น๐พ,๐ p๐ต๐q | ยด ๐๐พ,๐q |๐ข๐พ ยด โ๐พp๐ขq|2 โ ๐ฟ2 ` ๐ฟ3 ` ๐ฟ4 ` ๐ฟ5
with obvious notations. Immediately, using the hypothesis on ๐ and ๐ we have:
๐ฟ0 ฤ๐ผ๐ยด
2
รฟ
๐พP๐ฏ|๐พ| p๐ข๐พ ยด โ๐พp๐ขqq
2.
Then, noticing that the second member of the sum over ๐พ P ๐ฏ of (6.6) can be decomposed into three terms๐1 ` ๐2 ` ๐3 with obvious notations and combining the above results, we have that
ล
๐พP๐ฏ โ๐พ โ ๐1 ` ๐2 ` ๐3
is equivalent to ||๐ข ยด โโp๐ขq||2โ ฤ ๐1 ` ๐2 ` ๐3 ยด ๐ฟ1 ยด ๐ฟ4 ยด ๐ฟ5 as ๐ฟ0 ` ๐ฟ2 ` ๐ฟ3 exactly gives the square of the
|| ยจ ||โ norm. Using the fact that both ๐ข๐ and โ๐p๐ขq cancels for ๐ P โฑ๐ext,in, we obtain:
๐1 ` ๐2 โรฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑint,๐น๐พ,๐p๐ต๐qฤ0
1|๐|
ลผ
๐
๐น๐พ,๐ p๐ต๐q p๐ขยด โup๐ p๐ขqq pp๐ข
up๐ ยด โup
๐ p๐ขqq ยด p๐ข๐พ ยด โ๐พp๐ขqqq
`รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑext,๐น๐พ,๐p๐ต๐qฤ 0
1|๐|
ลผ
๐
๐น๐พ,๐ p๐ต๐q p๐ขยด โ๐พp๐ขqq p๐ข๐พ ยด โ๐พp๐ขqq
`รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑext,๐น๐พ,๐p๐ต๐qฤ0
1|๐|
ลผ
๐
๐น๐พ,๐ p๐ต๐q p๐ขยด โup๐ p๐ขqq pp๐ข๐พ ยด โ๐พp๐ขqq ยด p๐ข
up๐ ยด โup
๐ p๐ขqqq
which concludes the proof.
1938 J. COATLEVEN
6.3. Estimation of the residual terms
Establishing Proposition 6.1 now just consists in estimating each term appearing in the second member ofthe above global estimate. From Lemma 6.5, we see that ||๐ขยด โโp๐ขq||
2โ ฤ
ล7๐โ1 ๐๐ with obvious notations. The
first three terms account for the polynomial approximation error, while the four other terms account for theconsistency error of the flux. We now estimate those two families of residual terms separately.
Proof of Proposition 6.1: terms accounting for polynomial approximation error. Using CauchyโSchwarzinequality and the fact that p๐ขup
๐ ยด โup๐ p๐ขqq ยด p๐ข๐พ ยด โ๐พp๐ขqq is constant on ๐ gives:
|๐1|2 ฤ
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑint,๐น๐พ,๐p๐ต๐qฤ0
หลผ
๐
1|๐||๐น๐พ,๐ p๐ต๐q | ||๐ขยด โup
๐ p๐ขq|2
ห
||๐ขยด โโp๐ขq||2โ.
As the gradient operator underlying the fluxes is consistent, we know from Proposition 4.5 that there exists๐ถ ฤ 0 such that:
ห
ห
ห
ห
1|๐|
๐น๐พ,๐ p๐ต๐q
ห
ห
ห
ห
ฤ ||1|๐|
๐น๐พ,๐ p๐ต๐q ยด ๐โ๐ ยจ ๐๐พ,๐||๐ฟ8p๐q ` ||๐โ๐ ยจ ๐๐พ,๐||๐ฟ8p๐q
ฤ ๐ถโ||๐||๐ 1,8pฮฉq||๐||๐ 2,8pฮฉq ` ||๐||๐ฟ8pฮฉq||๐||๐ 2,8pฮฉq.
As soon as โ ฤ โ0 for some fixed โ0 this leads to the existence of some ๐ถ ฤ 0 independent on โ such that
|๐1|2 ฤ ๐ถ||๐||๐ 1,8pฮฉq||๐||๐ 2,8
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑint,๐น๐พ,๐p๐ต๐qฤ0
หลผ
๐
|๐ขยด โup๐ p๐ขq|
2
ห
||๐ขยด โโp๐ขq||2โ.
Then, using the definition of โโp๐ขq, we getลผ
๐
|๐ขยด โup๐ p๐ขq|
2 ฤ ๐ถโ๐พup๐|๐ข|2๐ป1p๐พup
๐ q
where, if ๐ P โฑint, ๐พup๐ โ ๐พ if ๐น๐พ,๐ p๐ต๐q ฤ 0 and ๐พup
๐ โ ๐ฟ if ๐น๐พ,๐ p๐ต๐q ฤ 0, and ๐พup๐ โ ๐พ
if ๐ P โฑext. From [7], we know that assumption (A1) implies that card pโฑ๐พq is bounded by a constant๐โฑ depending on the mesh parameters but independent on โ, and thus there exists ๐ถ ฤ 0 such that|๐1|
2 ฤ 2โ๐ถ||๐||๐ 1,8pฮฉq||๐||๐ 2,8 |๐ข|2๐ป1pฮฉq||๐ขยดโโp๐ขq||2โ. For ๐2, obviously we have using CauchyโSchwarz inequal-
ity:
|๐2|2 ฤ
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑext,๐น๐พ,๐p๐ต๐qฤ 0
หลผ
๐
1|๐||๐น๐พ,๐ p๐ต๐q | ||๐ขยด โ๐พp๐ขq|
2
ห
||๐ขยด โโp๐ขq||2โ
and thus, proceeding the same way than for ๐1, we obtain |๐2|2 ฤ 2โ๐ถ||๐||๐ 1,8pฮฉq||๐||๐ 2,8 |๐ข|2๐ป1pฮฉq||๐ขยดโโp๐ขq||
2โ.
The situation is more delicate for ๐3. Indeed, โup๐ p๐ขq โ 0 for ๐ P โฑ๐ext,in, while ๐ข โ 0 on ๐ P โฑext,in, thus โup
๐ p๐ขq
is not directly a good approximation of ๐ข. For any ๐ P โฑ๐ext,inXโฑext,in, โup๐ p๐ขq โ ๐ข โ 0 and thus the contribution
to ๐3 is zero. Then, it just remains to consider the case where ๐ P โฑ๐ext,in and ๐ R โฑext,in. In this case we haveยดล
๐๐โ๐ ยจ ๐๐พ,๐ ฤ 0 while ๐น๐พ,๐ p๐ต๐q ฤ 0. However, by definition of the residual
ห
ห๐น๐พ,๐ p๐ต๐q `ล
๐๐โ๐ ยจ ๐๐พ,๐
ห
ห ฤ
๐ ๐พ,๐ which immediately implies thatห
ห
ล
๐๐โ๐ ยจ ๐๐พ,๐
ห
ห ฤ ๐ ๐พ,๐ and |๐น๐พ,๐ p๐ต๐q| ฤ ๐ ๐พ,๐. Consequently, we getusing CauchyโSchwarz inequality:
|๐3| ฤ
ยจ
ห
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑext,๐น๐พ,๐p๐ต๐qฤ0
1|๐|
๐ ๐พ,๐
ลผ
๐
๐ข2
ห
โ
12
ห
ยจ
ห
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑext,๐น๐พ,๐p๐ต๐qฤ0
๐ ๐พ,๐ pp๐ข๐พ ยด โ๐พp๐ขqq ยด p๐ขup๐ ยด โup
๐ p๐ขqqq2
ห
โ
12
.
SOME MULTIPLE FLOW DIRECTION ALGORITHMS 1939
Using the definition of ๐พpโq and the trace inequality on ฮฉ, this leads to:
๐3 ฤ ๐พpโq12
ยจ
ห
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑext,๐น๐พ,๐p๐ต๐qฤ0
1|๐|
๐ ๐พ,๐
ลผ
๐
๐ข2
ห
โ
12
||๐ขยดโโp๐ขq||โ ฤ ๐ถโ12๐พpโq12||๐ข||๐ป1pฮฉq||๐ขยดโโp๐ขq||โ.
It finally remains to estimate the terms accounting for the consistency error on the fluxes.
Proof of Proposition 6.1: terms accounting for flux consistency error. For ๐4, let us first remark that:
๐4 โ ยดรฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑint
ลผ
๐
ห
1|๐|
๐น๐พ,๐ p๐ต๐q ` ๐โ๐ ยจ ๐๐พ,๐
ห
๐ข pp๐ข๐พ ยด โ๐พp๐ขqq ยด p๐ขup๐ ยด โup
๐ p๐ขqqq
ยดรฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑext
ลผ
๐
ห
1|๐|
๐น๐พ,๐ p๐ต๐q ` ๐โ๐ ยจ ๐๐พ,๐
ห
๐ข p๐ข๐พ ยด โ๐พp๐ขqq .
Using the fact that both ๐ข๐ and โ๐p๐ขq cancel for ๐ P โฑ๐พ X โฑext and ๐น๐พ,๐ p๐ต๐q ฤ 0, this rewrites:
๐4 โ ยดรฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
ลผ
๐
ห
1|๐|
๐น๐พ,๐ p๐ต๐q ` ๐โ๐ ยจ ๐๐พ,๐
ห
๐ข pp๐ข๐พ ยด โ๐พp๐ขqq ยด p๐ขup๐ ยด โup
๐ p๐ขqqq
ยดรฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑext,๐น๐พ,๐p๐ต๐qฤ 0
ลผ
๐
ห
1|๐|
๐น๐พ,๐ p๐ต๐q ` ๐โ๐ ยจ ๐๐พ,๐
ห
๐ข p๐ข๐พ ยด โ๐พp๐ขqq โ ๐4,1 ` ๐4,2.
CauchyโSchwarz inequality then leads to:
|๐4,1| ฤ
ยจ
ห
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
1|๐|
๐ ๐พ,๐
ลผ
๐
๐ข2
ห
โ
12
ห
ยจ
ห
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
๐ ๐พ,๐
ยด
p๐ข๐พ ยด โ๐พp๐ขqq ยด p๐ขup๐ ยด โup
๐ p๐ขqq2ยฏ
ห
โ
12
.
Using (6.3), (2.2) and the definition of ๐พpโq, we easily get:
|๐4,1| ฤ ๐ถ๐พpโq12
ยจ
ห
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
โ๐พ
ลผ
๐
๐ข2
ห
โ
12
||๐ขยด โโp๐ขq||โ ฤ ๐ถ๐พpโq12||๐ข||๐ป1pฮฉq||๐ขยด โโp๐ขq||โ.
In the same way, CauchyโSchwarz inequality and this time the trace inequality on ฮฉ directly gives:
|๐4,2| ฤ ๐ถ๐พpโq12
ยจ
ห
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑext,๐น๐พ,๐p๐ต๐qฤ 0
โ๐พ
ลผ
๐
๐ข2
ห
โ
12
||๐ขยด โโp๐ขq||โ ฤ ๐ถ๐พpโq12โ12||๐ข||๐ป1pฮฉq||๐ขยด โโp๐ขq||โ.
From Lemma 6.4, we deduce that:
|๐5| ฤรฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
๐ ๐พ,๐
ยด
p๐ข๐พ ยด โ๐พp๐ขqq2ยด p๐ขup
๐ ยด โup๐ p๐ขqq
2ยฏ
`รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑext,๐น๐พ,๐p๐ต๐qฤ 0
๐ ๐พ,๐ p๐ข๐พ ยด โ๐พp๐ขqq2.
1940 J. COATLEVEN
Using p๐2 ยด ๐2q โ p๐ยด ๐qp๐` ๐q and CauchyโSchwarz inequality, we obtain:
|๐5| ฤ ๐พpโq12||๐ขยด โโp๐ขq||โ
ยจ
ห
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
๐ ๐พ,๐ pp๐ข๐พ ยด โ๐พp๐ขqq ` p๐ขup๐ ยด โup
๐ p๐ขqqq2
ห
โ
12
` ๐พpโq12||๐ขยด โโp๐ขq||โ
ยจ
ห
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑext,๐น๐พ,๐p๐ต๐qฤ 0
๐ ๐พ,๐ p๐ข๐พ ยด โ๐พp๐ขqq2
ห
โ
12
.
Using (6.3) and assumption (A1), we immediately obtain that:รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
๐ ๐พ,๐ p๐ข๐พ ยด โ๐พp๐ขqq2ฤ ๐ถ๐โฑ
รฟ
๐พP๐ฏ|๐พ| p๐ข๐พ ยด โ๐พp๐ขqq
2
and in the same way, using the fact that ๐ขup๐ โ and โup
๐ p๐ขq โ 0 on any ๐ P โฑext such that ๐น๐พ,๐ p๐ต๐q ฤ 0 andthe commensurability of โ๐พ and โ๐ฟ if โฑ๐พ X โฑ๐พ โฐ H under assumption (A1):
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
๐ ๐พ,๐ p๐ขup๐ ยด โup
๐ p๐ขqq2ฤ ๐ถ
รฟ
๐ฟP๐ฏ
รฟ
๐Pโฑ๐ฟ,๐น๐ฟ,๐p๐ต๐qฤ 0
|๐|โ๐ฟ pโ๐ฟ ยด โup๐ฟ p๐ขqq
2
ฤ ๐ถ๐โฑรฟ
๐พP๐ฏ|๐พ| p๐ข๐พ ยด โ๐พp๐ขqq
2
and also:รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑext,๐น๐พ,๐p๐ต๐qฤ 0
๐ ๐พ,๐ p๐ข๐พ ยด โ๐พp๐ขqq2ฤ ๐ถ
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑext,๐น๐พ,๐p๐ต๐qฤ 0
|๐|โ๐พ p๐ข๐พ ยด โ๐พp๐ขqq2
ฤ ๐ถ๐โฑรฟ
๐พP๐ฏ|๐พ| p๐ข๐พ ยด โ๐พp๐ขqq
2.
Gathering those intermediate results, we get that |๐5| ฤ ๐ถ๐พpโq12||๐ข ยด โโp๐ขq||2โ. Finally, we have by definition
of ๐พpโq and || ยจ ||โ that |๐6 ` ๐7| ฤ ๐ถ๐พpโq12||๐ข ยด โโp๐ขq||2โ. Then, using the hypothesis that ๐พpโq goes to zero
when โ goes to zero, for any ๐ถ ฤ 0 independent on โ there exists โ0 ฤ 0 such that 1ยด๐ถ max pโ, ๐พpโqq12ฤ 12,
and the result then immediately follows.
6.4. Error estimate for the water discharge
To establish Corollary 6.2, we will need an auxiliary discrete stability result:
Lemma 6.6. The following estimates holds:
12
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ
๐น๐พ,๐ p๐ต๐q๐ข2๐พ `
12
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑext,๐น๐พ,๐p๐ต๐qฤ 0
๐น๐พ,๐ p๐ต๐q๐ข2๐พ
ยด12
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
p๐ข๐พ ยด ๐ขup๐ q
2๐น๐พ,๐ p๐ต๐q โ
รฟ
๐พP๐ฏ|๐พ|๐๐พ๐ข๐พ . (6.7)
Proof. To establish (6.7), using the definition of ๐ขup๐ , we obtain multiplying (5.1) by ๐ข๐พ and summing over
๐พ P ๐ฏ :รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ 0
๐น๐พ,๐ p๐ต๐q๐ข2๐พ `
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
๐น๐พ,๐ p๐ต๐q๐ขup๐ ๐ข๐พ โ
รฟ
๐พP๐ฏ|๐พ|๐๐พ๐ข๐พ .
SOME MULTIPLE FLOW DIRECTION ALGORITHMS 1941
Remark that ๐ขup๐ ๐ข๐พ โ 1
2
ยด
๐ข2๐พ ` ๐ข๐ข๐2
๐
ยฏ
ยด 12 p๐ข๐พ ยด ๐ขup
๐ q2 which immediately leads to
รฟ
๐พP๐ฏ|๐พ|๐๐พ๐ข๐พ โ
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ 0
๐น๐พ,๐ p๐ต๐q๐ข2๐พ `
12
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
๐น๐พ,๐ p๐ต๐q๐ข2๐พ
`12
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
๐น๐พ,๐ p๐ต๐q๐ข๐ข๐2
๐ ยด12
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
๐น๐พ,๐ p๐ต๐q p๐ข๐พ ยด ๐ขup๐ q
2.
Using the fact that:รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
๐น๐พ,๐ p๐ต๐q๐ข๐ข๐2
๐ โ ยดรฟ
๐ฟP๐ฏ
รฟ
๐Pโฑ๐ฟXโฑint,๐น๐ฟ,๐p๐ต๐qฤ 0
๐น๐ฟ,๐ p๐ต๐q๐ข2๐ฟ
we immediately get (6.7) as:
12
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ 0
๐น๐พ,๐ p๐ต๐q๐ข2๐พ `
12
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
๐น๐พ,๐ p๐ต๐q๐ข๐ข๐2
๐
โ12
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑext,๐น๐พ,๐p๐ต๐qฤ 0
๐น๐พ,๐ p๐ต๐q๐ข2๐พ .
Proof of Corollary 6.2. Let us denote p๐๐q0ฤ๐ฤ1 the canonical basis of R2. Assume that โ ฤ โ0, where โ0 isdefined in Proposition 6.1 and consider:
๐พ โรฟ
๐Pโฑ๐พ
๐ข๐พ
|๐พ|๐น๐พ,๐ p๐ต๐q p๐ฅ๐ ยด ๐ฅ๐พq .
We begin by establishing thatห
รฟ
๐พP๐ฏ|| ยด ๐๐ขโ๐ยด ๐พ ||
2๐ฟ2p๐พq2
ยธ12
ฤ ๐ถ maxpโ, ๐พpโqq12||๐ข||๐ป1pฮฉq. (6.8)
Indeed, we have:ลผ
๐พ
๐๐พB๐๐ โ
ลผ
๐พ
๐๐พโ๐โ p๐ฅ๐ ยด ๐ฅ๐พ,๐q โ ยด
ลผ
๐พ
๐๐พโ๐ p๐ฅ๐ ยด ๐ฅ๐พ,๐q `รฟ
๐Pโฑ๐พ
ลผ
๐
๐๐พโ๐ ยจ ๐๐พ,๐ p๐ฅ๐ ยด ๐ฅ๐พ,๐q .
Using again the consistency estimate of Proposition 4.5 for the fluxes, we get:ห
ห
ห
ห
ห
รฟ
๐Pโฑ๐พ
ลผ
๐
ห
ยด๐๐พโ๐ ยจ ๐๐พ,๐ ยด1|๐|
๐น๐พ,๐ p๐ต๐q
ห
p๐ฅ๐ ยด ๐ฅ๐พ,๐q
ห
ห
ห
ห
ห
ฤ ๐ถ|๐|card pโฑ๐พqโ2๐พ ||๐||๐ 1,8pฮฉq||๐||๐ 2,8pฮฉq.
And thus, using assumption (A1) we get that:ห
ห
ห
ห
ยด๐ข๐พ๐๐พ1|๐พ|
ลผ
๐พ
โ๐ยด ๐พ
ห
ห
ห
ห
ฤ ๐ถโ๐พ ||๐||๐ 1,8pฮฉq||๐||๐ 2,8pฮฉq|๐ข๐พ |.
Using the triangle inequality:
|| ยด ๐๐ขโ๐ยด ๐พ ||2๐ฟ2p๐พq2 ฤ || p๐ยด ๐๐พq๐ขโ๐||2๐ฟ2p๐พq2 ` ||๐๐พ p๐ขยด ๐ข๐พqโ๐||2๐ฟ2p๐พq2
` ||๐๐พ๐ข๐พ
ห
โ๐ยด1|๐พ|
ลผ
๐พ
โ๐
ห
||2๐ฟ2p๐พq2 ` || ยด ๐๐พ๐ข๐พ1|๐พ|
ลผ
๐พ
โ๐ยด ๐พ ||2๐ฟ2p๐พq2 .
1942 J. COATLEVEN
PoincareโWirtingerโs inequality applied component by component (see [7]) gives:
||โ๐ยด1|๐พ|
ลผ
๐พ
โ๐||๐ฟ2p๐พq2 ฤ ๐ถ๐ โ๐พ ||๐||๐ป2p๐พq
where ๐ถ๐ ฤ 0 is independent on โ under assumption (A1), and (6.8) follows from Proposition 6.1 and the aboveestimate. To conclude, it suffices to remark that by definition:
|๐พ|ยด
๐พ ยด๐๐พ
ยฏ
โรฟ
๐Pโฑ๐พXโฑint,๐น๐พ,๐p๐ต๐qฤ0
p๐ข๐พ ยด ๐ข๐ฟq p๐ฅ๐ ยด ๐ฅ๐พq๐น๐พ,๐ p๐ต๐q
`รฟ
๐Pโฑ๐พXโฑext,๐น๐พ,๐p๐ต๐qฤ0
๐ข๐พ p๐ฅ๐ ยด ๐ฅ๐พq๐น๐พ,๐ p๐ต๐q .
Thus applying CauchyโSchwarz inequality, we get:
รฟ
๐พP๐ฏ|๐พ|
ห
ห
ห๐พ ยด๐๐พ
ห
ห
ห
2
ฤ 2
ยจ
ห
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑint,๐น๐พ,๐p๐ต๐qฤ0
||๐ฅ๐ ยด ๐ฅ๐พ ||2
|๐พ||๐น๐พ,๐ p๐ต๐q|
`รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑext,๐น๐พ,๐p๐ต๐qฤ0
||๐ฅ๐ ยด ๐ฅ๐พ ||2
|๐พ||๐น๐พ,๐ p๐ต๐q|
ห
โ||๐ข||โ,ห
ฤ ๐ถ
ยจ
ห
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
|๐|โ2๐พ
|๐พ|
ห
โฤ ๐ถโ|ฮฉ|||๐ข||โ,ห
using the bound on the flux from the proof of Proposition 6.1, and where:
||๐ข||2โ,ห โ12
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พXโฑext,๐น๐พ,๐p๐ต๐qฤ 0
๐น๐พ,๐ p๐ต๐q |๐ข๐พ |2 ยด
12
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ ,๐น๐พ,๐p๐ต๐qฤ0
๐น๐พ,๐ p๐ต๐q p๐ข๐พ ยด ๐ขup๐ q
2.
From (6.7) we know that, still denoting ๐ข the cellwise constant function of ๐ฟ2pฮฉq taking the value ๐ข๐พ in cell ๐พ:
||๐ข||2โ,ห ฤ ||๐ ||๐ฟ2pฮฉq||๐ข||๐ฟ2pฮฉq ยดรฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ
๐น๐พ,๐ p๐ต๐q๐ข2๐พ .
However, one has
รฟ
๐พP๐ฏ
รฟ
๐Pโฑ๐พ
๐น๐พ,๐ p๐ต๐q๐ข2๐พ โ
รฟ
๐พP๐ฏ
ห
ลผ
๐พ
โ๐`รฟ
๐Pโฑ๐พ
๐น๐พ,๐ p๐ต๐q
ยธ
๐ข2๐พ ยด
รฟ
๐พP๐ฏ
ลผ
๐พ
โ๐๐ข2๐พ .
Using Stokesโ formula, the consistency of the flux (4.5) and assumption (A1), we get:ห
ห
ห
ห
ห
ลผ
๐พ
โ๐`รฟ
๐Pโฑ๐พ
๐น๐พ,๐ p๐ต๐q
ห
ห
ห
ห
ห
โ
ห
ห
ห
ห
ห
รฟ
๐Pโฑ๐พ
๐น๐พ,๐ p๐ต๐q `
ลผ
๐
๐โ๐ ยจ ๐๐พ,๐
ห
ห
ห
ห
ห
ฤ ๐ถรฟ
๐Pโฑ๐พ
|๐|โ๐พ ฤ ๐ถ|๐พ|.
Thus, we have ||๐ข||2โ,ห ฤ ||๐ ||๐ฟ2pฮฉq||๐ข||๐ฟ2pฮฉq ` ๐ถ||๐ข||2๐ฟ2pฮฉq, and using the triangle inequality ||๐ข||๐ฟ2pฮฉq ฤ ||๐ข ยด
๐ข||๐ฟ2pฮฉq ` ||๐ข||๐ฟ2pฮฉq, we get that:
||๐ข||2โ,ห ฤ ๐ถยด
1`maxpโ, ๐พpโqq12ยฏยด
||๐ ||๐ฟ2pฮฉq `
ยด
1`maxpโ, ๐พpโqq12ยฏ
||๐ข||๐ป1pฮฉq
ยฏ
||๐ข||๐ป1pฮฉq
which concludes the proof.
SOME MULTIPLE FLOW DIRECTION ALGORITHMS 1943
Figure 3. Example of meshes for the 2dDelaunay and 2dDualDelaunay mesh sequences.
7. Numerical exploration
To assess the asymptotic behavior of the method, we use a very simple analytical solution on which the positiv-ity of the Laplacian is guaranteed (contrary to typical geological applications). Let us consider the square domainฮฉ โs0, ๐ฟrหs0, ๐ฟr associated with the topography ๐p๐ฅ, ๐ฆq โ ๐0 ยด ๐ฟ
ยด
p๐ฅยด ๐ฅ0q2` p๐ฆ ยด ๐ฆ0q
2ยฏ
where ๐ฅ0 โ ๐ฆ0 โ ๐ฟ2,
for which ยดโ๐ โ 4๐ฟ. Consider the water height ๐ข defined by ๐ขp๐ฅ, ๐ฆq โ ๐ข0 expยด
ยด๐ผยด
p๐ฅยด ๐ฅ0q2` p๐ฆ ยด ๐ฆ0q
2ยฏยฏ
.
Defining ๐p๐ฅ, ๐ฆq โ 1, we obtain ยดdivp๐๐ขโ๐q โ ๐ with ๐p๐ฅ, ๐ฆq โ 4๐ฟ๐ขp๐ฅ, ๐ฆqยด
1ยด ๐ผยด
p๐ฅยด ๐ฅ0q2` p๐ฆ ยด ๐ฆ0q
2ยฏยฏ
. Thecorresponding water discharge ๐ โ ||๐๐ขโ๐|| is given by
๐ โ 2๐ฟ๐ขp๐ฅ, ๐ฆqยด
p๐ฅยด ๐ฅ0q2` p๐ฆ ยด ๐ฆ0q
2ยฏ12
.
In practice we take ๐ข0 โ 1, ๐ผ โ 2, ๐ฟ โ 12, ๐0 โ 2 and ๐ฟ โ 1. We consider seven types of mesh sequences. Thefirst sequence consists in classical Delaunay meshes (2dDelaunay). The second one (2dDualDelaunay) is obtainedby considering the dual meshes of a sequence of Delaunay meshes (Fig. 3). The third sequence (2dVoronoi) ismade of Voronoi meshes, possessing the mesh orthogonality property. The fourth sequence (2dKershawBox ) isa sequence of Kershaw meshes, while the fifth one (2dCheckerBoardBox ) is a sequence of checkerboard meshes.These two sequences have only quadrangular cells which are distorted for the sequence 2dKershawBox, whilethe sequence 2dCheckerBoardBox allows to test the behavior of the method in presence of non conformities.The sixth sequence, named 2dSquareCart, is simply a uniform cartesian mesh with square cells, while theseventh one named 2dRectCart is a uniform cartesian mesh with rectangular cells. These two last sequences aremainly intended to serve as reference. For the generalized MFD algorithm, we consider two consistent gradientreconstructions coming from finite volumes: the first one is the hybrid finite volume gradient operator of [12],that uses faces values for the topography ๐, while the second one is the vertex based finite volume gradientoperator (VVM) of [5].
As it is the quantity of practical interest, we will express our convergence results directly in terms of the waterdischarge ๐ instead of using the water height. Let us begin by some results for sequences 2dSquareCart, 2dRect-Cart and 2dVoronoi. Being the only ones that satisfy the mesh orthogonality requirement (see Figs. 4 and 5),we expect that the TPFA scheme will also converge on those sequences. The corresponding results are displayedon Figures 6 and 7, while the associated approximate convergence orders for each scheme are given in Table 1.
1944 J. COATLEVEN
Figure 4. Example of meshes for the 2dVoronoi and 2dKershawBox mesh sequences.
Figure 5. Example of meshes for the 2dCheckerBoardBox and 2dRectCart mesh sequences.
Clearly, on the basic cartesian cases 2dSquareCart and 2dRectCart, all the schemes give identical results. Acloser look at our generalized flux formula immediately reveals that this is perfectly normal as the flux ๐น๐พ,๐
degenerate into the two-point flux formula on those cartesian meshes where the barycenter of each internal faceis exactly the intersection point between the face and the segment joining the cell centers on each side of theface (see [12]). Moreover, Table 1 reveals that all the methods are superconvergent on the three orthogonal meshsequences. In [20] it is indeed established in the case of exact fluxes that the approximation of the water heightsuperconverges at least on rectangular meshes, which explains that this observed superconvergence is probablynot anomalous. Next, we turn to the other mesh sequences, for which we cannot expect convergence for theTPFA scheme. The corresponding results are displayed on Figures 8 and 9, while the associated approximateconvergence orders for each scheme are given in Table 2.
As expected, the TPFA scheme does not converge anymore, however the generalized MFD algorithm isconverging for both the hybrid and virtual volume (VVM) gradients. The results for those two variants are infact relatively close to each other. In particular, the behavior of the method on sequence 2dCheckerBoardBoxconfirms its ability to deal with non conformities and thus local mesh refinement. Remark that superconvergence
SOME MULTIPLE FLOW DIRECTION ALGORITHMS 1945
Figure 6. Convergence curves for sequences 2dSquareCart and 2dRectCart.
Table 1. Approximate orders of convergence for the three schemes on orthogonal meshes.
2dSquareCart 2dRectCart 2dVoronoi
TPFA 0.943 0.942 1.147Hybrid 0.943 0.942 1.061VVM 0.943 0.942 1.061
Figure 7. Convergence curves for sequence 2dVoronoi.
1946 J. COATLEVEN
Figure 8. Convergence curves for sequences 2dDelaunay and 2dDualDelaunay.
Figure 9. Convergence curves for sequences 2dKershawBox and 2dCheckerBoardBox.
Table 2. Approximate orders of convergence for the three schemes on general meshes.
2dDelaunay 2dDualDelaunay 2dKershawBox 2dCheckerBoardBox
Hybrid 0.991 1.081 0.941 1.032VVM 0.971 1.084 1.213 1.032
SOME MULTIPLE FLOW DIRECTION ALGORITHMS 1947
Figure 10. Comparison between influx and water discharge for the TPFA scheme on orthog-onal meshes (exact water discharge is bottom right).
is also achieved again on those test cases. To conclude, on Figure 10, we represent on the bottom-right the correctwater discharge, while the top-left figure is the water influx r๐๐พ for sequence 2dSquareCart, the top-right figure isthe water influx for sequence 2dRectCart and the bottom-left figure is the water influx for sequence 2dVoronoi.These very basic examples illustrate our point on the water influx r๐๐พ : it is clearly a mesh dependent quantity,that cannot be expected to reproduce the behavior of the discharge correctly, even on those very simple cases. Ifone does not consider the ManningโStrickler framework, one of the tricky features of r๐๐พ is that on very commoncartesian mesh sequences such as 2dSquareCart and 2dRectCart it has a relatively nice qualitative behavior.It even seems to converge, although to a mesh sequence dependent quantity. This explains why it has beenused inadvertently in the hydrogeology community, despite of its mesh dependent nature: on the widely usedcartesian meshes, it has an acceptable behavior that cannot be easily discarded without a reference continuousmodel, in particular for complex topographies. However, on more general meshes, the situation is very different,even for the TPFA scheme as the case of sequence 2dVoronoi reveals. We display on Figure 11 the water influxr๐๐พ for the four other mesh sequences, in the case of the hybrid gradient. It is completely obvious on thosesequences that r๐๐พ should not be considered as a physically relevant quantity.
8. Conclusion
We have established the equivalence between a classical family of multiple flow direction algorithms and theTPFA scheme applied to a family of stationary ManningโStrickler models. From this equivalence we have pro-posed an improvement of the derivation of the discrete water discharge based on a consistent flux reconstruction
1948 J. COATLEVEN
Figure 11. Water influx for hybrid gradient on meshes 2dDelaunay, 2dDualDelaunay, 2dKer-shawBox and 2dCheckerBoardBox.
rather than directly using the water influx intermediate unknown that is not only non convergent in general, butdoes not approximate the water discharge in any case. Then, using more advanced flux reconstruction schemes,we have proposed a multiple flow direction algorithm adapted to general meshes. A convergence theory thatcovers both cases was developed, and numerical experiments illustrate the good behavior of the method. Remarkthat the extension to the case of ManningโStrickler tensors rather than coefficients is straightforward, as wouldbe the extension of the present analysis to multiple flow direction models that use powers of the topographicslope.
Acknowledgements. The author would like to thank G. Enchery and L. Agelas for our numerous discussions on thecontents of this article.
References
[1] C. Bardos, Problemes aux limites pour les equations aux derivees partielles du premier ordre a coefficients reels; theoremesdโapproximation; application a lโequation de transport. Ann. Sci. Ec. Norm. Sup. Ser. 4 3 (1970) 185โ233.
[2] H. Beirao Da Veiga, Existence results in Sobolev spaces for a stationary transport equation. Ricerche Mat. Suppl. XXXVI(1987) 173โ184.
[3] S. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods, 3rd ed. Springer (2008).[4] J. Coatleven, Semi hybrid method for heterogeneous and anisotropic diffusion problems on general meshes. ESAIM: M2AN
49 (2015) 1063โ1084.[5] J. Coatleven, A virtual volume method for heterogeneous and anisotropic diffusion-reaction problems on general meshes.
ESAIM: M2AN 51 (2017) 797โ824.[6] B. Cockburn, B. Dong and J. Guzman, Optimal convergence of the original DG method for the transport-reaction equation
on special meshes. SIAM J. Numer. Anal. 46 (2008) 1250โ1265.
SOME MULTIPLE FLOW DIRECTION ALGORITHMS 1949
[7] D.A. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods. Springer (2012).
[8] R.J. DiPerna and P.L. Lions, Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 (1989)511โ547.
[9] R.H. Erskine, T.R. Green, J.A. Ramirez and L.H. MacDonald, Comparison of grid-based algorithms for computing upslopecontributing area. Water Resour. Res. 42 (2006) W09416.
[10] R. Eymard, T. Gallouet and R. Herbin, Finite volume methods, edited by P.G. Ciarlet and J.-L. Lions. In: Handbook ofNumerical Analysis: Techniques of Scientific Computing, Part III. North-Holland, Amsterdam (2000) 713โ1020.
[11] R. Eymard, T. Gallouet and R. Herbin, A new finite volume scheme for anisotropic diffusion problems on general grids:convergence analysis. C. R. Math. Acad. Sci. Paris 344 (2007) 403โ406.
[12] R. Eymard, T. Gallouet and R. Herbin, Discretisation of heterogeneous and anisotropic diffusion problems on general noncon-forming meshes SUSHI: a scheme using stabilisation and hybrid interfaces. IMA J. Numer. Anal. 30 (2010) 1009โ1043.
[13] R. Eymard, C. Guichard and R. Herbin, Benchmark 3D: the vag scheme. In: Vol. 2 of Springer Proceedings in Mathematics,FVCA6, Prague (2011) 213โ222.
[14] R. Eymard, C. Guichard and R. Herbin, Small-stencil 3D schemes for diffusive flows in porous media. ESAIM: M2AN 46(2011) 265โ290.
[15] E. Fernandez-Cara, F. Guillen and R.R. Ortega, Mathematical modeling and analysis of visco-elastic fluids of the oldroyd kind,edited by P.G. Ciarlet and J.L. Lions. In: Vol. VIII of Handbook of Numerical Analysis: Numerical Methods for Fluids, Part2. North-Holland, Amsterdam (2002) 543โ661.
[16] T.G. Freeman, Calculating catchment area with divergent flow based on a regular grid. Comput. Geosci. 17 (1991) 413โ422.
[17] V. Girault and L. Tartar, ๐๐ and ๐ค1,๐ regularity of the solution of a steady transport equation. C. R. Acad. Sci. Paris, Ser. I348 (2010) 885โ890.
[18] P. Holmgren, Multiple flow direction algorithms for runoff modelling in grid based elevation models: an empirical evaluation.Hydrol. Process. 8 (1994) 327โ334.
[19] C. Johnson and J. Pitkaranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math.Comput. 46 (1986) 1โ26.
[20] P. Lesaint and P.A. Raviart, On a finite element method for solving the neutron transport equation. Publ. Math. Inf. RennesS4 (1974) 1โ40.
[21] C. Qin, A.-X. Zhu, T. Pei, B. Li, C. Zhou and L. Yang, An adaptive approach to selecting a flow-partition exponent for amultiple-flow-direction algorithm. Int. J. Geog. Inf. Sci. 21 (2007) 443โ458.
[22] P. Quinn, K. Beven, P. Chevallier and O. Planchon, The prediction of hillslope flow paths for distributed hydrological modellingusing digital terrain models. Hydrol. Process. 5 (1991) 59โ79.
[23] A. Richardson, C.N. Hill and J.T. Perron, IDA: an implicit, parallelizable method for calculating drainage area. Water Resour.Res. 50 (2013) 4110โ4130.
[24] J. Seibert and B.L. McGlynn, A new triangular multiple flow direction algorithm for computing upslope areas from griddeddigital elevation models. Water Resour. Res. 43 (2007) W04501.
[25] D.G. Tarboton, A new method for the determination of flow directions and upslope areas in grid digital elevation models.Water Resour. Res. 33 (1997) 309โ319.
[26] D.M. Wolock and G.J. McCabe Jr., Comparison of single and multiple flow direction algorithms for computing topographicparameters in topmodel. Water Resour. Res. 31 (1995) 1315โ1324.
[27] Q. Zhou, P. Pilesjo and Y. Chen, Estimating surface flow paths on a digital elevation model using a triangular facet network.Water Resour. Res. 47 (2011) W07522.