journal of mathematical analysis and...

J. Math. Anal. Appl. 429 (2015) 901–923

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Journal of Mathematical Analysis and Applications

www.elsevier.com/locate/jmaa

Reduced-order finite difference extrapolation model based on

proper orthogonal decomposition for two-dimensional shallow

water equations including sediment concentration ✩

Zhendong Luo a,∗, Junqiang Gao a, Zhenghui Xie b

a School of Mathematics and Physics, North China Electric Power University, Beijing 102206, Chinab LASG, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 4 February 2015Available online 11 April 2015Submitted by Goong Chen

Keywords:Error estimateNumerical simulationProper orthogonal decompositionReduced-order finite difference extrapolating modelShallow water equations including sediment concentration

In this study, we employ a proper orthogonal decomposition (POD) method to establish a POD-based reduced-order finite difference (FD) extrapolating model with very few degrees of freedom for two-dimensional shallow water equations that include the sediment concentration. We provide estimates of the error between the accurate solution and classical FD solutions, as well as those between the accurate solution and the POD-based reduced-order FD solutions. Moreover, we present two numerical simulation experiments to demonstrate that the POD-based reduced-order FD extrapolating model can greatly reduce the computational load. Thus, we validate both the feasibility and efficiency of the POD-based reduced-order FD extrapolating model.

© 2015 Elsevier Inc. All rights reserved.

1. Introduction

A system of shallow water equations (SWEs) can be used to describe the propagation and transformation of short waves in shallow waters, which are also referred to as the Saint-Venant system (see [13]). SWEs have extensive applications in ocean, environmental, hydraulic, and coastal engineering, such as open channel flows in rivers and reservoirs, tidal flows in estuaries and coastal water regions, bore wave propagation, and stationary hydraulic jumps and rivers, as mentioned in [16]. SWEs comprise a system of nonlinear partial differential equations (PDEs), so they generally have no analytical solutions, and thus we have to rely on numerical solutions.

Many previous studies have considered the numerical solutions for two-dimensional (2D) SWEs that only include the continuity equation and the momentum equations, i.e., that only include the water depth and

✩ This research was jointly supported by the National Science Foundation of China (11271127) and the Strategic Priority Research Program of the Chinese Academy of Sciences (No. XDA05110102).* Corresponding author. Fax: +86 10 61772167.

E-mail address: [email protected] (Z. Luo).

http://dx.doi.org/10.1016/j.jmaa.2015.04.0240022-247X/© 2015 Elsevier Inc. All rights reserved.

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the velocity of fluid, such as the finite volume (FV) method on unstructured triangular meshes proposed by Anatasiou and Chan in [1], the upwind methods described by Bermudez and Vazquez in [5], the parallel block preconditioning techniques given by Cai and Navon in [8], the optimal control technique with a finite element (FE) limited-area proposed by Chen and Navon in [11], the least-squares FE method of Liang and Hsu in [22], the finite difference (FD) Lax–Wendroff weighted essentially non-oscillatory (WENO) schemes proposed by Lu and Qiu in [25], the FE simulation technique of Navon in [39], the FD WENO schemes described by Qiu and Shu in [41], the Roe’s approximate Riemann solver technique of Rogers et al.in [44], the essentially non-oscillatory and WENO schemes with an exact conservation property proposed by Vukovic and Sopta in [55], the explicit multi-conservation FD scheme of Wang in [56], the composite FV method on unstructured meshes given by Wang and Liu in [58], the high order FD WENO schemes of Xing and Shu in [61], the high order well-balanced FV WENO schemes and discontinuous Galerkin (DG) methods of Xing and Shu in [62], the positivity-preserving high order well-balanced DG methods of Xing et al. in [63], the dispersion-correction FD scheme of Yoon et al. in [66], the non-oscillatory FV method proposed by Yuan and Song in [67], the surface gradient method of Zhou et al. in [69], and the total variation diminishing FD scheme given by Wang et al. in [59]. However, the transport and sedimentation of silt and sand are important processes in changing natural environments, such as formation and evolution of deltas, the expansion of alluvial plains, and the migration of rivers. In addition, some serious problems need to be carefully considered in many hydraulic problems, such as irrigation systems, transportation channels, hydroelectric stations, ports, and other coastal engineering works. A model for 2D SWEs that includes the sediment concentration was established in [68] and some numerical methods have also been presented based on the optimal control approach (see [70]) and mixed FE technique (see [35,36]).

It is well known that the model based on classical FD scheme in [68] is the simplest and most convenient method for solving 2D SWEs including the sediment concentration, but it also contains many degrees of freedom (i.e., unknown quantities). Therefore, this method can cause many difficulties in real-life engineering computation, e.g., the accumulation of truncated errors in the computing process will increase very rapidly so the classical FD solutions may appear to deviate greatly after several computational steps. Therefore, it is extremely important to build a reduced-order FD scheme with sufficiently high accuracy and very few degrees of freedom in order to alleviate the accumulation of truncated errors and to reduce the computational load, as well as decreasing the time required to make the calculations and resource demands during the computational process, thereby obtaining more accurate simulations of the development of alluvial plains in an estuary and dam-break floods.

The proper orthogonal decomposition (POD) technique (see [18,20]) is one of the most effective means for reducing the degrees of freedom in numerical models of time-dependent complex and nonlinear problems. POD is based in statistics (see [14,24]) but it has been used widely in the study of coherent structures in turbulent flows (see [2,4,26,38,43,47]). In the past 30 years, the applications of the POD technique have developed greatly (for example, see [27,2,4,9,37,38,43,45–47]). In particular, over the past 20 years, it has been applied to the construction of numerical computational models for time-dependent PDEs, or reduced-basis models for parameterized PDEs, e.g., some Galerkin POD methods for a general equation in fluid dynamics proposed by Kunisch and Volkwein in [21], POD-based reduced-order FD schemes described by Luo’s group in [31–34,50–52], POD-based reduced-order FE methods given by Luo et al. in [29,28,30], explicit reduced-order models for the stabilized FE approximation of the incompressible Navier–Stokes equations proposed by Baiges et al. in [3], an artificial viscosity POD given by Borggaard et al. in [6], an error estimation method for use in POD-based dynamic reduced-order thermal modeling of data centers described by Ghosh and Joshi in [15], the extrapolation-based acceleration of iterative solvers for application to simulations of 3D flows proposed by Grinberg and Karniadakis in [17], some iterative methods for model reduction by domain decomposition given by Buffoni et al. in [7], feedback control based on low-order modeling of the laminar flow past a bluff body by Weller et al. in [60], some reduced-basis models for parameterized PDEs given by Patera’s group in [40,53,54,65] and by Rozza et al. in [42], a variational

Z. Luo et al. / J. Math. Anal. Appl. 429 (2015) 901–923 903

multiscale POD for Navier–Stokes equations proposed by Iliescu and Wang in [19], some POD closure models for turbulent flows given by Wang et al. in [57], and a space-time Petrov–Galerkin certified reduced basis method for the Boussinesq equations proposed by Yano in [64].

However, as mentioned above, most existing POD-based reduced-order numerical methods employ nu-merical solutions of classical numerical methods on the total time span [0, T ] to formulate the POD basis and to build POD-based reduced-order models, before recomputing the numerical solutions on the same time span [0, T ], which are unrewarding repeated computations.

Some POD-based reduced-order models for 2D SWEs have been proposed (e.g., [10,48,49,71]), but these POD-based reduced-order models for 2D SWEs do not included the sediment concentration and they also employ the numerical solutions obtained from classical numerical methods on the total time span [0, T ] to formulate the POD basis, to build POD-based reduced-order models, and recompute the solutions on the same time span [0, T ], which also leads to repeated computations.

Therefore, in this study, we improve the existing methods by only employing the first few given classical FD numerical solutions for 2D SWEs on a very short time span [0, T0] (T0 � T ) as snapshots to formulate the POD basis and build a POD-based reduced-order FD extrapolating model, before finding the numerical solutions on the total time span [0, T ] via extrapolation and iteration, as well as POD basis updating. Thus, we fully exploit the advantages of the POD method, i.e., we employ the given data (on a very short time span [0, T0]) to predict future physical phenomena (on a time span [T0, T ]). Thus, our proposed method could have a very important role if it can be extended to real-life numerical computations. In particular, our POD-based reduced-order FD extrapolating model is completely different from the existing POD-based reduced-order models that we mentioned previously, where it is improved greatly by various innovations.

The remainder of this paper is organized as follows. In Section 2, we formulate snapshots and the POD basis from the classical FD solutions for 2D SWEs including the sediment concentration, and we establish the POD-based reduced-order FD extrapolating model. We also provide error estimates for the solutions and we describe the implementation of the algorithm for the POD-based reduced-order FD extrapolating model. In Section 3, we provide two numerical simulation examples that verify the reliability and effectiveness of the POD-based reduced-order FD extrapolating model. Finally, we give our main conclusions and some discussion in Section 4.

2. Classical FD scheme and POD-based reduced-order FD extrapolating model, as well as error estimates and algorithm implementation

2.1. Governing equations and classical FD scheme for 2D SWEs including the sediment concentration

Let Ω ⊂ R2 be a bounded and connected domain. The governing equations for 2D SWEs including the sediment concentration are defined as follows (see [16] and [68], but they are modified appropriately)

∂Z

∂t+ ∂(Zu)

∂x+ ∂(Zv)

∂y= γ

(∂2Z

∂x2 + ∂2Z

∂y2

), (x, y, t) ∈ Ω × (0, T ), (1)

∂u

∂t+ u∂u

∂x+ v∂u

∂y− fv = A

(∂2u

∂x2 + ∂2u

∂y2

)− g

∂(Z + zb)∂x

− CDu√u2 + v2

Z, (x, y, t) ∈ Ω × (0, T ), (2)

∂v

∂t+ u∂v

∂x+ v∂v

∂y+ fu = A

(∂2v

∂x2 + ∂2v

∂y2

)− g

∂(Z + zb)∂y

− CDv√u2 + v2

Z, (x, y, t) ∈ Ω × (0, T ), (3)

∂S

∂t+ u∂S

∂x+ v∂S

∂y= ε

(∂2S

∂x2 + ∂2S

∂y2

)+ αω(S − S∗)

Z, (x, y, t) ∈ Ω × (0, T ), (4)

∂zb + gb

(∂u + ∂v

)= αω(S − S∗)

, (x, y, t) ∈ Ω × (0, T ), (5)
∂t ∂x ∂y ρ


Fig. 1. Water profile.

where γ and A are two coefficients of viscosity, (u, v) is the vector of velocity, Z = z− zb is the water depth, z is the surface height, zb is the height of the bed (see Fig. 1), f is the Coriolis constant, g is the gravitational constant, CD is the coefficient of bottom drag, ε is the diffusion coefficient of sand, ω is the falling speed of suspended sediment particles, S is the concentration of sediment in water, S∗ = K[(u2 + v2)3/2/(gωZ)]mis the capacity for sediment transport in the bottom bed (a given empirical function), ρ is the density of dry sand (which can be taken as a constant), gb = Γ(u2 + v2)3/2Zpdq[1 − vc/(u2 + v2)1/2] is also a given empirical function, vc is the velocity of sediment mass transport (also a given function), d is the diameter of sediment, and K, m, Γ, p, and q are all empirical constants.

The boundary conditions are assumed to be as follows:

Z(x, y, t) = Z0(x, y, t), u(x, y, t) = u0(x, y, t), v(x, y, t) = v0(x, y, t),

S(x, y, t) = S0(x, y, t), zb(x, y, t) = zb0(x, y, t), (x, y, t) ∈ ∂Ω × (0, T ), (6)

where Z0(x, y, t), u0(x, y, t), v0(x, y, t), S0(x, y, t), and zb0(x, y, t) are all given functions. The initial condi-tions are assumed to be as follows:

Z(x, y, 0) = Z0(x, y), u(x, y, 0) = u0(x, y), v(x, y, 0) = v0(x, y),

S(x, y, 0) = S0(x, y), zb(x, y, 0) = z0b (x, y), x ∈ ∂Ω × (0, T ), (7)

where Z0(x, y), u0(x, y), v0(x, y), S0(x, y), and z0b (x, y) are also given functions.

Let �t be the time step, �x and �y comprise the spatial step, and N = [T/�t]. By discretizing (1), (4), and (5) at reference point (xj , yk, tn), (2) at reference point (xj+ 1

2, yk, tn), and (3) at reference point

(xj , yk+ 12, tn), we obtain the classical FD scheme for 2D SWEs including the sediment concentration as

follows:

Zn+1j,k − Zn

j,k

�t= γ

(Znj+1,k − 2Zn

j,k + Znj−1,k

�x2 +Znj,k+1 − 2Zn

j,k + Znj,k−1

�y2

)

−unj+ 1

2 ,kZnj+ 1

2 ,k− un

j− 12 ,k

Znj− 1

2 ,k

�x−

vnj,k+ 1

2Znj,k+ 1

2− vn

j,k− 12Znj,k− 1

2

�y, (8)

un+1j+ 1

2 ,k− un

j+ 12 ,k

�t= A

(unj+ 3

2 ,k− 2un

j+ 12 ,k

+ unj− 1

2 ,k

�x2 +unj+ 1

2 ,k+1 − 2unj+ 1

2 ,k+ un

j+ 12 ,k−1

�y2

)

−unj+ 1

2 ,k(un

j+1,k − unj,k)

�x−

vnj+ 1

2 ,k(un

j+ 12 ,k+ 1

2− un

j+ 12 ,k− 1

2)

�y

− gZnj+1,k + znb,j+1,k − Zn

j,k − znb,j,k
�x


+ fvnj+ 12 ,k

−CDun

j+ 12 ,k

√(un

j+ 12 ,k

)2 + (vnj+ 1

2 ,k)2

Znj+ 1

2 ,k

, (9)

vn+1j,k+ 1

2− vn

j,k+ 12

�t= A

(vnj+1,k+ 1

2− 2vn

j,k+ 12

+ vnj−1,k+ 1

2

�x2 +vnj,k+ 3

2− 2vn

j,k+ 12

+ vnj,k− 1

2

�y2

)

−unj,k+ 1

2(vn

j+ 12 ,k+ 1

2− vn

j− 12 ,k+ 1

2)

�x−

vnj,k+ 1

2(vnj,k+1 − vnj,k)�y

− gZnj,k+1 + znb,j,k+1 − Zn

j,k − znb,j,k�y

− funj,k+ 1

2−

CDvnj,k+ 1

2

√(un

j,k+ 12)2 + (vn

j,k+ 12)2

Znj,k+ 1

2

, (10)

Sn+1j,k − Sn

j,k

�t= ε

(Snj+1,k − 2Sn

j,k + Snj−1,k

�x2 +Snj,k+1 − 2Sn

j,k + Snj,k−1

�y2

)

−unj,k(Sn

j+ 12 ,k

− Snj− 1

2 ,k)

�x−

vnj,k(Snj,k+ 1

2− Sn

j,k− 12)

�y+

αω(Snj,k − S∗n

j,k)Znj,k

, (11)

zn+1b,j,k − znb,j,k

�t= −

gnb,j,k(unj+ 1

2 ,k− un

j− 12 ,k

)�x

−gnb,j,k(vnj,k+ 1

2− vn

j,k− 12)

�y+

αω(Snj,k − S∗n

j,k)ρ

, (12)

where n = 1, 2, . . . , N , J = max(x1,y),(x2,y)∈Ω |x1 − x2|, and K = max(x,y1),(x,y2)∈Ω |y1 − y2|.Under the conditions that �t · (|u| + |v|) � min{4γ, 4ε, 4A} and 4�t max{γ, A, ε} � min{�x2, �y2}, by

using the stability analysis technique of FD schemes (see [12] or [23]), it has been proved that the classical FD scheme (8)–(12) is locally stable, and the proof is provided in Appendix A. By employing Taylor’s formula to expand (8), (11), and (12) at reference point (xj , yk, tn), (9) at reference point (xj+ 1

2, yk, tn),

and (10) at reference point (xj , yk+ 12, tn), we can obtain the following error estimates:

|Z(xj , yk, tn) − Znj,k| + |u(xj+ 1

2, yk, tn) − un

j+ 12 ,k

| + |v(xj , yk+ 12, tn) − vnj,k+ 1

2|

+ |S(xj , yk, tn) − Snj,k| + |zb(xj , yk, tn) − znb,j,k|

= O(Δt,Δx2,Δy2), 1 � n � N, 1 � j � J, 1 � k � K. (13)

Remark 1. The classical FD scheme (8)–(12) is only first-order accurate in time. If we want to achieve higher order time approximate accuracy, it is necessary to change the time difference coefficients on the left-hand sides in (8)–(12) into higher order ones (e.g., time central difference coefficients or time second-order difference coefficients).

Remark 2. Provided that we are given the Coriolis constant f , the acceleration due to gravity g, the coefficients of viscosity γ and A, the coefficient of bottom drag CD, the diffusion coefficient of sand ε, the falling speed of suspended sediment particles ω, the velocity of sediment mass transport vc, the diameter of sediment d; the empirical constants K, m, n, p, and q; the boundary value functions Z0(x, y, t), u0(x, y, t), v0(x, y, t), S0(x, y, t), and zb0(x, y, t); the initial value functions Z0(x, y), u0(x, y), v0(x, y), S0(x, y), and z0b (x, y); the time step increment Δt, and the spatial step increments Δx and Δy, we can obtain the

classical FD solutions unj+ 1

2 ,k, vn

j,k+ 12, Sn

j,k, Znj,k, and znb,j,k (0 � j � J , 0 � k � K, 1 � n � N) for 2D

SWEs including the sediment concentration by solving the FD schemes (8)–(12).


2.2. Establishing the POD-based reduced-order FD extrapolating model

In order to formulate the POD basis, we extract the first L sequence of solutions (ulj+ 1

2 ,k, vl

j,k+ 12, Sl

j,k, Zlj,k,

zlb,j,k) (l = 1, 2, . . . , L) for the classical FD scheme (8)–(12) as snapshots and put uli = ul

j+ 12 ,k

, vli = vlj,k+ 1

2,

Sli = Sl

j,k, Zli = Zl

j,k, and zlbi = zlb,j,k (i = kJ + j + 1, 1 � i � m, m = JK, 0 � j � J − 1, 0 � k � K − 1), respectively. Thus, we formulate five m × L matrices Ar = (rli)m×L (r = u, v, S, Z, zb) denoted by

Ar =

⎛⎜⎜⎜⎜⎝

r11 r2

1 · · · rL1r12 r2

2 · · · rL2...

.... . .

...r1m r2

m · · · rLm

⎞⎟⎟⎟⎟⎠ .

It is obvious that the number of mesh points m is much larger than that of the extracted snapshots L. Therefore, the degree m for the matrices ArA

Tr is much larger than the degree L for the matrices AT

r Ar, but their positive eigenvalues are identical. Therefore, we first find the eigenvalues λr1 � λr2 � . . . � λrMr

> 0(Mr = rankAr) for the matrices AT

r Ar and corresponding eigenvectors ϕrj . Then, by the relationship

φrj = Arϕrj/√

λrj , j = 1, 2, . . . , Mr, r = u, v, S, Z, zb,

we formulate the eigenvectors φrj (j = 1, 2, . . . , Mr) corresponding to the nonzero eigenvalues for the matrix ArA

Tr (r = u, v, S, Z, zb).

We use the first Mr (0 < Mr � Mr � L) columns of the eigenmatrices Ur = (φr1, φr2, · · · , φrMr) to

formulate five set of orthonormal POD bases (see [32] or [33]) Φr = (φr1, φr2, · · · , φrMr) (r = u, v, S, Z, zb).

Put

rnm = (rn1 , rn2 , · · · , rnm)T , n = 1, 2, . . . , N, r = u, v, S, Z, zb. (14)

It is easy to obtain the following error inequalities (see [32] or [33])

‖rlm − ΦrΦTr r

lm‖2 �

√λr(Mr+1) l = 1, 2, . . . , L, r = u, v, S, Z, zb, (15)

where ‖a‖2 = (∑m

i=1 a2i )1/2 is the standard norm of vector a = (a1, a2, . . . , am)T . Thus, the classical FD

scheme (8)–(12) can be rewritten in the following vector form

(un+1m ,vn+1

m ,Sn+1m ,Zn+1

m , zn+1b,m

)T

=(unm,vn

m,Snm,Zn

m, znb,m

)T + F (unm,vn

m,Snm,Zn

m, znb,m), n = 1, 2, · · · , N − 1, (16)

where F is defined by the classical FD scheme (8)–(12). In fact, when we solve the classical FD scheme, it is also usually necessary to write the classical FD scheme in vector form.

Let

(u∗nm ,v∗n

m ,S∗nm ,Z∗n

m , z∗nb,m

)T =(Φuβ

nMu

,ΦvβnMv

,ΦSβnMS

,ΦZβnMZ

,ΦzbβnMzb

)T

, (17)

where r∗nm = (r∗n1 , r∗n2 , · · · , r∗nm )T (r = u, v, S, Z, zb) are five column vectors corresponding to r (r =u, v, S, Z, zb), respectively. If we substitute u∗n

m , v∗nm , S∗n

m , Z∗nm , and z∗n

b,m in (17) for unm, vn

m, Snm, Zn

m, and zn

b,m in (16) (n = 0, 1, 2, . . . , N) and note that five matrices Φr (r = u, v, S, Z, zb) are formed with the


orthonormal vectors, then we obtain the POD-based reduced-order FD extrapolating model for 2D SWEs including the sediment concentration, as follows

βnMu

= ΦTuu

nm,βn

Mv= ΦT

v vnm,βn

MS= ΦT

SSnm,βn

MZ= ΦT

ZZnm,βn

Mzb= ΦT

zbznb,m, 1 � n � L; (18)

(βn+1Mu

,βn+1Mv

,βn+1MS

,βn+1MZ

,βn+1Mzb

)T

=(βnMu

,βnMv

,βnMS

,βnMZ

,βnMzb

)T

+ (ΦTu ,ΦT

v ,ΦTS ,ΦT

Z ,ΦTzb

)T F (ΦuβnMu

,ΦvβnMv

,ΦSβnMS

,ΦZβnMZ

,ΦzbβnMzb

), L � n � N − 1, (19)

which only includes Mu +Mv +MS +MZ +Mzb (Mu, Mv, MS , MZ , Mzb � L � m) degrees of freedom on each time level and it has no repeated computations.

If we obtain βnMu

, βnMv

, βnMS

, βnMZ

, and βnMzb

from (18) and (19), then we can determine the POD-based reduced-order FD solutions as follows

u∗nm = Φuβ

nMu

,v∗nm = Φvβ

nMv

,S∗nm = ΦSβ

nMS

,Z∗nm = ΦZβ

nMZ

, z∗nb,m = Φzbβ

nMzb

, n = 1, 2, . . . , N. (20)

Furthermore, we obtain the component forms of the POD-based reduced-order FD solutions, denoted by u∗nj+ 1

2 ,k= u∗n

i , v∗nj,k+ 1

2= v∗ni , S∗n

j,k = S∗ni , Z∗n

j,k = Z∗ni , and z∗nb,j,k = z∗nb,i (0 � j � J − 1, 0 � k � K − 1,

i = k(J + 1) + j + 1, 1 � i � m = KJ).

Remark 3. It is easy to see that the classical FD scheme (8)–(12) on each time level contains 5m unknown quantities, whereas the system of equations (18)–(20) on each time level (when n > L) only contains Mu + Mv + MS + MZ + Mzb unknown quantities (Mu, Mv, MS , MZ , Mzb � L � m, e.g., in Section 3, L = 20, Mu = Mv = MS = MZ = Mzb = 6, but m = 7000 or 25 × 106). Therefore, the POD-based reduced-order FD extrapolating model (18)–(20) includes very few degrees of freedom, does not involve repeated computations, and it is completely different from the existing POD-based reduced-order models mentioned in Section 1. In this study, we extract the snapshots from the first L classical FD solutions, but if we want to solve real-life problems, we may extract snapshots from the samples obtained in experimental analyses of physical system trajectories.

2.3. Error estimates of the solutions obtained by the POD-based reduced-order FD extrapolating model

In the following, we provide error estimates for the solutions obtained by the POD-based reduced-order FD extrapolating model.

The error estimates between the reduced-order FD solutions for the POD-based reduced-order FD ex-trapolating model (18)–(20) and the classical FD solutions for the classical FD scheme (8)–(12) are as follows, for which the proof is provided in Appendix B.

Theorem 1. If (unm, vn

m, Snm, Zn

m, znb,m)T (n = 1, 2, . . . , N) are the solution vectors formed from the classical

FD scheme (8)–(12) and (u∗nm , v∗n

m , S∗nm , Z∗n

m , z∗nb,m)T (n = 1, 2, . . . , N) are the reduced-order FD solutions

for the POD-based reduced-order FD extrapolating model (18)–(20), then we have the following error estimate

‖(unm,vn

m,Snm,Zn

m, znb,m) − (u∗n

m ,v∗nm ,S∗n

m ,Z∗nm , z∗n

b,m)‖2

� C(δn)[√

λu(Mu+1) +√

λv(Mv+1) +√λS(MS+1) +

√λZ(MZ+1) +

√λzb(Mzb

+1)

], n = 1, 2, . . . , N,

where C(δn) =1 (1 � n � L), C(δn) = (1 +δ)n−L (L +1 � n � N), and δ=�tmax{γ,A, ε}/min{Δx2,Δy2}.


It is well known that the absolute value of each component of the vector does not exceed its standard norm. Thus, by combining (13) with Theorem 1, we obtain the following result.

Theorem 2. The accurate solution for 2D SWEs and the reduced-order FD solutions obtained from the POD-based reduced-order FD extrapolating model (18)–(20) have the following error estimate

|u(xj+ 12, yk, tn) − u∗n

j+ 12 ,k

| + |v(xj , yk+ 12, tn) − v∗nj,k+ 1

2| + |S(xj , yk, tn) − S∗n

j,k| + |Z(xj , yk, tn) − Z∗nj,k|

+ |zb(xj , yk, tn) − z∗nb,j,k| = O(C(δn)

[√λu(Mu+1) +

√λv(Mv+1) +

√λS(MS+1) +

√λZ(MZ+1)

+√

λzb(Mzb+1)

],Δt,Δx2,Δy2

), 1 � n � N. (21)

Remark 4. The error terms √λu(Mu+1) +

√λv(Mv+1) +

√λS(MS+1) +

√λZ(MZ+1) +

√λzb(Mzb

+1) in The-orems 1 and 2 are due to the POD-based reduced-order in the classical FD scheme, which can be used to select the number of the POD basis, i.e., it is necessary to take Mu, Mv, MS , MZ , and Mzb such that √λu(Mu+1) +

√λv(Mv+1) +

√λS(MS+1) +

√λZ(MZ+1) +

√λzb(Mzb

+1) = O(Δt, Δx2, Δy2). By contrast, C(δn) = (1 + δ)n−L (L + 1 � n � N) are due to extrapolating iteration, which can be used to guide the re-newal of the POD basis, i.e., if C(δn)[

√λu(Mu+1)+

√λv(Mv+1)+

√λS(MS+1)+

√λZ(MZ+1)+

√λzb(Mzb

+1)] >max(Δt, Δx2, Δy2), then it is necessary to update the POD basis.

2.4. Implementation of the algorithm for the POD-based reduced-order FD extrapolating model

The implementation of the algorithm for the POD-based reduced-order FD extrapolating model (18)–(20)comprises the following five steps.

Step 1. Solving the classical FD scheme (8)–(12) in the first few L steps (in the following, we take L = 20) yields the classical FD solutions un

j+ 12 ,k

, vnj,k+ 1

2, Sn

j,k, Znj,k, and znb,j,k (0 � j � J , 0 � k � K, 1 � n � L) and

it also forms a set of snapshots {uli, v

li, S

li, Z

li , z

lb,i}Ll=1 (1 � i � m) with L ×m elements, where un

i = unj+ 1

2 ,k,

vni = vnj,k+ 1

2, Sn

i = Snj,k, Zn

i = Znj,k, and znb,i = znb,j,k (i = kJ + j + 1, 1 � i � m, m = JK, 0 � j � J − 1,

0 � k � K − 1), respectively.

Step 2. Formulate snapshot matrices Ar = (rli)m×L (r = u, v, S, Z, zb), and compute the eigenvalues λr1 �λr2 � . . . � λrMr

> 0 (Mr = rankAr) and the eigenvectors ϕrj (j = 1, 2, . . . , Mr, r = u, v, S, Z, zb) of matrices AT

r Ar, respectively.

Step 3. For the tolerance error μ = O(Δt, Δx2, Δy2), determine the number Mr (Mr � Mr, r = u, v, S, Z, zb) of POD bases such that

√λu(Mu+1) +

√λv(Mv+1) +

√λS(MS+1) +

√λZ(MZ+1) +

√λzb(Mzb

+1) � μ, and

formulate the POD bases Φr = (φr1, φr2, . . . , φrMr) (where φsj = Arϕrj/

√λrj , j = 1, 2, . . . , Mr, r =

u, v, S, Z, zb).

Step 4. Solving the POD-based reduced-order extrapolating FD model (18)–(20) yields the reduced-order solution vectors u∗n

m = (u∗n1 , u∗n

2 , . . . , u∗nm ), v∗n

m = (v∗n1 , v∗n2 , . . . , v∗nm ), S∗nm = (S∗n

1 , S∗n2 , . . . , S∗n

m ), Z∗nm =

(Z∗n1 , Z∗n

2 , . . . , Z∗nm ), and z∗n

b,m = (z∗nb,1, z∗nb,2, . . . , z∗nb,m), as well as the component forms u∗nj+ 1

2 ,k= u∗n

i , v∗nj,k+ 1

2=

v∗ni , S∗nj,k = S∗n

i , Z∗nj,k = Z∗n

i and z∗nb,j,k = z∗nb,i (0 � j � J , 0 � k � K, i = k(J +1) + j+1, 1 � i � m = KJ).

Step 5. Put δ = �tmax{γ,A, ε}/min{Δx2,Δy2}. If

(1 + δ)n−L[√

λu(Mu+1) +√λv(Mv+1) +

√λS(MS+1) +

√λZ(MZ+1) +

√λzb(Mz +1)

]� μ,

b


then u∗nm = (u∗n

1 , u∗n2 , . . . , u∗n

m ), v∗nm = (v∗n1 , v∗n2 , . . . , v∗nm ), S∗n

m = (S∗n1 , S∗n

2 , . . . , S∗nm ), Z∗n

m = (Z∗n1 , Z∗n

2 , . . . ,Z∗nm ), and z∗n

b,m = (z∗nb,1, z∗nb,2, . . . , z∗nb,m) (n = 1, 2, . . . , N) are simply the solution vectors for the POD-based reduced-order extrapolating FD model (18)–(20) that satisfy the accuracy requirement. Else, i.e., if

(1 + δ)n−L[√

λu(Mu+1) +√λv(Mv+1) +

√λS(MS+1) +

√λZ(MZ+1) +

√λzb(Mzb

+1)

]> μ,

put (rl1, rl2, . . . , rlm) = (r∗l1 , r∗l2 , . . . , r∗lm) (r = u, v, S, Z, zb; l = n − L, n − L − 1, . . . , n − 1), return to Step 2.

3. Numerical experiments

In the following, we present two numerical experiments that demonstrate the feasibility and efficiency of the POD-based reduced-order extrapolating FD model for 2D SWEs including the sediment concentra-tion.

3.1. Example simulation of sediment transport and flow in an estuary

The computational domain is Ω = {(x, y) : 23 − 23x/25 � y � 27 + 33x/25, 0 � x � 25} ∪ {(x, y) : 25 �x � 40, 0 � y � 50} (the unit of x and y is km). The depth at the entrance is 10 m (i.e., Z0|x=0 = 0.01 km). The sediment thickness at the entrance is 2 m (i.e., zb0|x=0 = 0.002 km). The velocity u0 of the fluid at the x-direction from the entrance is 2 m/s (i.e., u0|x=0 = 7.2 km/h), but v0 = 0. The sediment concentration in the water flow is 1.2 kg/m3 (i.e., S0 = S0 = 1.2 × 10−3 kg/km3). The change in the bottom topography every 100 km falls 1 m along the flow direction (i.e., zb0 = z0

b = 10−5x + 2, 0 � x � 40). The bilateral boundaries of the water flow are two solid borders, i.e., u0 = v0 = 0 on set {(x, y) : y = 23 − 23x/25, 0 �x � 25} ∪ {(x, y) : y = 27 + 33x/25, 0 � x � 25} ∪ {(x, 0) : 25 � x � 40} ∪ {(x, 50) : 25 � x � 40}. The time step Δt = 3600 s = 1 h. The spatial step Δx = Δy = 200 m = 0.2 km. According to [68], we take f = 1.1 × 10−4, γ = 0.001, A = 7.5 × 10−3, CD = 0.01, d = 0.001, ω = 0.01, vc = 0, α = 0.3, K = 0.35, Γ = 5, m = 0.92, p = −0.25, q = 0.25, and ρ = 1.5 × 103.

By means of the classical FD scheme (8)–(12), we obtained the classical FD solutions unj+ 1

2 ,kand vn

j,k+ 12,

Snj,k, and Zn

j,k for the velocity u in the x-direction and v in the y-direction, the sediment concentration S, and the water depth Z (the change in zb was very small, so it is not described) when n = 8760, 26 280, and 43 800(i.e., in the first year, third year, and fifth year, respectively), which are depicted graphically in the left column in Figs. 2, 3, and 4, respectively.

This was achieved by computing √λu7 +

√λv7 +

√λS7 +

√λZ7 � 4.5 × 10−3 when L = 20. Thus, it was

only necessary to choose the first six POD bases. The changes (black link lines) in the 20 dominant singular eigenvalues in Fig. 5 also verify this fact. In addition, by comparing the 20 singular dominant eigenvalues (see Fig. 5) for a short period (the first time interval 0 � t � 20 h) and those for the full period (total time interval 0 � t � 43 800 h), we found that the dominant singular eigenvalues were smaller for the short period than those for the full period. This implies that if we take the same modes (e.g., six POD bases), the accuracy of the solutions obtained by the POD-based reduced-order FD extrapolating model (18)–(20) is higher than that of those obtained by the usual POD FD scheme, where the usual POD FD scheme implies the use of all the classical numerical solutions on the time interval [0, T ] to form snapshots and to formulate the POD basis, as well as repeated computation of the POD-based reduced-order numerical solutions on the same time interval [0, T ], as described in [31–33].

Thus, we obtained the POD-based reduced-order FD solutions of the velocity u in the x-direction and vin the y-direction, the sediment concentration S, and the water depth Z (the change in zb is very small, so it is omitted) when n = 8760, 26 280, and 43 800 (i.e., at time points in the first year, third year, and fifth year) by implementing the five steps of the algorithm for the POD-based reduced-order FD extrapolating


Fig. 2. Velocity, sediment concentration, and water depth in the delta of the estuary in the first year. (For interpretation of the colors in this figure, the reader is referred to the web version of this article.)


Fig. 3. Velocity, sediment concentration, and water depth in the delta of the estuary in the third year. (For interpretation of the colors in this figure, the reader is referred to the web version of this article.)


Fig. 4. Velocity, sediment concentration, and water depth in the delta of the estuary in the fifth year. (For interpretation of the colors in this figure, the reader is referred to the web version of this article.)


Fig. 5. Changes in the 20 dominant eigenvalues in a short period and the first 20 dominant eigenvalues in the full period in terms of the velocity, sediment concentration, and water depth in the delta of the estuary.

model (18)–(20), as described in Section 2.4, where we needed to update the POD basis automatically four times, as shown in the right-hand columns in Figs. 2, 3, and 4, respectively. The corresponding left and right columns in Figs. 2, 3, and 4, respectively, exhibit quasi-identical similarity.

The relative deviations (computed by the formula [rk−∑N

k=1(rk/N)]/ ∑N

k=1(rk/N), r = u, v, S, Z, zb) of the POD-based reduced-order FD solutions on the starting time-span were slightly larger than those of the classical FD solutions, but the POD-based reduced-order FD extrapolating model on each time-level only included 5 ×6 degrees of freedom whereas the classical FD scheme had 5 ×7000 degrees of freedom, i.e., the POD-based reduced-order FD extrapolating model had far less degrees of freedom than the classical FD scheme, so the POD-based reduced-order FD extrapolating model can greatly reduce the accumulation of truncated errors during the computational process, decrease the calculation load, reduce the time required for the calculations, and improve the actual accuracy of the numerical solutions. Therefore, after a specific time, the numerical relative deviations of the solutions obtained by the POD-based reduced-order FD extrapolating model were lower than those with the classical FD scheme (see Fig. 6). In fact, Fig. 6 shows the accumulated truncated errors on 0 � t � 5 (year), where the relative deviations of the classical FD solutions were far greater than those of the reduced-order solutions obtained from the POD-based reduced-order FD extrapolating model. According to the changes in the relative deviations of classical FD solutions, the classical FD solutions appeared to have greater deviations than the POD-based reduced-order solutions after some computational steps, whereas the error accumulation with the POD-based reduced-order FD extrapolating model was very slow, and thus it could simulate the continuous development of water flow. We also found that the POD-based reduced-order FD extrapolating model was highly effective in searching for the numerical solutions of SWEs including the sediment concentration.

In order to quantify the efficiency of the POD-based reduced-order FD extrapolating model, we employed the root mean squared error (RMSE) and the correlation coefficient (CORCOE) between the usual POD FD solutions and the POD-based reduced-order FD extrapolating solutions in the first year, third year, and


Fig. 6. Changes in the relative deviations of the numerical solutions for the velocity, sediment concentration, and water depth in the delta of the estuary in years [0, 5].

fifth year. The RMSE and CORCOE are obtained using the following formulae:

RMSE(rj) =

√√√√ 1N

N∑n=1

|rnj − rndj |2, r = u, v, S, Z, j = 1, 3, 5;

CORCOE(rj) =

N∑n=1

(rnj − ¯rnj )(rndj − rndj)√N∑

n=1(rnj − ¯rnj )2

N∑n=1

(rndj − rndj)2, r = u, v, S, Z, j = 1, 3, 5,

where rjn (r = u, v, S, Z, j = 1, 3, 5) are the j-th year usual POD solutions, rndj are the j-th year POD-based reduced-order FD extrapolating POD solutions, and N = 8760, 26 280, and 43 800. Tables 1and 2 show the RMSEs and CORCOEs, respectively, between the usual POD FD solutions and the POD-based reduced-order FD extrapolating solutions in the first year, third year, and fifth year, i.e., N = 8760, 26 280, and 43 800, with six POD bases. Table 1 shows that the numerical computing RMSEs are consistent with the theoretical errors, although they increase with the number of time nodes. Table 2 also shows that the CORCOEs of the numerical solutions for the usual POD FD solutions and the POD-based reduced-order FD extrapolating solutions became increasing smaller with time, which is reasonable because the POD-based reduced-order FD extrapolating model only takes the first 20 solutions as snapshots. How-ever, the errors are within the range of tolerance. Therefore, the POD-based reduced-order FD extrapolating model performs better than the usual POD FD scheme.

We compared the classical FD scheme with the POD-based reduced-order FD extrapolation model con-taining six bases by implementing the numerical simulation computations when t = 5 years, where we found that the classical FD scheme included 5 ×7000 degrees of freedom on each time level and the computing time required was 120 minutes on a ThinkPad E530 PC, whereas the POD-based reduced-order FD extrapolation model with six POD bases on each time level only had 5 × 6 degrees of freedom and the corresponding time


Table 1RMSEs between the usual POD solutions and the POD-based extrapolating solutions.

u v S Z

N = 8760 1.48E−4 1.24E−4 1.37E−4 1.68E−4N = 26 280 1.12E−3 1.35E−3 1.64E−3 1.72E−3N = 26 280 4.38E−3 5.23E−3 6.36E−3 6.82E−3

Table 2CORCOEs for the usual POD solutions and the POD-based extrapolating solutions.

u v S Z

N = 8760 1.57E−4 2.38E−4 2.46E−4 2.59E−4N = 26 280 1.46E−6 2.35E−6 2.65E−6 2.71E−6N = 26 280 1.43E−8 2.24E−8 2.58E−8 2.86E−8

was about 30 seconds on the same PC, i.e., the computational time with the classical FD scheme was about 240 times more than that with the POD-based reduced-order FD extrapolation model with six POD bases. We also showed that the POD-based reduced-order FD extrapolation model greatly reduced the accumula-tion of the truncated errors in the computational process, decreased the calculation load, reduced the time required for the calculations, and improved the actual accuracy of the numerical solutions.

3.2. Example simulation of a dam-break flow

A dam-break flow is an uncontrolled release of water when a vertical barrier is removed suddenly and it is the simplest available model for many important phenomena, such as break-out floods, sheet flow events, and the formative stages of lahars or debris flows.

An idealized model of the dam-break flow may show that the barrier at x = 50 and 0 � y � 100 divides fluids of different depths 10 m and 5 m, until time t = 0, when a gate of width 15 m (i.e., on x = 50 and 50 � y � 75) in the barrier is removed instantaneously and fluid (depth 10 m) floods into the shallower region (depth 5 m). Thus, the computational domain for the dam-break flow is a square of area 100 ×100 m2, i.e., Ω = [0, 100] × [0, 100], which holds water depths of 10 m on the sub-domain [0, 50] × [0, 100] and 5 m on the sub-domain [50, 100] × [0, 100]. Since zb = 0, then znb,j,k = 0 in (12)–(18) and correspondingly z∗nb,m = 0 in (18)–(20).In order to solve the classical FD scheme (12)–(18) and the POD-based reduced-order FD extrapolating

FD model for 2D SWEs including the sediment concentration, it is necessary to take the time step Δt =0.01 s and the spatial step Δx = Δy = 0.02 m, and to designate all of the parameters: f = 1.1 × 10−4, γ = 0.001, A = 7.5 × 10−3, CD = 0.01, ω = 0.01, α = 0.3, K = 0.35, m = 0.92, and ρ = 1.5 × 103 (see [67,68]).

By means of the classical FD scheme (8)–(12), we obtained the classical FD solutions of the dam-break flow and the sediment concentration (zb = 0, so it is not described) when n = 100, 300, and 500 (i.e., at 1 s, 3 s, and 5 s), as shown in the left columns in Figs. 7, 8, and 9, respectively. We also obtained the classical FD solutions for u and v at t = 5 s, as shown in the left column in Fig. 10.

Similarly, when L = 20, this was achieved by computing √λu7 +

√λv7 +

√λS7 +

√λZ7 � 3.5 × 10−3.

Thus, it is also only necessary to choose the first six POD bases. Then, by implementing the five steps of the algorithm for the POD-based reduced-order FD extrapolating model (18)–(20), as described in Section 2.4, but without needing POD-basis renewal, we obtained the POD-based reduced-order FD solutions for the dam-break flow and sediment concentration (zb = 0, so it is omitted) at n = 100, 300, and 500 (i.e., at 1 s, 3 s, and 5 s), as shown in the right columns in Figs. 7, 8, and 9, respectively. We also obtained the POD-based reduced-order solutions for u and v at t = 5 s, as shown in the right column in Fig. 10. The corresponding charts in the left and right columns of Figs. 7, 8, 9, and 10 exhibit quasi-identical similarity. The classical


Fig. 7. The left and right charts show the classical FD solutions and the reduced-order FD solutions for the dam-break flow and the sediment concentration at t = 1 s, respectively. (For interpretation of the colors in this figure, the reader is referred to the web version of this article.)

FD scheme included 4 × 25 × 106 degrees of freedom on each time level in the numerical simulation of the dam-break flow, whereas the POD-based reduced-order extrapolating model with six POD bases on each time level had 4 × 6 degrees of freedom, so the POD-based reduced-order FD extrapolating model greatly reduced the accumulation of truncated errors in the computational process, decreased the time required, reduced the calculation load, and improved the actual computational accuracy. We also found that the POD-based reduced-order FD extrapolating model was very effective in simulating the dam-break flow.

4. Conclusion and discussion

In the current study, we employed the POD method to establish a POD-based reduced-order FD extrap-olating model for 2D SWEs including the sediment concentration. We determined the error estimates of the POD-based reduced-order FD to facilitate the selection of the number of the POD basis and for updating the POD basis. We presented two numerical examples to demonstrate that the POD-based reduced-order FD extrapolating model is highly effective in finding the numerical solutions of SWEs including the sed-iment concentration. As mentioned in Section 1, the POD-based reduced-order FD extrapolating model for 2D SWEs including sediment concentration is completely different from existing POD-based reduced-


Fig. 8. The left and right charts show the classical FD solutions and the reduced-order FD solutions for the dam-break flow and the sediment concentration at t = 3 s, respectively. (For interpretation of the colors in this figure, the reader is referred to the web version of this article.)

order models for SWEs (e.g., [10,48,49]). Our new model performs better than existing models and other POD-based reduced-order methods or reduced-basis methods, as mentioned in Section 1.

Appendix A

In this Appendix A, we deduce the local stability of solutions for classical FD schemes (8)–(12). Thus, it is necessary to introduce the following Discrete Gronwall Lemma (see [36]).

Lemma 3 (Discrete Gronwall Lemma). If {an}, {bn}, and {cn} are three positive sequences, and {cn} is monotone, and they satisfy

an + bn � cn + λn−1∑i=0

ai (λ > 0); a0 + b0 � c0,

then

an + bn � cn exp(nλ) n = 0, 1, 2, . . . .


Fig. 9. The left and right charts shows the classical FD solutions and the reduced-order FD solutions for the dam-break flow and the sediment concentration at t = 5 s, respectively. (For interpretation of the colors in this figure, the reader is referred to the web version of this article.)

The proof of the stability of the solutions for classical FD schemes (8)–(12) is as follows.If γ�t/�x2 � 1/4 and γ�t/�y2 � 1/4 as well as 4�t(|u| + |v|) � min{γ, ε, A}, which implies that

4�t(‖u‖∞ + ‖v‖∞) � min{γ, ε, A}, then by (8), we have

|Zn+1j,k | �

(1 − 2γ�t

�x2 − 2γ�t

�y2

)|Zn

j,k| +γ�t

�x2 (|Znj+1,k| + |Zn

j−1,k|) + γ�t

�y2 (|Znj,k+1| + |Zn

j,k−1|)

+ �t

�x(|un

j+ 12 ,k

| · |Znj+ 1

2 ,k| + |un

j− 12 ,k

| · |Znj− 1

2 ,k|) + �t

�y(|vnj,k+ 1

2| · |Zn

j,k+ 12| + |vnj,k− 1

2| · |Zn

j,k− 12|)

�(

1 + 2�t

�x‖u‖∞ + 2�t

�y‖v‖∞

)‖Zn‖∞ �

(1 + γ

2�x+ γ

2�y

)‖Zn‖∞, (22)

where ‖ · ‖∞ is the norm in L∞(Ω). Thus, from (22), we obtain

‖Zn+1‖∞ �(

1 + γ + γ)‖Zn‖∞, n = 0, 1, 2, · · · . (23)
2�x 2�y


Fig. 10. The left and right charts show the classical FD solutions and the reduced-order FD solutions for the dam-break flow velocity u and v at t = 5 s, respectively. (For interpretation of the colors in this figure, the reader is referred to the web version of this article.)

By summing (23) from 0 to n, we obtain

‖Zn+1‖∞ � ‖Z0‖∞ +(

γ

2�x+ γ

2�y

) n∑j=0

‖Zn‖∞, n = 0, 1, 2, · · · , N. (24)

By applying Discrete Gronwall Lemma 3 to (24), we obtain

‖Zn+1‖∞ � ‖Z0‖∞ exp(

nγ

2�x+ nγ

2�y

), n = 0, 1, 2, · · · , N, (25)

which shows that the series {Zn+1} is locally stable when the time interval [0, T ] is finite. Furthermore, it

is convergent from the stability theories of FD schemes (see [12] or [23]). The water depth is positive, so there are two positive constants β1 and β2 such that

β1 � ‖Zn‖∞ � β2, n = 0, 1, 2, · · · , N. (26)

If 4A�t � min{�x2, �y2} and 4ε�t � min{�x2, �y2}, then by using the same technique employed to prove (23), from (9)–(12) and by (26), we obtain


‖un+1‖∞ �(

1 + A

2�x+ A

2�y

)‖un‖∞ + 2g�t

�x‖Zn‖∞ + 2g�t

�x‖znb ‖∞ + �t|f |‖vn‖∞

+ CDA

4β1(‖un‖∞ + ‖vn‖∞), n = 0, 1, 2, · · · , N, (27)

‖vn+1‖∞ �(

1 + A

2�x+ A

2�y

)‖vn‖∞ + 2g�t

�y‖Zn‖∞ + 2g�t

�y‖znb ‖∞ + �t|f |‖un‖∞

+ CDA

4β1(‖un‖∞ + ‖vn‖∞), n = 0, 1, 2, · · · , N, (28)

‖Sn+1‖∞ �(

1 + ε

2�x+ ε

2�y

)‖Sn‖∞ + αω�t

β1(‖Sn‖∞ + ‖S∗n‖∞), n = 0, 1, 2, · · · , N, (29)

‖zn+1b ‖∞ � 2�t‖gnb ‖∞

(‖un‖∞�x

+ ‖vn‖∞�y

)+ αω�t

ρ(‖Sn‖∞ + ‖S∗n‖∞), n = 0, 1, 2, · · · , N. (30)

Note that ‖S∗n‖∞ � K[(‖un‖2∞ + ‖vn‖2

∞)3/2/(gωβ1)]m � K[(A/�t)2m/(gωβ1)m](‖un‖∞ + ‖vn‖∞) and ‖gnb ‖∞ � Γβp

2dq(‖un‖2

∞ + ‖vn‖2∞)3/2 � Γβp

2dq(A/�t)3. Put � = max{K[(A/�t)2m/(gωβ1)m]αω�t/

(β1 +ρ) +A/(2�x+2�y) +�t|f | +2CDA/(4β1) +2Γβp2d

qA3/(�x�t2), 2g�t/(�x+�y) +K[(A/�t)2m/

(gωβ1)m]αω�t/(β1+ρ), A/(2�x+�t|f | +2�y) +2Γβp2d

qA3/(�y�t2) +2CDA/(4β1), ε/(2�x) +ε/(2�y) +αω�t/(β1 + ρ)}. By (27)–(30), we obtain

‖un+1‖∞ + ‖vn+1‖∞ + ‖Sn+1‖∞ + ‖zn+1b ‖∞ � (1 + �)(‖un‖∞ + ‖vn‖∞ + ‖Sn‖∞ + ‖znb ‖∞)

+(

2g�t

�x+ 2g�t

�y

)‖Z0‖∞ exp

(nγ

2�x+ nγ

2�y

), n = 0, 1, 2, · · · , N. (31)

By summing (31) from 0 to n and using Discrete Gronwall Lemma 3, we obtain

‖un+1‖∞ + ‖vn+1‖∞ + ‖Sn+1‖∞ + ‖zn+1b ‖∞ � (‖u0‖∞ + ‖v0‖∞ + ‖S0‖∞ + ‖z0

b‖∞) exp(n�)

+(

2gn�t

�x+ 2gn�t

�y

)‖Z0‖∞ exp

(nγ

2�x+ nγ

2�y

)exp(n�), n = 0, 1, 2, · · · , N. (32)

When the time interval [0, T ] is finite, the right-hand side of (32) is bounded. Thus, from the stability theories of FD schemes (see [12] or [23]) and (32), we conclude that the sequence {un, vn, Sn, znb } is locally stable.

Appendix B

The proof of Theorem 1 is as follows.By (20), we may write the POD-based reduced-order FD extrapolating model (18) and (19) in the

following vector form

u∗nm = ΦuΦT

uunm,v∗n

m = ΦvΦTv v

nm,S∗n

m = ΦSΦTSS

nm,Z∗n

m = ΦZΦTZZ

nm, z∗n

b,m = ΦzbΦTzbznb,m,

n = 1, 2, . . . , L; (33)(u∗n+1m ,v∗n+1

m ,S∗n+1m ,Z∗n+1

m , z∗n+1b,m

)T

=(u∗nm ,v∗n

m ,S∗nm ,Z∗n

m , z∗nb,m

)T + F (u∗nm ,v∗n

m ,S∗nm ,Z∗n

m , z∗nb,m),

L � n � N − 1, (34)

which has the same stability conditions as those in (16), i.e., (8)–(12). Put en = (unm, vn

m, pnm)T −

(u∗nm , v∗n

m , p∗nm )T . By (15) and Eq. (33), we have


‖en‖2 = ‖(unm,vn

m,Snm,Zn

m, znb,m)T − (ΦuΦT

uunm,ΦvΦT

v vnm,ΦSΦT

SSnm,ΦZΦT

ZZnm,ΦzbΦT

zbznb,m)T ‖2

�√

λu(Mu+1) +√

λv(Mv+1) +√λS(MS+1) +

√λZ(MZ+1) +

√λzb(Mzb

+1), n = 1, 2, . . . , L. (35)

By (16) and (34), as well as (35), we obtain

‖en‖2 � ‖en−1‖2 + δ‖en−1‖2 = (1 + δ)‖en−1‖2 � . . . � (1 + δ)n−L‖eL‖2

� (1 + δ)n−L[√

λu(Mu+1) +√λv(Mv+1) +

√λS(MS+1) +

√λZ(MZ+1) +

√λzb(Mzb

+1)

],

n = L + 1, . . . , N, (36)

where δ = �tmax{γ,A, ε}/min{Δx2,Δy2} and δ � 1/4 under the stability conditions (8)–(12) and (18)–(20). Synthesizing the discussion above yields the result of Theorem 1.

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