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Noname manuscript No.(will be inserted by the editor)

New Multi-implicit space-time spectral element methods1

for advection-diffusion-reaction problems2

Chaoxu Pei · Mark Sussman ·3

M. Yousuff Hussaini4

5

Received: date / Accepted: date6

Abstract Novel multi-implicit space-time spectral element methods are de-7

scribed for approximating solutions to advection-diffusion-reaction problems8

characterized by multiple time scales. The new methods are spectrally accurate9

in space and time and they are designed to be easy to implement and robust.10

In other words, given an existing stable low order operator split method for11

approximating solutions to PDEs exhibiting multiple scales, the algorithms12

described in this article enable one to easily extend a low order method to be13

a robust space-time spectrally accurate method. In space, two spectrally accu-14

rate advective flux reconstructions are proposed: extended element-wise flux15

reconstruction and non-extended element-wise flux reconstruction. In time, for16

the Hyperbolic term(s), a low-order explicit I-stable building block time inte-17

gration scheme is introduced in order to obtain a stable and efficient building18

block for the spectrally accurate space-time scheme. In this article, multiple19

spectrally accurate space discretization strategies, and multiple spectrally ac-20

curate time discretization strategies are compared to one another. It is found21

that all methods described are spectrally accurate with each method having22

distinguishing properties.23

Keywords Space-time · Operator splitting · Coupling strategy · Multiple24

time scales · Spectral accuracy25

Mathematics Subject Classification (2000) 65B05 · 65M7026

C. PeiDepartment of Mathematics, Florida State University, Tallahassee, FL, 32306, USA.E-mail: [email protected]

M. Sussman?

Department of Mathematics, Florida State University, Tallahassee, FL, 32306, USA.E-mail: [email protected]

M. Yousuff HussainiDepartment of Mathematics, Florida State University, Tallahassee, FL, 32306, USA.E-mail: [email protected]

2 Chaoxu Pei et al.

1 Introduction1

In this article, novel space-time spectrally accurate numerical methods are2

presented for approximating solutions to time dependent partial differential3

equations exhibiting multiple temporal scales. Our new methods are distinct4

in that they are demonstrated to be space-time spectrally accurate, easy to5

implement, and result in methods which are robust to varying initial and6

boundary conditions and varying stiffness of source terms.7

8

Some applications in which one solves a multiple time scales partial differ-9

ential equation are combustion [6] and the transport of air pollutants [23].10

11

Many of the existing numerical methods that approximate solutions to mul-12

tiple time scales problems can be classified as either “divide and conquer”[37,13

8,38,18,24,7,20,4,6,31,32,30,19,2,28,14,15,25] or “monolithic”[21,39,36,35,14

34,33,11].15

Consider a time dependent partial differential equation of the following16

form:17

∂w

∂t= F1(w) + F2(w) + . . .+ Fm(w). (1)

A “Monolithic” method will simultaneously integrate in time all the terms18

in (1). These “Monolithic” methods [21,39,36,35,34,33,11] are space-time19

spectrally accurate and there are no splitting errors, but the performance of a20

“monolithic” method relies on the resulting large sparse system of equations21

not being ill-conditioned. For many time dependent partial differential equa-22

tions, there is no guarantee that a “monolithic method” will not break down23

(the sparse system solver will not diverge).24

25

A “divide and conquer” method sequentially integrates in time each force26

term in (1),27

∂w

∂t= Fj(w), j = 1, . . . ,m, (2)

where distinct (e.g. explicit vs implicit) time integration schemes are imple-28

mented for each distinct force term Fj , j = 1, . . . ,m. For the “divide and29

conquer” algorithm, the practitioner can pick and choose the optimal time in-30

tegration approach for a given force term Fj . “Divide and conquer” methods31

overcome the need to solve large sparse, ill-conditioned matrix systems, but32

these methods are less accurate than “monolithic” approaches. The low order33

“Divide and conquer” methods will incur splitting errors.34

35

A semi implicit spectral deferred correction (SISDC) method [30,19,2] or36

a multi-implicit spectral deferred correction (MISDC) method [6,31,32] is a37

“divide and conquer” method that is derived from a given low order “building38

block” operator split method. The (SISDC) or (MISDC) methods enable one39

Multi-implicit space-time spectral element method 3

to derive a new operator split temporal integration scheme of arbitrary order1

from a low order operator split method. In contrast to “monolithic” methods,2

(SISDC) and (MISDC) methods do not necessitate that one solves a possi-3

bly ill-conditioned sparse system of equations. It is reported in the (MISDC)4

literature [31,32], that as one increases the temporal order of accuracy, the5

splitting errors are decreased.6

7

Inspired by the work in [31,32], we have developed novel multi-implicit8

space-time spectral element methods for solving problems associated with two9

or more processes with differing characteristic time scales. The key distinction10

between the present work and [31,32] is that not only can one arbitrarily select11

the temporal order of accuracy with our new methods, but also one can select12

an arbitrarily high spatial order of accuracy. In this article we demonstrate13

spectral convergence for increasing temporal and spatial order of accuracy.14

We have developed a space-time spectrally accurate “divide and conquer”15

method which has the advantages of both “monolithic” and “divide and con-16

quer” methods (no splitting errors, high order, robust, and easy to implement),17

without having any of the disadvantages (splitting errors, ill-conditioned sparse18

systems, difficult to implement).19

2 Problem formulation20

We consider the following problem that exhibits multiple time scales

wt +∇ · (uw) = ∇ · (D∇w) + fr(w), x ∈ Ω ⊂ R2, (3)

where u = (u, v)> is the velocity vector, D is a symmetric matrix of diffusion21

coefficients, and fr(w) is the reaction term which could be stiff.22

3 Overview of our new space-time spectral element method for the23

advection-diffusion-reaction problem24

For the spatial discretization of ∇ · (uw) found in (3), we have developed25

two spectrally accurate methods for the discretization of the advective fluxes:26

(i) extended element-wise flux reconstruction (EEFR) and (ii) non-extended27

element-wise flux reconstruction (NEEFR). In the extended element-wise flux28

reconstruction, the Extended Gauss points, constructed by adding 2 points29

from the neighbors to the existing Gauss points in one spatial element (see30

Fig. 5), become the interpolation stencil. For the non-extended element-wise31

flux reconstruction algorithm, only the existing Gauss points in one spatial32

element are employed as the interpolation stencil. The idea of reconstructing33

fluxes with an extended stencil has been reported in an article by Dumbser et34

al. [10]. Dumbser et al.’s method was called the discontinuous Galerkin PnPm35

scheme (m ≥ n), where Pn represents a DG solution as a piecewise polynomial36

of degree of n, and Pm is a reconstructed polynomial solution of degree of m37

4 Chaoxu Pei et al.

that is used to compute the fluxes. Later, Luo et al. [29] proposed an improved1

scheme that is referred to as the reconstructed discontinuous Galerkin method.2

Distinct from the previous work in [10,29] that involve all information from the3

neighboring elements, our extended element-wise flux reconstruction scheme4

includes only one point from its neighbors, and thus it is compact and simple.5

6

In order for the overall (MISDC) temporal integration method to be spec-7

trally accurate, it is necessary that the building block temporal integration8

scheme(s) be at least first order accurate. In our spectrally accurate method,9

the diffusion and reaction source terms in (3) are discretized in time with10

the first order implicit backwards Euler method for the low order building11

block. A first order, explicit, I-stable building block time integration scheme12

is introduced as an explicit building block for the advection term in order13

to obtain a stable, accurate and efficient space-time scheme. The first order14

I-stable building block time integration scheme is a two-stage scheme. High or-15

der I-stable time integration scheme can be found in Bao and Jin [3], e.g., 3rd16

order Runge-Kutta method and 4th order Runge-Kutta method. We also refer17

the reader to [12,13] and the references therein for high order strong stability18

preserving time discretization methods which would also be good candidates19

for the advective temporal building block. The key principle of a spectral de-20

ferred correction (SDC) method is to construct an arbitrary high-order time21

scheme from a low-order time scheme by a series of deferred correction steps.22

In the present multi-implicit spectral deferred correction coupling method, the23

solutions obtained by the temporally low order operator splitting method are24

used as provisional solutions, and then such provisional solutions are corrected25

by a series of deferred correction steps. Compared to standard low order oper-26

ator splitting methods, the splitting error in the MISDC method is eliminated27

by exploiting an iterative coupling strategy in the deferred correction proce-28

dure. Two different spectrally accurate space-time coupling strategies for the29

MISDC time integration scheme are investigated. Also since the low order I-30

stable building block time integration scheme is a two-stage scheme, this gives31

rise to two more combinations of correction strategies.32

33

In all, there are a total of eight combinations of spectrally accurate space-34

time discretization methods that we have developed and analyzed in this ar-35

ticle. All of the methods are space-time spectrally accurate, but each method36

has advantages and disadvantages. For example, the EEFR method is more37

accurate than the NEEFR method, but the EEFR method is not extendable38

to unstructured grids. The (MISDC) correction strategy in (24) through (26)39

is expected to be more accurate than the correction strategy in (27a) through40

(27c), but more “intrusive.”41


4 Space-time discretization1

In a space-time discretization, we introduce a space-time domain E as E =2

Ω × [t0, T ]. A point in the space-time domain, x ∈ E , has coordinates (x, t).3

First, we partition the time interval [t0, T ] uniformly by the time levels 0 =4

t0 < t1 < ... < tE(t)

= T . The space-time domain E is then divided into5

E(t) space-time slabs. The n-th space-time slab is denoted as En = E ∩ In,6

where In = [tn, tn+1] is the n-th time interval with length ∆t = tn+1 − tn.7

Next, we divide the spatial domain Ω into E(x)×E(y) non-overlapping spatial8

elements. Let Ωne and Ωn+1e be the spatial element e at time level tn and tn+1,9

respectively. A space-time element Kne is then obtained by connecting Ωne and10

Ωn+1e . The tessellation of the space-time domain is denoted as Th.11

In each space-time slab En, the time interval In = [tn, tn+1] is divided12

into p(t) subintervals by choosing Legendre Gauss-Lobatto points tnm for m =13

0, ..., p(t), that is, tn = tn0 < tn1 < ... < tnp(t)

= tn+1. In the following, the14

length ∆t (∆t = tn+1 − tn) is referred to as a time step size while ∆tm15

(∆tm = tnm+1−tnm) is referred to as a time substep size. Fig. 1 is an illustration16

of a space-time slab.

-

6

x

y

t

tn0 = tn

tn1

tn2

tn3

tn4 = tn+1

∆t1 = tn2 − tn1

6

?

6

?

∆t = tn+1 − tnEn

Ωne

Ωn+1e

W

Fig. 1 An illustration of a space-time slab En. The time interval [tn, tn+1] is divided into 4subintervals by choosing Legendre Gauss-Lobatto points tnm for m = 0, ..., 4. ∆t = tn+1− tnis referred to as a time step size while ∆tm = tnm+1 − tnm is referred to as a time substepsize. The small rectangle cube is the space-time element Kn

e constructed by connecting Ωne

and Ωn+1e .

17

6 Chaoxu Pei et al.

4.1 Basis functions1

A discontinuous spectral element method in the collocation form [17,1] is used2

as the spatial discretization in a staggered grid. The approximate solution lo-3

cated at the Gauss-Gauss nodes, and the velocity vector defined on the marker-4

and-cell (MAC) grid. The MAC grid consist of Gauss-Lobatto–Gauss points5

for the velocity component u in the x-direction and Gauss–Gauss-Lobatto6

points for the velocity component v in the y-direction. The locations of the7

Gauss-Gauss points and the MAC grid points are illustrated in Fig. 2.

Fig. 2 An illustration of the Gauss-Gauss nodes and the MAC grid nodes in the spatialelement Ωe. Closed circles: Gauss-Gauss points for the approximate solution w. The velocityu = (u, v)> is on the MAC grid nodes, that is, open square: Gauss-Lobatto–Gauss points forvelocity component u, closed square: Gauss–Gauss-Lobatto points for velocity componentv.

8

The basis functions are Lagrange interpolation polynomials defined as fol-lows,

`gi (s) =

r∏k=0k 6=i

s− sgksgi − s

gk

, `gli (s) =

r+1∏k=0k 6=i

s− sglksgli − s

glk

, (4)

where sgi i=0,...,r are the roots of the (r + 1)-th order Legendre Gauss poly-

nomials, and sgli i=0,...,r+1 are the roots of the (r + 2)-th order LegendreGauss-Lobatto polynomials. Then, the discrete solution and the discrete ve-


locity vector in the space-time slab En are defined as follows,

wh(x, t)|Ωe =

p(x),p(y),p(t)∑i,j,k=0

wi,j,kφgi (x)φgj (y)φglk (t), (5a)

uh(x, t)|Ωe =


ui,j,kφgli (x)φgj (y)φglk (t), (5b)

vh(x, t)|Ωe =


vi,j,kφgi (x)φglj (y)φglk (t), (5c)

where

φgi (x) = `gi (s), with x = xa +∆x(sg + 1)/2, (6a)

φgli (x) = `gli (s), with x = xa +∆x(sgl + 1)/2, (6b)

φgj (y) = `gj (s), with y = ya +∆y(sg + 1)/2, (6c)

φglj (y) = `glj (s), with y = ya +∆y(sgl + 1)/2, (6d)

φglk (t) = `glk (s), with t = tn +∆t(sgl + 1)/2. (6e)

Here xa (or ya) denotes the lower bound of the spatial element Ωe in the x-1

direction (or y-direction), and ∆x (or ∆y) is the length of the spatial element2

Ωe in the x-direction (or y-direction).3

5 Multi-implicit space-time spectral element method4

Multi-implicit space-time spectral element methods are proposed for approx-5

imating the solutions of PDEs that have multiple source terms and multiple6

time scales, i.e., advection-diffusion-reaction problems. In this section, two new7

spectrally accurate spatial discretization algorithms and accompanying two8

different coupling strategies in the multi-implicit spectral deferred correction9

method are developed. Moreover, the efficiency of two correction strategies for10

integrating the hyperbolic force term are discussed.11

5.1 Discontinuous spectral element method in space12

We use a discontinuous spectral element method in the collocation form [17,1]13

for the spatial discretization. First, we describe the spatial discretization of the14

viscous term ∇· (D∇w) in the model equation (3) by two steps: discretization15

of the gradient operator and discretization of the divergence operator.16

– Discretization of the gradient operator. In each spatial element, the ap-17

proximate solution is located at the Gauss-Gauss points (see Fig. 2). The18

x derivative of the gradient, ∂∂x , is located at the Gauss-Lobatto–Gauss19

8 Chaoxu Pei et al.

points, and the y derivative of the gradient, ∂∂y , is located at the Gauss–1

Gauss-Lobatto points (see the MAC grid points in Fig. 2). Since the pro-2

cedures of computing the x derivative and the y derivative are the same,3

we describe the approximation of the x derivative as an example. We en-4

hance the approximate solution stencil in a spatial element, which are5

referred to as Extended-Gauss points, by adding 2 points to the existing6

p(x) Gauss points (see the Extended-Gauss points in Fig. 3). The solution7

values are then interpolated from the set of Extended-Gauss points onto8

the Gauss-Lobatto points of (p(x) + 2) that has the same number points of9

Extended-Gauss points. The gradient values on the Gauss-Lobatto points10

of (p(x) + 1) are obtained by differentiating the solution interpolated from11

the (p(x) + 2) Gauss-Lobatto points and then evaluating at the (p(x) + 1)12

Gauss-Lobatto points. An illustration of nodes used in computing the x13

derivative is displayed in Fig. 3. For an element adjacent to the domain14

boundary, the set of Extended-Gauss points are constructed by adding the15

point on the domain boundary and the point from the neighbor element16

to enhance the stencil, see Fig. 4.

Fig. 3 An illustration of nodes used in computing the x derivative in the x-direction.Extended-Gauss points (p(x) + 2) denotes the Gauss points together with two additionalpoints from the neighbors, Gauss-Lobatto points (p(x)+2) denotes the Gauss-Lobatto pointswith the same number points of Extended-Gauss, and Gauss-Lobatto points (p(x) + 1) de-notes the MAC grid points in the x-direction.

17

– Discretization of the divergence operator. In each spatial element, the diver-18

gence operator is approximated at the Gauss-Gauss points which coincides19

with the location of the approximate solution (see Fig. 2). Due to the dis-20

continuity of the gradient values at inter-element boundaries, the viscous21

flux is double-valued across element boundaries. In order to define a unique22

flux at inter-element boundaries, we replace the double valued flux with23

the average of the coincident gradient values coming from each side of the24

inter-element face. The product of the continuous gradient values ∇w and25

the viscous coefficients D on the MAC grid points are used to compute26


Fig. 4 An illustration of nodes used in computing the x derivative in the x-direction for theelement near the wall. Extended-Gauss points (p(x) + 2) denotes the Gauss points plus thepoint on the wall and a point from the neighbor, Gauss-Lobatto points (p(x)+2) denotes theGauss-Lobatto points with the same number points of Extended-Gauss, and Gauss-Lobattopoints (p(x) + 1) denotes the MAC grid points in the x-direction.

the differentiation. The values of the divergence operator are obtained by1

differentiating the product on the MAC grid points and then evaluating at2

the Gauss-Gauss points.3

For the spatial discretization of the advective term ∇ · (uw), we present4

two approaches for reconstructing the advective flux: extended element-wise5

flux reconstruction and non-extended element-wise flux reconstruction. These6

two approaches for computing the x derivative are illustrated in Fig. 5 and are7

explained in details as follows.8

– Extended element-wise flux reconstruction (EEFR). To reconstruct the ad-9

vective flux on the MAC grid points, the values of the approximate solution10

need to be interpolated from the Gauss-Gauss points onto the MAC grid11

points. Take the x-component for example. We enhance the approximate12

solution stencil in a spatial element, which are referred to as Extended-13

Gauss points, by adding 2 points to the existing p(x) Gauss points (see the14

Extended-Gauss points in Fig. 5). The solution values are then interpolated15

from the set of Extended-Gauss points of (p(x) +2) onto the Gauss-Lobatto16

points of (p(x) + 1). Due to the discontinuity of the solution values, the17

advective flux values are double-valued across the inter-element faces. The18

endpoints are then overwritten by either the average of the solution values19

on both sides of the inter-element faces or the upwind values at the inter-20

element faces. The values of the derivative are obtained by differentiating21

the solution values on the (p(x) + 1) Gauss-Lobatto points and then evalu-22

ating at the p(x) Gauss points. Since the method uses the Extended-Gauss23

points, it is referred to as extended element-wise flux reconstruction.24

– Non-extended element-wise flux reconstruction (NEEFR). In each spatial25

element, the values of the approximate solution on the Gauss-Gauss points26

are interpolated directly onto the MAC grid points. The endpoints, where27

the flux values are double-valued, are then overwritten by either the average28

10 Chaoxu Pei et al.

of the solution values on either side of the inter-element faces or the upwind1

values at the inter-element faces. The values of the derivatives are obtained2

by differentiating the solution values on the MAC grid points and then3

evaluating at the Gauss-Gauss points (see Fig. 5). As it uses the Gauss4

points in one spatial element, this method is referred to as non-extended5

element-wise flux reconstruction.6

Fig. 5 An illustration of two advective flux reconstructions, extended element-wise flux re-construction and non-extended element-wise flux reconstruction, of computing the x deriva-tive in the x-direction. Extended-Gauss points (p(x) + 2) denotes the Gauss points plus twoadditional points from the neighbors, Gauss-Lobatto points (p(x) + 2) denotes the Gauss-Lobatto points with the same number points of Extended-Gauss, and Gauss-Lobatto points(p(x) + 1) denotes the MAC grid points in the x-direction.

To obtain the space-time spectral element discretization, we substitute theapproximate solution wh in Eq. (5a) and the velocity vector uh = (uh, vh)>

in Eqs. (5b) and (5c) into the model equation (3), and then impose zero resid-ual at Gauss-Gauss points. After applying spectral element discretizations inspace, we obtain a system of ODEs as follows

p(t)∑k=0

wi,j,k(φglk )′(t) = (wi,j)t = FA(wi,j , t) + FD(wi,j , t) + FR(wi,j , t), (7)

i = 0, ..., p(x), j = 0, ..., p(y),

where FA(wi,j , t) is defined as follows for the extended element-wise flux re-construction,

FA(wi,j , t) =−p(t)∑k=0

p(x)+1∑ii=0

uii,j,k2wuii,j,k(φglii )

′(xi)φglk (t)

−p(t)∑k=0

p(y)+1∑jj=0

vi,jj,k2wvi,jj,k(φgljj)

′(yj)φglk (t). (8)


with

2wuii,j,k =

p(y),p(t)∑j′,k′=0

p(x)+1∑i′=−1

wi′,j′,k′φgi′(xii)φ

gj′(yj)φ

glk′(tk), (9)

2wvi,jj,k =

p(x),p(t)∑i′,k′=0

p(y)+1∑j′=−1

wi′,j′,k′φgi′(xi)φ

gj′(yjj)φ

glk′(tk), (10)

or for the non-extended element-wise flux reconstruction,

FA(wi,j , t) =−p(t)∑k=0

p(x)+1∑ii=0

uii,j,k1wuii,j,k(φglii )

′(xi)φglk (t)

−p(t)∑k=0

p(y)+1∑jj=0

vi,jj,k1wvi,jj,k(φgljj)

′(yj)φglk (t). (11)

with

1wuii,j,k =

p(x),p(y),p(t)∑i′,j′,k′=0


gj′(yj)φ

glk′(tk), (12)

1wvi,jj,k =

p(x),p(y),p(t)∑i′,j′,k′=0


gj′(yjj)φ

glk′(tk), (13)

FD(wi,j , t) is defined as follows,

FD(wi,j , t) =

p(t)∑k=0

p(x)+1∑ii=0

(D11)ii,j,k2wxii,j,k(φglii )

′(xi)φglk (t)

+

p(t)∑k=0

p(y)+1∑jj=0

(D22)i,jj,k2wyi,jj,k(φgljj)

′(yj)φglk (t)

+

p(t)∑k=0

p(x)+1∑ii=0

(D12)ii,j,k2wyii,j,k(φglii )

′(xi)φglk (t)

+

p(t)∑k=0

p(y)+1∑jj=0

(D21)i,jj,k2wxi,jj,k(φgljj)

′(yj)φglk (t), (14)

and FR(wi,j , t) is defined by

FR(wi,j , t) = fr(

p(t)∑k=0

wi,j,kφglk (t)). (15)


Note that the terms 2wx and 2wy in Eq. (5.1) are computed as follows,

2wx =

p(y),p(t)∑j,k=0

p(x)+2∑ii=0

2wxii,j,k(φglii )′(x)φgj (y)φglk (t), (16)

2wy =

p(x),p(t)∑i,k=0

p(y)+2∑jj=0

2wyi,jj,kφgi (x)(φgljj)

′(y)φglk (t), (17)

with

2wxii,j,k =

p(x)+1∑i′=−1

p(y),p(t)∑j′,k′=0


gj′(yj)φ

glk′(tk), (18)

2wyi,jj,k =

p(y)+1∑j′=−1

p(x),p(t)∑i′,k′=0


gj′′(yjj)φ

glk′(tk). (19)

The points xip(x)

i=0 , yjp(y)

j=0 are sets of Gauss points; xiip(x)+1i=0 , yjjp

(y)+1j=0 ,1

and tkp(t)

k=0 are sets of Gauss-Lobatto points; xi′p(x)+1i′=−1 and yj′p

(y)+1j′=−1 are2

sets of Extended-Gauss points described in Fig. 3. Note that the flux values at3

the endpoints in each spatial element for both the hyperbolic term and viscous4

term are defined as follows.5

– The EEFR. The values at the endpoints in each spatial element, 2wu0,j,k,6

2wup(x)+1,j,k

, 2wvi,0,k, and 2wvi,p(y)+1,k

, are overwritten by either the average7

values or the upwind values at the inter-element faces.8

– The NEEFR. The values at the endpoints in each spatial element, 1wu0,j,k,9

1wup(x)+1,j,k

, 1wvi,0,k, and 1wvi,p(y)+1,k

, are overwritten by the average values10

on either side of the inter-element faces for conservation laws (see Table 1),11

and the endpoints are overwritten by either the average values or the up-12

wind values at the inter-element faces for parabolic problems.13

– Viscous flux. The values of 2wx and 2wy at the endpoints in each spatial14

element are overwritten by the average values on either side of the inter-15

element faces.16

Remark 1 Huynh [16] reported on the stability of a number of spectral ele-17

ment approaches for solving the conservation laws (see section V I in [16]), in18

which the scheme of NEEFR-upwind with the Chebyshev-Gauss points and19

the Chebyshev-Gauss-Lobatto points defined on a staggered grid is found to20

be mildly unstable. We have found that the scheme of NEEFR-upwind with21

the Legendre-Gauss points and the Legendre-Gauss-Lobatto points defined on22

a staggered grid is also mildly unstable for solving a linear advection equation23

(see section 6), which is consistent with the results reported by Huynh [16].24

We list the stability of several numerical schemes for solving conservation laws25

in Table 1.26


Table 1 The stability of numerical schemes for solving the conservation laws.

Numerical method Stability Nodes Extended stencilEEFR-average yes Legendre yesEEFR-upwind yes Legendre yes

NEEFR-average yes Legendre noNEEFR-upwind no Legendre/Chebyshev noDG-upwind [9] yes Legendre/Chebyshev no

Flux reconstruction [16] yes Legendre no

5.2 Multi-implicit spectral deferred correction method1

The MISDC method introduced by Bourlioux et al. [5] is a variant of the classi-2

cal spectral deferred correction (SDC) method. SDC methods are derived from3

representing an evolution equation as an integral in time, and approximating4

this integral by high order quadrature rules. To reduce the integration error,5

a series of correction equations are designed and solved by a low-order time-6

integration scheme. These correction equations can be applied iteratively to7

achieve arbitrary high-order accuracy in time. The choices of correction equa-8

tions and the efficiency of variances of the SDC methods have been discussed9

in great detail by Layton [26,27].10

A key feature of the MISDC method is that it iteratively couples all phys-ical processes together by including the effects of each process during the in-tegration of any particular process. In contrast, traditional operator-splittingmethods ignore the effects of other processes, that is, each process is discretizedin isolation. After the spatial discretization of the advection-diffusion-reactionequation, the resulting ODE system (7) is rewritten as

∂w

∂t= FA(w) + FD(w) + FR(w) = F (w(t), t), (20)

where FA(w) denotes the spatial discretization of the advection term ∇·(uw),FD denotes the spatial discretization of the diffusion term ∇ · (D∇w) and FRdenotes the spatial discretization of the reaction term fr(w). Since the advec-tion, diffusion and reaction processes are on distinct time scales, one would liketo use an explicit treatment for advection while an implicit treatment for bothdiffusion and reaction in order to avoid a stringent time step constraint. Acomplicated system of different processes can be solved in a decoupled fashionby applying an operator splitting, but the overall order of accuracy is restrictedby the splitting error. For example, the solution advanced in one time step by


an operator splitting method can be obtained as follows,

wA(tn+1) = w(tn) +

∫ tn+1

tnFA(wA(τ))dτ, (21)

wD(tn+1) = wA(tn+1) +

∫ tn+1

tnFD(wD(τ))dτ, (22)

w(tn+1) = wD(tn+1) +

∫ tn+1

tnFR(w(τ))dτ. (23)

One can show that the above approximation is O(∆t) globally unlessthe operators associated with FA, FD and FR commute. In comparison, theMISDC method can achieve higher order of accuracy by the iterative cou-pling strategy which reduces both the splitting error and the integration er-ror. Pazner et al. [32] proposed an iterative coupling strategy in the deferredcorrection procedure as follows,

wn+1,k+1A = wn,k+1 +

∫ tn+1

tn[FA(wk+1

A )− FA(wk)]dτ

+

∫ tn+1

tn[FA(wk) + FD(wk) + FR(wk)]dτ, (24)

wn+1,k+1AD = wn,k+1 +

∫ tn+1

tn[FA(wk+1

A )− FA(wk)]dτ

+

∫ tn+1

tn[FA(wk) + FD(wk) + FR(wk)]dτ

+

∫ tn+1

tn[FD(wk+1

AD )− FD(wk)]dτ, (25)

wn+1,k+1 = wn,k+1 +

∫ tn+1

tn[FA(wk+1

A )− FA(wk)]dτ

+

∫ tn+1

tn[FA(wk) + FD(wk) + FR(wk)]dτ

+

∫ tn+1

tn[FD(wk+1

AD )− FD(wk)]dτ

+

∫ tn+1

tn[FR(wk+1)− FR(wk)]dτ. (26)

Overall fourth order of accuracy has been demonstrated by Pazner et al. [32] on1

one-dimensional low Mach number flow. They pointed out that either reducing2

the time step size or increasing the number of iterations per step can achieve3

the fourth-order accuracy. For example, 8 iterations per step was sufficient to4

achieve the fourth-order accuracy in all tests presented in [32].5


5.2.1 Two different coupling strategies1

Based on the idea of iterative coupling, we propose a new multi-implicit SDCcoupling scheme as follows,

wn+1,k+1A = wn,k+1 +

∫ tn+1

tn[FA(wk+1

A )− FA(wkA)]dτ +

∫ tn+1

tn[FA(wk)]dτ,

(27a)

wn+1,k+1AD = wn+1,k+1

A +

∫ tn+1

tn[FD(wk+1

AD )− FD(wkAD)]dτ +

∫ tn+1

tn[FD(wk)]dτ,

(27b)

wn+1,k+1 = wn+1,k+1AD +

∫ tn+1

tn[FR(wk+1)− FR(wk)]dτ +

∫ tn+1

tn[FR(wk)]dτ.

(27c)

In order to construct an arbitrary high-order time integration scheme, thesecond integrals in (27) are evaluated by a higher-order quadrature rule, e.g.,the Gauss quadrature. Let Im+1

m (F (wk)) be a numerical quadrature, e.g., theGauss quadrature, an approximation to∫ tnm+1

tnm

F (wk(τ), τ)dτ. (28)

On the other hand, the first integrals are discretized by a low-order time2

integration scheme, e.g., a first order time integration scheme.3

One might notice that both flux reconstructions, including informationfrom all neighbors, are analogous to high order centered difference schemes [3].Along with the spatial discretization of high-order center differences, Bao andJin [3] applied a fourth order I-stable method in time that allows for a verylarge cell Reynolds number when solving a system of convection-diffusion equa-tions with a small viscosity. The region of absolute stability of their I-stablemethod includes part of the imaginary axis. Motivated by such a temporalapproach, we introduce a first-order I-stable building block time integrationscheme along with either the EEFR or the NEEFR as an explicit treatmentfor the advection process.

w∗i,j = (wi,j)m +∆tmFA((wi,j)m, tnm), (29a)

(wi,j)m+1 = (wi,j)m +∆tmFA((wi,j)∗, tnm+1). (29b)

The region of absolute stability (or simply the stability region) of an I-stable4

method contains part of the imaginary axis. In Fig.6, the stability regions of5

both I-stable scheme (black) and forward Euler (red) are illustrated.6

The discretization of the present multi-implicit space-time spectral element7

method with new coupling strategy is described in Algorithm 1 (indices i and8

j in Eq. (7) are suppressed for simplicity), in which the extended element-wise9


Fig. 6 An illustration of the stability regions of I-stable scheme (black) and forward Euler(red). Both time integration schemes are first order.

flux reconstruction is employed for treating the advection term. The coupling1

strategy applied in Algorithm 1 is referred to as coupling 1.2

In addition, a second coupling strategy, by extending the method of Pazneret al. [32], is introduced as follows

wn+1,k+1A = wn,k+1 +

∫ tn+1

tn[FA(wk+1

A )− FA(wkA)]dτ +

∫ tn+1

tn[FA(wk)]dτ,

(30a)

wn+1,k+1ADR = wn+1,k+1

A +

∫ tn+1

tn[FD(wk+1

ADR)− FD(wk)]dτ

+

∫ tn+1

tn[FD(wk) + FR(wk)]dτ, (30b)

wn+1,k+1 = wn+1,k+1ADR +

∫ tn+1

tn[FR(wk+1)− FR(wk)]dτ. (30c)

The procedure of such a coupling strategy is described in the Algorithm 23

(indices i and j in Eq. (7) are suppressed for simplicity), which is referred to4

as coupling 2. Comparing Algorithm 1 to Algorithm 2, we see that steps 175

and 20 in Algorithm 2 are different from those in Algorithm 1 due to different6

coupling strategies employed.7

With a coupling strategy (coupling 1 or coupling 2) used in a series ofdeferred correction steps, the temporal splitting error is eliminated iteratively.Thus, we conjecture that our multi-implicit space-time spectral element meth-ods, Algorithm 1 and Algorithm 2, have an overall order of accuracy

minp(x) + 1, p(y) + 1, p(t) + 1,K, (31)


Algorithm 1 The multi-implicit space-time spectral element method usingthe extended element-wise reconstruction with I-stable building block scheme– Coupling 1.

1: for m = 0, .., p(t) − 1 do2: FA(w0

m, tnm) = 0.

3: FA(w0,∗, tnm) = 0.4: Im+1

m (FA(w0)) = 0.5: FD((wAD)0m+1, t

nm+1) = 0.

6: Im+1m (FD(w0)) = 0.

7: FR(w0m+1, t

nm+1) = 0.

8: Im+1m (FR(w0)) = 0.

9: end for10: for k = 1, ..,K do11: wk

0 = w(tn).

12: for m = 0, .., p(t) − 1 do13: Compute FA(wk

m, tnm) by Eq. (8). . Advection process (13-16)

14: (wA)k,∗ = wkm +∆tm[FA(wk

m, tnm)− FA(wk−1

m , tnm)] + Im+1m (FA(wk−1)).

15: Compute FA(wk,∗, tnm) by Eq. (8).16: (wA)km+1 = wk

m +∆tm[FA((wA)k,∗, tnm+1)− FA((wA)k−1,∗, tnm+1)]

+Im+1m (FA(wk−1)).

17: (wAD)km+1 = (wA)km+1 +∆tm[FD((wAD)km+1, tnm+1)−

FD((wAD)k−1m+1, t

nm+1)] + Im+1

m (FD(wk−1)).

18: (wAD)km+1 is computed by an iterative method.

19: Compute FD((wAD)km+1, tnm+1) by Eq. (14). . Diffusion process (17-19)

20: wkm+1 = (wAD)km+1 +∆tm[FR(wk

m+1, tnm+1)− FR(wk−1

m+1, tnm+1)]

+Im+1m (FR(wk−1)).

21: wkm+1 is computed by Newton’s method.

22: Compute FR(wkm+1, t

nm+1) by Eq. (15). . Reaction process (20-22)

23: end for24: Compute Im+1

m (FA(wk)), Im+1m (FD(wk)), and Im+1

m (FR(wk)) with the updatedsolutions wk.

25: end for

where K is the number of iterations in the multi-implicit SDC method, p(x),1

p(y) and p(t) is the polynomial order in the x-direction, the y-direction and2

the temporal direction, respectively.3

5.3 Efficiency of the low order building block in the deferred correction4

procedure5

The first-order I-stable building block time integration scheme in Eq. (29) isa two-stage explicit Runge-Kutta method for integrating the advection term.In section 5.2.1, each stage of the first-order I-stable building block time inte-gration scheme is corrected in the deferred correction phase of the space-timealgorithm, and the discretization is illustrated in steps 14 and 16 in both Al-gorithm 1 and Algorithm 2. This correction strategy in Eq. (32) is referredto as the “intrusive correction” because the predictor time integration scheme


Algorithm 2 The multi-implicit space-time spectral element method usingthe extended element-wise reconstruction with I-stable building block schemeand the coupling strategy obtained by extending the method of Pazner etal. [32] – Coupling 2.

1: for m = 0, .., p(t) − 1 do2: FA(w0

m, tnm) = 0.

3: FA(w0,∗, tnm) = 0.4: Im+1

m (FA(w0)) = 0.5: FD((wADR)0m+1, t

nm+1) = 0.

6: Im+1m (FD(w0)) = 0.

7: FR(w0m+1, t

nm+1) = 0.

8: Im+1m (FR(w0)) = 0.

9: end for10: for k = 1, ..,K do11: wk

0 = w(tn).

12: for m = 0, .., p(t) − 1 do13: Compute FA(wk

m, tnm) by Eq. (8). . Advection process (13-16)

14: (wA)k,∗ = wkm +∆tm[FA(wk

m, tnm)− FA(wk−1

m , tnm)] + Im+1m (FA(wk−1)).

15: Compute FA(wk,∗, tnm) by Eq. (8).16: (wA)km+1 = wk

m +∆tm[FA((wA)k,∗, tnm)− FA((wA)k−1,∗, tnm)]

+Im+1m (FA(wk−1)).

17: (wADR)km+1 = (wA)km+1 +∆tm[FD((wADR)km+1, tnm+1)−

FD(wk−1m+1, t

nm+1)] + Im+1

m (FD(wk−1)) + Im+1m (FR(wk−1)).

18: (wADR)km+1 is computed by an iterative method.

19: Compute FD((wADR)km+1, tnm+1) by Eq. (14). . Diffusion process (17-19)

20: wkm+1 = (wADR)km+1 +∆tm[FR(wk

m+1, tnm+1)− FR(wk−1

m+1, tnm+1)].

21: wkm+1 is computed by Newton’s method.

22: Compute FR(wkm+1, t

nm+1) by Eq. (15). . Reaction process (20-22)

23: end for24: Compute Im+1

m (FA(wk)), Im+1m (FD(wk)), and Im+1

m (FR(wk)) with the updatedsolutions wk.

25: end for

cannot be treated as a “black box.”

(wA)k,∗ = wkm +∆tm[FA(wkm, tnm)− FA(wk−1m , tnm)] + Im+1

m (FA(wk−1)),(32a)

(wA)km+1 = wkm +∆tm[FA((wA)k,∗, tnm)− FA((wA)k−1,∗, tnm)] + Im+1m (FA(wk−1)).

(32b)

We have found that an alternate, “non-intrusive,” correction algorithm isjust as accurate as the “intrusive” algorithm and is more memory and cpuefficient. The “non-intrusive” approach simply ignores the correction of thefirst stage in the two stage I-scheme. Such a correction strategy in Eq. (33)is referred to as “non-intrusive correction”. We analyze the “non-intrusive”


method in section 6.

(wA)k,∗ = wkm +∆tm[FA(wkm, tnm)], (33a)

(wA)km+1 = wkm +∆tm[FA((wA)k,∗, tnm)− FA((wA)k−1,∗, tnm)] + Im+1m (FA(wk−1)).

(33b)

6 Numerical tests1

In this section, we test the multi-implicit space-time spectral element methodfor solving problems with multiple time scales in two spatial dimensions. Thenumerical results are compared with the exact solutions in order to examinethe spectral accuracy in both space and time. An explicit time integrationscheme is applied on the advection process, which leads to the following CFLcondition for the time-substep size

max∆tm = C(min∆xn|u|max

+min∆yn|v|max

), (34)

where ∆tm is the time substep size (max∆tm ≤ ∆t), C is a constant that is2

less than 1.0, and min∆xn (or min∆yn) is the smallest distance between3

two Gauss points that is proportional to the inverse of the square of polynomial4

order, i.e., 1/(p(x))2 (or 1/(p(y))2) in the x-direction (or y-direction) [22]. The5

discrete system obtained from the diffusion process is well conditioned and6

can be solved in a very efficient way via the biconjugate gradient stabilized7

method (BiCGSTAB), even without the use of any preconditioner.8

In the following tests, we keep the number of iterations K in the multi-implicit SDC method as K = p(t) unless the value of K is prescribed. In the

last space-time slab EE(t)−1, the errors are measured in the discrete L∞ norms

at time tE(t)

= T

‖Errw‖∞ = maxi=0,...,p(x),

j=0,...,p(y)

‖wE(t)

i,j − (wh)E(t)

i,j ‖∞, (35)

where T is the finial computational time, wE(t)

i,j denotes the exact solution eval-9

uated at the node (xi, yj), and (wh)E(t)

i,j is the approximate solution evaluated10

at the same node.11

6.0.1 Treatments for the advection process12

We consider the following advection problem in a Cartesian domain Ω =[0, 1]× [0, 1]

wt +∇ · (uw) = 0, x ∈ Ω, (36)

with the exact solution

w(x, t) = sin(2π(x− ut)) sin(2π(y − vt)). (37)


Periodic boundary conditions are imposed, and the velocity vector u = (u, v)>1

is set to be (1, 1)>.2

First, we demonstrate the spectral accuracy of our method by plotting the3

maximum errors of the solution as a function of polynomial orders, (p(x), p(y), p(t)) =4

(p, p, p(t)). The computational domain is divided into 4 × 4 and 8 × 8 spatial5

elements, and the final time of the computation is T = 0.5. In Fig. 7, we6

see that both the extended element-wise flux reconstruction (EEFR) and the7

non-extended element-wise flux reconstruction (NEEFR) exhibit spectral ac-8

curacy in space when the number of space-time slab is fixed, i.e., E(t) = 160.9

The errors of all three schemes decrease as the resolution in space increases.10

The results of the NEEFR with the upwind values are not shown because11

the scheme is mildly unstable for solving conservation laws (see Remark 1),12

which will be discussed in detail later. In addition, we compare two correction

1 2 3 4 5 6 7 8 91011

10−14

10−13

10−12

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Poly. Order p

||E

RR

||m

ax

p(t)

=p;EEFR−average;

4 × 4

8 × 8

1 2 3 4 5 6 7 8 9101110

−15

10−14

10−13

10−12

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Poly. Order p

||E

RR

||m

ax

p(t)

=p;EEFR−upwind;

4 × 4

8 × 8

1 2 3 4 5 6 7 8 9101110

−14

10−13

10−12

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Poly. Order p

||E

RR

||m

ax

p(t)

=p;NEEFR−average;

4 × 4

8 × 8

Fig. 7 Errors in the solution as a function of polynomial order in space p for all threeschemes: the EEFR-average, the EEFR-upwind and the NEEFR-average. The simulation iscomputed up to 0.5 with E(t) = 160. The polynomial order in time p(t) is chosen to thesame as the one in space p. The left one is the results obtained by the EEFR-average withtwo different spatial tessellations (4 × 4 and 8 × 8), the middle one is the results obtainedby the EEFR-upwind with two different spatial tessellations, and the right one is the resultsobtained by the NEEFR-average with two different spatial tessellations.

13

strategies, “intrusive correction” and “non-intrusive correction”. In Fig. 8, we14

see that both options have the same error behavior.15

We then compare the performance of all three schemes: the EEFR with the16

average values (EEFR-average), the EEFR with the upwind values (EEFR-17

upwind), and the NEEFR with the average values (NEEFR-average). The18

comparison results are shown in Fig. 9. The schemes with the EEFR (EEFR-19

average and EEFR-upwind) are more accurate than the scheme with the20


1 2 3 4 5 6 7 8 9 101110

−14

10−13

10−12

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Poly. Order p

||E

RR

||m

ax

p(t)

=p; EEFR−average;

Intrusive

Non−intrusive

1 2 3 4 5 6 7 8 9 101110

−14

10−13

10−12

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Poly. Order p

||E

RR

||m

ax

p(t)

=p; EEFR−upwind;

Intrusive

Non−intrusive

1 2 3 4 5 6 7 8 9 101110

−12

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Poly. Order p

||E

RR

||m

ax

p(t)

=p; NEEFR−average;

Intrusive

Non−intrusive

Fig. 8 Errors in the solution as a function of polynomial order in space p for all threeschemes: the EEFR-average, the EEFR-upwind and the NEEFR-average. The simulationis computed up to 0.5 with (E(x) × E(y), E(t)) = (4 × 4, 160). The polynomial order intime p(t) is chosen to the same as the one in space p. The left one is the results obtainedby the EEFR-average with two different correction strategies (“intrusive correction” and“non-intrusive correction”), the middle one is the results obtained by the EEFR-upwindwith two different correction strategies, and the right one is the results obtained by theNEEFR-average with two different correction strategies.

NEEFR (NEEFR-average) when using the same number of degrees of freedom.1

Thus, the schemes with the EEFR as the treatment for the advection process2

are more accurate and efficient than those with the NEEFR. Furthermore,3

there is no much difference in the error behavior between the EEFR-average4

and the EEFR-upwind.5

Next, we examine the performance of our multi-implicit space-time method6

with respect to the number of space-time slab E(t). Fixing the mesh in space7

to be 8 × 8, we test the model equation (36) with E(t) = 80 and E(t) = 160.8

Note that the minimum value of the number of space-time slab is limited9

by the CFL condition because the explicit I-stable time integration scheme10

is employed. In Fig. 10, we see that all three schemes (the EEFR-average,11

the EEFR-upwind and the NEEFR-average) exhibit spectral accuracy in time12

with the polynomial order in space equal to the one in time: p = p(t). In13

addition, the comparison among all three schemes is shown in Fig. 11. In14

addition, two correction strategies, “intrusive correction” and “non-intrusive15

correction”, are compared. The results are shown in Fig. 12. It is clear that16

both correction strategies have the same error behavior.17

Now, we report results which indicate that the NEEFR scheme using the18

upwind values at the endpoints (NEEFR-upwind) in each spatial element is19


2 3 4 5 6 7 8 9 10 1110

−14

10−13

10−12

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Poly. Order p

||E

RR

||m

ax

p(t)

=p; 4 × 4;

EEFR−average

EEFR−upwind

NEEFR−average

2 3 4 5 6 7 8 9 10 1110

−15

10−14

10−13

10−12

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Poly. Order p

||E

RR

||m

ax

p(t)

=p; 8 × 8;

EEFR−average

EEFR−upwind

NEEFR−average

Fig. 9 Errors in the solution as a function of polynomial order in space p for two differentspatial tessellations: 4× 4 and 8× 8. The simulation is computed up to 0.5 with E(t) = 160.The polynomial order in time p(t) is chosen to the same as the one in space p. The left oneis the comparison among the EEFR-average, the EEFR-upwind and the NEEFR-averagewith 4×4 spatial elements, while the right one is the comparison among these three schemeswith 8× 8 spatial elements.

1 2 3 4 5 6 7 8 91011

10−15

10−14

10−13

10−12

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

Poly. Order p(t)

||E

RR

||m

ax

p=p(t)

;EEFR−average;

E(t)

=80

E(t)

=160

1 2 3 4 5 6 7 8 91011

10−15

10−14

10−13

10−12

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

Poly. Order p(t)

||E

RR

||m

ax

p=p(t)

;EEFR−upwind;

E(t)

=80

E(t)

=160

1 2 3 4 5 6 7 8 9101110

−14

10−13

10−12

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Poly. Order p(t)

||E

RR

||m

ax

p=p(t)

;NEEFR−average;

E(t)

=80

E(t)

=160

Fig. 10 Errors in the solution as a function of polynomial order in time p(t) for all threeschemes: the EEFR-average, the EEFR-upwind and the NEEFR-average. The simulation iscomputed up to 0.5 with 8× 8 numbers of spatial elements. The pair of polynomial ordersin space, (p(x), p(y)) = (p, p), is chosen to the same as the one in time p(t). The left oneis the results obtained by the EEFR-average with two different numbers of space-time slab(E(t) = 80 and E(t) = 160), the middle one is the results obtained by the EEFR-upwindwith two different numbers of space-time slab, and the right one is the results obtained bythe NEEFR-average with two different numbers of space-time slab.


2 3 4 5 6 7 8 9 10 1110

−15

10−14

10−13

10−12

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Poly. Order p(t)

||E

RR

||m

ax

p=p(t)

; E(t)

=80;

EEFR−average

EEFR−upwind

NEEFR−average

2 3 4 5 6 7 8 9 10 1110

−15

10−14

10−13

10−12

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Poly. Order p(t)

||E

RR

||m

ax

p=p(t)

; E(t)

=160;

EEFR−average

EEFR−upwind

NEEFR−average

Fig. 11 Errors in the solution as a function of polynomial order in time p(t) for two differentnumbers of space-time slab: E(t) = 80 and E(t) = 160. The simulation is computed up to 0.5with 8× 8 spatial tessellation. The pair of polynomial orders in space, (p(x), p(y)) = (p, p),is chosen to the same as the one in time p(t). The left one is the comparison among theEEFR-average, the EEFR-upwind and the NEEFR-average with E(t) = 80, while the rightone is the comparison among these three schemes with E(t) = 160.

mildly unstable for solving conservation laws. We compare the EEFR-average,1

the EEFR-upwind, the NEEFR-average and the NEEFR-upwind with 8 × 82

spatial tessellation for solving the linear advection equation (36). The compu-3

tation is carried out up to T = 1.0, and the results are shown in Fig. 13. On the4

left panel, four schemes are compared when E(t) = 200. The stable schemes,5

the EEFR-average, the EEFR-upwind and the NEEFR-average, exhibit the6

spectral accuracy: the error decays exponentially fast until reaching 10−147

and staying around there as the polynomial order increases. By contrast, the8

mildly unstable scheme, the NEEFR-upwind, has an error behavior of first9

exponentially decaying until reaching 10−11 at polynomial order p = 9 and10

increasing afterwords. In order to exclude the possibility that the CFL con-11

straint for the NEEFR-upwind is smaller, we also test it with more space-time12

slabs, i.e., E(t) = 500. The result is shown on the right panel of the Fig. 13.13

The error in both cases exhibit the same behavior, that is, it increases after14

reaching 10−11. In addition, two correction strategies, “intrusive correction”15

and “non-intrusive correction”, are compared. The results of four schemes are16

shown in Fig. 14. It is clear that both correction strategies have the same error17

behavior.18


1 2 3 4 5 6 7 8 9 1011

10−15

10−14

10−13

10−12

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

Poly. Order p(t)

||E

RR

||m

ax

p=p(t)

; EEFR−average;

Intrusive

Non−intrusive

1 2 3 4 5 6 7 8 9 1011

10−15

10−14

10−13

10−12

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

Poly. Order p(t)

||E

RR

||m

ax

p=p(t)

; EEFR−upwind;

Intrusive

Non−intrusive

1 2 3 4 5 6 7 8 9 101110

−14

10−13

10−12

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Poly. Order p(t)

||E

RR

||m

ax

p=p(t)

; NEEFR−average;

Intrusive

Non−intrusive

Fig. 12 Errors in the solution as a function of polynomial order in space p for all threeschemes: the EEFR-average, the EEFR-upwind and the NEEFR-average. The simulation iscomputed up to 0.5 with (E(x)×E(y), E(t)) = (8×8, 80). The polynomial order in time p(t)

is chosen to the same as the one in space p. The left one is the results obtained by the EEFR-average with two different correction strategies, the middle one is the results obtained bythe EEFR-upwind with two different correction strategies (“intrusive correction” and “non-intrusive correction”), and the right one is the results obtained by the NEEFR-average withtwo different correction strategies.

6.0.2 Advection-diffusion-reaction problems1

On a computational domain Ω = [0, 1] × [0, 1], we consider the followingadvection-diffusion-reaction problem

wt +∇ · (uw) = ν∆w + λw, (38)

with an exact solution

w(x, t) = 3e−(2(2π)2ν−λ)t sin(2π(x− ut)) sin(2π(y − vt)). (39)

Periodic boundary conditions are imposed, and the velocity vector u = (u, v)>2

is set to be (1, 1)>. We also choose ν = 0.04 and λ = 5.0 in the following tests.3

The performance of the two advective flux reconstructions is compared4

with each coupling strategies, coupling 1 (Algorithm 1) and coupling 2 (Al-5

gorithm 2), on the space-time tessellation, (E(x) × E(y), E(t)) = (5 × 5, 80).6

The final time of the computation is T = 1.0, and the results are shown in7

Fig. 15. The left panel is the comparison made among the EEFR-average, the8

EEFR-upwind, the NEEFR-average and the NEEFR-upwind with coupling9

1 (Algorithm 1), and the one on the right is the comparison made among10


2 3 4 5 6 7 8 9 10 11

10−14

10−13

10−12

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Poly. Order p

||E

RR

||m

ax

p(t)

=p;10 × 10;E(t)

=200;

EEFR−average

EEFR−upwind

NEEFR−average

NEEFR−upwind

2 3 4 5 6 7 8 9 10 1110

−12

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Poly. Order p

||E

RR

||m

ax

p(t)

=p;10 × 10;NEEFR−upwind;

E(t)

=200

E(t)

=500

Fig. 13 Errors in the solution of the advection equation (36) as a function of polynomialorder in space p. The simulation is computed up to 1.0 with 8 × 8 spatial tessellation.The polynomial order in time, p(t), is chosen to the same as the one in space p. The leftone is the comparison among the EEFR-average, the EEFR-upwind, the NEEFR-averageand the NEEFR-upwind when using E(t) = 200. The right one is the comparison of theNEEFR-upwind between different numbers of space-time slab: E(t) = 200 and E(t) = 500.

the EEFR-average, the EEFR-upwind, the NEEFR-average and the NEEFR-1

upwind with coupling 2 (Algorithm 2). Both coupling strategies with each2

treatment for the advective process exhibit spectral accuracy. Since the tol-3

erance is set to be 10−11 in the correction step, the error decreases until it4

reaches 10−10. Clearly, the schemes with the EEFR are more accurate and5

efficient than those with the NEEFR. In the current test, coupling strategies6

with the EEFR-upwind are slightly more accurate than those with the EEFR-7

average, even though their error behavior is indeed similar. In addition, two8

correction strategies, “intrusive correction” and “non-intrusive correction”, are9

compared. The results of four schemes with two different coupling strategies10

are shown in Fig. 16, in which it is shown that both correction strategies have11

the same error behavior.12

Next, we look closely into the performance of schemes with two different13

coupling strategies using the EEFR-upwind as the treatment for the advection14

process. We demonstrate the spectral accuracy of our method by plotting15

the maximum error as a function of the polynomial orders (p(x), p(y), p(t)) =16

(p, p, p(t)). The computational domain is divided into 5×5 and 10×10 spatial17

elements, respectively. The final time is T = 1.0. In Fig. 17, we see that both18

coupling strategies (Algorithm 1 and Algorithm 2) exhibit spectral accuracy in19

space when the number of space-time slab is fixed, i.e., E(t) = 160. The errors20

of both schemes decrease as the resolution in space increases. We also compare21

the performance of the two different coupling strategies, and the comparison22

is shown in Fig. 9. Two different spatial tessellations, 5 × 5 and 10 × 10, are23


2 3 4 5 6 7 8 9 10 11

10−14

10−13

10−12

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Poly. Order p

||E

RR

||m

ax

p(t)

=p; 10 × 10; E(t)

=200; EEFR−average

Intrusive

Non−intrusive

2 3 4 5 6 7 8 9 10 11

10−14

10−13

10−12

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Poly. Order p

||E

RR

||m

ax

p(t)

=p; 10 × 10; E(t)

=200; EEFR−upwind

Intrusive

Non−intrusive

2 3 4 5 6 7 8 9 10 11

10−14

10−13

10−12

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Poly. Order p

||E

RR

||m

ax

p(t)

=p; 10 × 10; E(t)

=200; NEEFR−average

Intrusive

Non−intrusive

2 3 4 5 6 7 8 9 10 11

10−14

10−13

10−12

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Poly. Order p

||E

RR

||m

ax

p(t)

=p; 10 × 10; E(t)

=200; NEEFR−upwind

Fully correction

Semi−correction

Fig. 14 Errors in the solution of the advection equation (36) as a function of polynomialorder in space p. The simulation is computed up to 1.0 with (E(x)×E(y), E(t)) = (8×8, 200).The polynomial order in time, p(t), is chosen to the same as the one in space p. On the toppanel, it is the comparison between two correction strategies (“intrusive correction” and“non-intrusive correction”) along with either the EEFR-average (left) or the EEFR-upwind(right). On the bottom panel, it is the comparison among two correction strategies alongwith either the NEEFR-average (left) or the NEEFR-upwind (right).

tested. We find that there is no much difference in the error behavior of the1

two different coupling strategies in the current test case.2

We then examine the performance of our multi-implicit space-time method3

with respect to the number of space-time slab E(t). With the 10× 10 spatial4

mesh, we test our method with E(t) = 80 and E(t) = 160, respectively. Note5

again that the minimum value of the number of space-time slab is dictated6

by the CFL condition due to the explicit treatment of the advection process.7

In Fig. 19, we see that both coupling strategies exhibit spectral accuracy in8

time with the polynomial order in space equal to the one in time: p = p(t).9

In addition, the comparison between the two different coupling strategies is10

shown in Fig. 20.11

In Tables 2, 3 and 4, we report on the performance of our two differ-ent space-time coupling strategies (Algorithm 1 and Algorithm 2), using theEEFR-upwind as the treatment for the advection process, by varying the spa-tial tessellation E(x)×E(y), the number of space-time slab E(t) and the numberof iterations K. The order of convergence, r, listed in tables are found by

r = log2(‖Errh‖∞‖Errh/2‖∞

). (40)


2 3 4 5 6 7 8 9 10 1110

−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

Poly. Order p

||E

RR

||m

ax

p(t)

=p; Coupling 1;

EEFR−average

EEFR−upwind

NEEFR−average

NEEFR−upwind

2 3 4 5 6 7 8 9 10 1110

−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

Poly. Order p

||E

RR

||m

ax

p(t)

=p; Coupling 2;

EEFR−average

EEFR−upwind

NEEFR−average

NEEFR−upwind

Fig. 15 Errors in the solution as a function of polynomial order in space p. The simulationis computed up to 1.0 with 5×5 spatial tessellation and E(t) = 80. The polynomial order intime, p(t), is chosen to the same as the one in space p. The left part is the comparison madeamong the EEFR-average, the EEFR-upwind, the NEEFR-average and the NEEFR-upwindwith coupling 1 (Algorithm 1), while the one on the right is the comparison made amongthe EEFR-average, the EEFR-upwind, the NEEFR-average and the NEEFR-upwind withcoupling 2 (Algorithm 2). The tolerance is set to be 10−11.

The results reported in Table 2 are in agreement with the analytic formula (31),1

in which K-th order of accuracy is obtained. By comparing the results in Table2

3 with Table 4, we see that either reducing the time step (i.e., increasing the3

number of space-time slab E(t)) or increasing the number of iterations per4

step (i.e., increasing K) can achieve the desired order of accuracy: p(t) + 1. By5

comparing the errors in Tables 3 and 4, the strategy of increasing the number6

of iterations per time step is more accurate than the strategy of reducing the7

time step.8

Now, we investigate the convergence rate in space and in time, separately.9

In order to examine the order of convergence in space, the polynomial order10

in time is set to be higher than the one in space, so that the temporal error11

is negligible. In Table 5, the temporal polynomial order and the number of12

iterations are set to be p(t) = 6 and K = 8. We test different polynomial13

orders in space from p = 4 to p = 7. The results in Table 5 show that p + 114

order of convergence in space is obtained.15

To test the order of convergence in time, the polynomial order in space is16

set to be higher than the one in time, so that the spatial error is negligible.17

The polynomial order in space is chosen to be either p = 8 or p = 9, and then18

combinations of temporal order and the number of iterations are varied. The19

results are shown in Table 6, and the global order of accuracy is observed to20

be minp(t) + 1,K.21


2 3 4 5 6 7 8 9 10 11

10−1410−1310−1210−1110−1010−910−810−710−610−510−410−310−210−1100

Poly. Order p

||ER

R|| m

ax

p(t)=p; EEFR−average; Coupling 1;

IntrusiveNon−intrusive

2 3 4 5 6 7 8 9 10 1110

−1010

−910

−810

−710

−610

−510

−410

−310

−210

−110

0

Poly. Order p

||ER

R|| m

ax

p(t)=p; EEFR−average; Coupling 2;


2 3 4 5 6 7 8 9 10 1110

−1010

−910

−810

−710

−610

−510

−410

−310

−210

−110

0

Poly. Order p

||ER

R|| m

ax

p(t)=p; EEFR−upwind; Coupling 1;


2 3 4 5 6 7 8 9 10 1110

−1010

−910

−810

−710

−610

−510

−410

−310

−210

−110

0

Poly. Order p

||ER

R|| m

ax

p(t)=p; EEFR−upwind; Coupling 2;


2 3 4 5 6 7 8 9 10 11

10−1410−1310−1210−1110−1010−910−810−710−610−510−410−310−210−1100

Poly. Order p

||ER

R|| m

ax

p(t)=p; NEEFR−average; Coupling 1;


2 3 4 5 6 7 8 9 10 1110

−1010

−910

−810

−710

−610

−510

−410

−310

−210

−110

0

Poly. Order p

||ER

R|| m

ax

p(t)=p; NEEFR−average; Coupling 2;


2 3 4 5 6 7 8 9 10 1110

−1010

−910

−810

−710

−610

−510

−410

−310

−210

−110

0

Poly. Order p

||ER

R|| m

ax

p(t)=p; NEEFR−upwind; Coupling 1;


2 3 4 5 6 7 8 9 10 1110

−1010

−910

−810

−710

−610

−510

−410

−310

−210

−110

0

Poly. Order p

||ER

R|| m

ax

p(t)=p; NEEFR−upwind; Coupling 2;


Fig. 16 Errors in the solution as a function of polynomial order in space p. The simulationis computed up to 1.0 with (E(x) ×E(y), E(t)) = (5× 5, 80). The polynomial order in time,p(t), is chosen to the same as the one in space p. The left part is the comparison madebetween two correction strategies (“intrusive correction” and “non-intrusive correction”)along with coupling 1 (Algorithm 1) and one of the four schemes: the EEFR-average, theEEFR-upwind, the NEEFR-average and the NEEFR-upwind; the one on the right is thecomparison made between two correction strategies along with coupling 1 (Algorithm 2)and one of the four schemes: the EEFR-average, the EEFR-upwind, the NEEFR-averageand the NEEFR-upwind. The tolerance is set to be 10−11.


2 3 4 5 6 7 8 9 10 1110

−12

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Poly. Order p

||E

RR

||m

ax

p(t)

=p; Coupling 1;

5 × 5

10 × 10

2 3 4 5 6 7 8 9 10 1110

−12

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Poly. Order p

||E

RR

||m

ax

p(t)

=p; Coupling 2;

5 × 5

10 × 10

Fig. 17 Errors in the solution as a function of polynomial order in space p for two differentcoupling strategies, Algorithm 1 and Algorithm 2, using the EEFR-upwind as the treatmentfor the advection process. The simulation is computed up to 1.0 and E(t) = 160. Thepolynomial order in time p(t) is chosen to the same as the one in space p. The left one is theresults obtained by coupling 1 (Algorithm 1) with two different spatial tessellations (5 × 5and 10 × 10), while the right one is the results obtained by coupling 2 (Algorithm 2) withtwo different spatial tessellations (5× 5 and 10× 10). The tolerance is set to be 10−13.

2 3 4 5 6 7 8 9 10 1110

−12

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Poly. Order p

||E

RR

||m

ax

p(t)

=p; 5 × 5;

Coupling 1

Coupling 2

2 3 4 5 6 7 8 9 10 1110

−12

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Poly. Order p

||E

RR

||m

ax

p(t)

=p; 10 × 10;

Coupling 1

Coupling 2

Fig. 18 Errors in the solution as a function of polynomial order in space p for two differentspatial tessellations: 5×5 and 10×10. The simulation is computed up to 1.0 and E(t) = 160.The polynomial order in time p(t) is chosen to the same as the one in space p. The left one isthe comparison between coupling 1 with the EEFR-upwind and coupling 2 with the EEFR-upwind on the 5× 5 spatial tessellation, while the right one is the comparison between twodifferent coupling strategies on 10×10 spatial tessellation. The tolerance is set to be 10−13.


2 3 4 5 6 7 8 9 10 1110

−12

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Poly. Order p(t)

||E

RR

||m

ax

p=p(t)

; Coupling 1;

E(t)

=80

E(t)

=160

2 3 4 5 6 7 8 9 10 1110

−12

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Poly. Order p(t)

||E

RR

||m

ax

p=p(t)

; Coupling 2;

E(t)

=80

E(t)

=160

Fig. 19 Errors in the solution as a function of polynomial order in time p(t) for two dif-ferent coupling strategies: coupling 1 (Algorithm 1) with the EEFR-upwind and coupling 2(Algorithm 2) with the EEFR-upwind. The simulation is computed up to 1.0 with 10× 10spatial tessellation. The pair of polynomial orders in space, (p(x), p(y)) = (p, p), is chosen tothe same as the one in time p(t). The left one is the results obtained by coupling 1 using theEEFR-upwind with two different numbers of space-time slab (E(t) = 80 and E(t) = 160),while the right one is the results obtained by coupling 2 using the EEFR-upwind with twodifferent numbers of space-time slab. The tolerance is set to be 10−13.

2 3 4 5 6 7 8 9 10 1110

−12

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Poly. Order p(t)

||E

RR

||m

ax

p=p(t)

; E(t)

=80;

Coupling 1

Coupling 2

2 3 4 5 6 7 8 9 10 1110

−12

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Poly. Order p(t)

||E

RR

||m

ax

p=p(t)

; E(t)

=160;

Coupling 1

Coupling 2

Fig. 20 Errors in the solution as a function of polynomial order in time p(t) for two differentnumbers of space-time slab: E(t) = 80 and E(t) = 160. The simulation is computed up to 1.0with 10×10 spatial tessellation. The pair of polynomial orders in space, (p(x), p(y)) = (p, p),is chosen to the same as the one in time p(t). The left one is the comparison between twodifferent coupling strategies with E(t) = 80, while the right one is the comparison twodifferent coupling strategies with E(t) = 160. The tolerance is set to be 10−13.


Table 2 Overall order of convergence of the present multi-implicit space-time spectralelement method for two different coupling strategies, coupling 1 (Algorithm 1) with theEEFR-upwind and coupling 2 (Algorithm 2) with the EEFR-upwind, to approximate theadvection-diffusion-reaction equation (38). The number of iterations K is set to be p(t). Thetolerance is set to be 10−13, and the final time is T = 1.0.

Overall order of convergence

(p(x), p(y)), (p(t),K) (E(x) × E(y), E(t)) Coupling 1 Coupling 2

(2, 2), (2, 2)(5× 5, 40) — —

(10× 10, 80) 2.1 2.1(20× 20, 160) 2.0 2.0

(3, 3), (3, 3)(5× 5, 40) — —

(10× 10, 80) 3.0 3.0(20× 20, 160) 3.0 3.0

(4, 4), (4, 4)(5× 5, 40) — —

(10× 10, 80) 4.1 4.1(20× 20, 160) 4.0 4.0

(5, 5), (5, 5)(5× 5, 40) — —

(10× 10, 80) 5.0 5.1(20× 20, 160) 5.0 5.0

(6, 6), (6, 6)(5× 5, 40) — —

(10× 10, 80) 6.2 6.2(20× 20, 160) 5.9 5.9

Table 3 Overall order of convergence of the present multi-implicit space-time spectralelement method for two different coupling strategies, coupling 1 (Algorithm 1) with theEEFR-upwind and coupling 2 (Algorithm 2) with the EEFR-upwind, to approximate theadvection-diffusion-reaction equation (38). The number of iterations K is set to be p(t). Thetolerance is set to be 10−13, and the final time is T = 1.0.


(p(x), p(y)), (p(t),K) (E(x) × E(y), E(t)) ‖Err‖∞ (Coupling 1) Coupling 1 Coupling 2

(2, 2), (2, 2)(5× 5, 80) 0.917 — —

(10× 10, 160) 0.151 2.6 2.6(20× 20, 320) 3.529E-002 2.0 2.1

(3, 3), (3, 3)(5× 5, 80) 8.738E-003 — —

(10× 10, 160) 5.349E-004 4.0 4.0(20× 20, 320) 5.938E-005 3.1 3.1

(4, 4), (4, 4)(5× 5, 80) 6.460E-004 — —

(10× 10, 160) 1.822E-005 5.1 5.2(20× 20, 320) 8.479E-007 4.4 4.4

(5, 5), (5, 5)(5× 5, 80) 1.763E-005 — —

(10× 10, 160) 3.372E-007 5.7 5.7(20× 20, 320) 6.874E-009 5.6 5.6

(6, 6), (6, 6)(5× 5, 80) 5.480E-007 — —

(10× 10, 160) 1.626E-009 8.3 8.3(20× 20, 320) 3.794E-011 5.4 5.4

7 Conclusions1

Novel multi-implicit space-time spectral element methods are proposed to solve2

problems associated with two or more processes with differing characteristic3

time scales. Different from the fully implicit space-time method [11,33], the4

method of lines approach is employed in order to avoid the effort of solving5

a large nonlinear system of equations constituted by the degrees of freedom6

in both space and time. The discontinuous spectral element method in the7


Table 4 Overall order of convergence of the present multi-implicit space-time spectralelement method for two different coupling strategies, coupling 1 (Algorithm 1) with theEEFR-upwind and coupling 2 (Algorithm 2) with the EEFR-upwind, to approximate theadvection-diffusion-reaction equation (38). The number of iterations K is set to be p(t) + 1.The tolerance is set to be 10−13, and the final time is T = 1.0.


(p(x), p(y)), (p(t),K) (E(x) × E(y), E(t)) ‖Err‖∞ (Coupling 1) Coupling 1 Coupling 2

(2, 2), (2, 3)(5× 5, 40) 0.316 — —

(10× 10, 80) 1.893E-002 4.0 4.1(20× 20, 160) 1.747E-003 3.4 3.4

(3, 3), (3, 4)(5× 5, 40) 2.002E-002 — —

(10× 10, 80) 1.158E-003 4.1 4.1(20× 20, 160) 7.643E-005 3.9 3.9

(4, 4), (4, 5)(5× 5, 40) 3.250E-004 — —

(10× 10, 80) 7.788E-006 5.3 5.7(20× 20, 160) 2.316E-007 5.0 5.1

(5, 5), (5, 6)(5× 5, 40) 1.520E-005 — —

(10× 10, 80) 2.313E-007 6.0 6.0(20× 20, 160) 3.923E-009 5.8 5.8

(6, 6), (6, 7)(5× 5, 40) 6.832E-007 — —

(10× 10, 80) 2.693E-009 7.9 8.0(20× 20, 160) 5.261E-011 5.6 5.5

Table 5 Order of convergence in space of the present multi-implicit space-time spectralelement method for two different coupling strategies: coupling 1 (Algorithm 1) with theEEFR-upwind and coupling 2 (Algorithm 2) with the EEFR-upwind. The tolerance is setto be 10−13, and the final time is T = 1.0.

Order of convergence in space


(4, 4), (6, 8)(3× 3, 100) — —(6× 6, 100) 5.7 5.7

(12× 12, 100) 6.0 6.0

(5,5) ,(6,8)(3× 3, 100) — —(6× 6, 100) 6.5 6.5

(12× 12, 100) 5.7 5.7

(6,6), (6, 8)(3× 3, 100) — —(6× 6, 100) 8.1 8.1

(12× 12, 100) 8.1 8.1

(7,7), (6, 8)(3× 3, 100) — —(6× 6, 100) 8.2 8.2

(12× 12, 100) 8.2 8.5

collocation form is applied on the spatial operators, and the resulting system of1

ODEs are solved by a new spectral multi-implicit spectral deferred correction2

(MISDC) method.3

Two flux reconstructions have been developed for discretizing the hy-4

perbolic terms: (i) extended element-wise flux reconstruction and (ii) non-5

extended element-wise flux reconstruction. For the hyperbolic terms, a low-6

order I-stable building block time integration scheme is introduced which leads7

to a stable and spectrally accurate space-time method when applying with the8

MISDC iteration process. In the new MISDC coupling method, the provisional9

solution is computed by the operator splitting method, and then this solution10

is corrected by a series of deferred correction steps; two different deferred cor-11


Table 6 Order of convergence in time of the present multi-implicit space-time spectralelement method for two different coupling strategies: coupling 1 (Algorithm 1) with theEEFR-upwind and coupling 2 (Algorithm 2) with the EEFR-upwind. Different combinationsof temporal polynomial order p(t) and the number of iterations K are investigated. Thetolerance is set to be 10−13, and the final time is T = 1.0.

Order of convergence in time


(8, 8), (5, 4)(5× 5, 40) — —(5× 5, 80) 4.0 4.0(5× 5, 160) 4.0 4.0

(8, 8), (6, 4)(5× 5, 40) — —(5× 5, 80) 4.0 4.0(5× 5, 160) 4.0 4.0

(8, 8), (5, 5)(5× 5, 40) — —(5× 5, 80) 4.9 4.9(5× 5, 160) 5.0 5.0

(8, 8), (4, 5)(5× 5, 40) — —(5× 5, 80) 5.0 5.0(5× 5, 160) 5.0 5.0

(9, 9), (5, 6)(5× 5, 40) — —(5× 5, 80) 6.1 6.1(5× 5, 160) 6.0 6.0

(9, 9), (6, 6)(5× 5, 40) — —(5× 5, 80) 6.0 6.0(5× 5, 160) 5.9 5.9

rection strategies have been studied. Compared to low order operator splitting1

methods that each process is discretized in isolation, the splitting error in the2

MISDC method is eliminated by exploiting an iterative coupling strategy in the3

deferred correction procedure. Since the low order I-stable building block time4

integration scheme is a two-stage scheme, two correction strategies, “intrusive5

correction” and “non-intrusive correction,” are introduced and compared with6

each other. In the “non-intrusive” correction strategy, the correction terms7

are only applied on the last stage of the low-order I-stable scheme. Thus, the8

“non-intrusive” correction strategy requires less memory and is faster than the9

“intrusive strategy.”10

Numerical tests in two spatial dimensions are presented to demonstrated11

the performance of the present multi-implicit space-time spectral element12

method. The spectral convergence in both space and time is demonstrated13

for advection-diffusion-reaction problems. Both treatments for the hyperbolic14

terms exhibit spectral accuracy in both space and time. The accuracy of the15

extended element-wise flux reconstruction was found to be superior to the non-16

extended element-wise flux reconstruction with the same number of degrees of17

freedom. Both MISDC coupling strategies that we investigated were found to18

be spectrally accurate in both space and time.19

With the new coupling strategies, our new spectrally accurate space-time20

MISDC method can extend previous spectrally accurate in space and low or-21

der in time operator splitting methods to be spectrally accurate space-time22

numerical schemes. We envision that our spectral space-time method will be23


of benefit to the engineering community because very high spatial and tem-1

poral resolution is needed for giving a correct description of the flow physics2

for the simulation of turbulent flows. Future work will concern the extension3

of the present space-time spectral element schemes to a hierarchical block4

structured space-time spectral element method to solve multiphase incom-5

pressible/compressible Navier-Stokes equations.6

Acknowledgements This work and the authors were supported in part by the National7

Science Foundation under contract DMS 1418983.8

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