A New Multi-dimensional Relaxation Scheme for

Nonlinear Hyperbolic Conservation Laws Based on

Bicharacteristics

K.R. Arun1, S.V. Raghurama Rao2, M. Lukacova-Medvid’ova,3 P. Prasad1

Abstract

A new genuinely multi-dimensional relaxation scheme is proposed. Based on

a new discrete velocity Boltzmann equation, which is an improvement over pre-

viously introduced relaxation systems in terms of isotropic coverage of the multi-

dimensional domain by the foot of the characteristic, a finite volume method is

developed in which the fluxes at the cell interfaces are evaluated in a genuinely

multi-dimensional way, in contrast to the traditional dimension-by-dimension treat-

ment. This algorithm is tested on some bench-mark test problems for hyperbolic

conservation laws.

1 Introduction

Finite volume methods have been popular for the numerical solution of hyperbolic conser-vation laws in the last three decades. Numerical schemes based on kinetic theory representinteresting alternatives to the classical Riemann solver based schemes. A review of up-

wind methods based on kinetic theory is given in [7] and [10]. Jin and Xin [11] haveintroduced a new category of upwind method called relaxation schemes. A relaxationsystem converts a nonlinear convection equation into linear convection equations withnonlinear source terms. The numerical methods based on a relaxation system are calledrelaxation schemes. These schemes avoid solution of Riemann problems. The diagonalform of the relaxation system can be viewed as a discrete velocity Boltzmann equation

[1]. The so called discrete kinetic schemes are based on the discrete velocity Boltzmannequation. Some of the numerical investigations using the discrete Boltzmann equationcan be found in [1], [3], [19]. For multi-dimensional flows, however, the traditional finitevolume methods are typically based on a dimension-by-dimension treatment using one-dimensional approximate Riemann solver. As a result of this inherently one-dimensionaltreatment, the discontinuities which are oblique to the coordinate directions are not beresolved accurately. Developing genuinely multi-dimensional algorithms has been a topicof intense research in the last decade and a half. The reader is referred to [6], [9], [12],[14], [15], [18], [20], [23] for some multi-dimensional schemes. In the present work, wehave developed a genuinely multi-dimensional relaxation scheme using a discrete velocityBoltzmann equation. In the relaxation system of Aregba-Driollet and Natalini [1], thefoot of the bicharacteristic curves are not distributed in an isotropic way. To overcome

1Department of Mathematics, Indian Institute of Science, Bangalore, email:prasad@math.iisc.ernet.in2Department of Aerospace Engineering, Indian Institute of Science, Bangalore, India,

email:raghu@aero.iisc.ernet.in3Institute of Numerical Simulation, Hamburg University of Technology, Hamburg, Germany, email:

lukacova@tu-harburg.de

1

this deficiency Manisha et.al [16] have given an isotropic relaxation system in which thefoot of the characteristic traverses all quadrants in an isotropic way. The goal of this paperis to derive a genuinely multi-dimensional finite volume scheme based on the relaxationsystem in [16], that follows the work of Lukacova, see [15] and the references therein.

2 Relaxation Systems for Hyperbolic Conservation

Laws

In this section we introduce relaxation systems for multi-dimensional hyperbolic conser-vation laws. Let u be a weak solution to the Cauchy problem,

∂tu +d∑

i=1

∂xigi(u) = 0, (2.1)

u(x, 0) = u0(x). (2.2)

where u : Rd × R+ → U ⊂ R

m. The flux functions gi : U → Rm are locally Lipschitz

continuous. Let Ai(u) = Dgi(u)/Du, i = 1, 2, . . . , d be the flux Jacobian matrices. Weassume that the system of conservation laws (2.1) is strictly hyperbolic, i.e., for anyω ∈ R

d, with |ω| = 1, the matrix pencil A(ω) :=∑d

i=1 ωiAi has real eigenvalues and afull set of right eigenvectors.We approximate the Cauchy problem (2.1)-(2.2) by the following system

∂tfǫ +

d∑

i=1

Λi∂xif ǫ =

1

ǫ(F (uǫ) − f ǫ), (2.3)

f ǫ(x, 0) = f ǫ0(x). (2.4)

Here f ǫ takes values in RL, for some L > m. The quantity ǫ ≪ 1 is a small positive

number called the relaxation parameter. The matrices Λi are diagonal L × L matrices,P is a constant m × L matrix, the function uǫ is defined by uǫ := Pf ǫ. The functionF is defined on U and is Lipschitz continuous. The equation (2.3) is analogous to theBoltzmann equation of kinetic theory of gases with the BGK model for the collision term,except that the molecular velocity is replaced by discrete values. Hence (2.3) is termedas discrete velocity Boltzmann equation. In (2.3) f ǫ represents the molecular velocitydistribution function. The quantity F (uǫ) is called the local Maxwellian and is chosen insuch a way that the following consistency conditions are satisfied

PF (u) = u,PΛiF (u) = gi(u).

(2.5)

The conditions (2.5) imply the consistency of (2.3) with the original nonlinear conservationlaw (2.1). If we multiply (2.3) by P we obtain a conservation law

∂tuǫ +

d∑

i=1

∂xi(PΛif

ǫ) = 0, (2.6)

which is satisfied by every solution of (2.3). Fix now the initial function u0(x) in (2.2).Consider a sequence f ǫ of solutions to (2.3) with f ǫ

0 = F (u0). Assume that the sequencef ǫ is uniformly bounded and there exists a function f0 such that

f ǫ → f 0, (2.7)

2

as ǫ → 0 in some strong topology. Then setting u0 = Pf0 we have

uǫ → u0, (2.8)

in the same topology and the following identities holds.

f 0 = F (u0), (2.9)

∂tu0 +

d∑

i=1

∂xi

(

PΛif0)

= 0. (2.10)

Hence from (2.9) and the first condition in (2.5) we can obtain

PΛif0 = PΛiF (u0) = gi(u

0), for i = 1, 2, . . . , d. (2.11)

Hence we conclude that u0 is a weak solution to the Cauchy problem (2.1)-(2.2). Equiv-alently we say that the conservation law (2.1) is the equilibrium limit of the relaxationsystem (2.3), see [17] for more details.

The main point concerning the relaxation system (2.3) is the question of stability of thesolution u of the reduced system (2.1). It is well known that the solutions of (2.1) candevelop singularities even in a finite time. The stability issue can be investigated in severaldifferent ways. The question of local or global smooth solutions to (2.3) can be studiedusing the energy methods. The existence of global smooth solutions to the discrete kineticsystems of the form (2.3) is given in [25], see also [8], [24] and the references therein.

The stability investigation of the equilibrium limit as ǫ → 0 has been started by T.P.Liuand his collaborators by formulating an entropy condition for the relaxation system, see [5],[13]. An entropy condition for a general class of relaxation systems is given by Chen, Lev-ermore and Liu [5]. The entropy condition is equivalent to the so called sub-characteristic

condition for a 2 × 2 relaxation system. The reader is referred to [21] for a different ver-sion of the entropy condition and some structural conditions on the Maxwellians and thedicrete velocities. A general framework of the entropy satisfying BGK models has beengiven by Bochut [4] by characterising the Maxwellians as the minimizers of the kineticentropy functional. The fundamental idea of stability is that a first order correction to(2.3) derived by a Chapman-Enskog type expansion should be dissipative in the sense ofthe entropies of the conservation law (2.1).

2.1 Chapman-Enskog Analysis

In this section we derive the stability conditions for the discrete kinetic system (2.3). Sincethe local equilibrium to the system (2.3) is the original nonlinear conservation law (2.1),we derive a diffusive first order correction to (2.3) by a Chapman-Enskog type expansion.This is an analoge of the compressible Navier-Stokes equations in the kinetic theory. Fora more comprehensive treatment the reader is advised to [1], [5].

Now onwards we drop the superscript ǫ from f ǫ for simplicity. Let f be a solution to(2.3) with F as the local Maxwellian. Denote vi = PΛif , i = 1, 2, . . . , d. Using the

3

consistency conditions (2.5) in (2.3) we obtain

∂tu +d∑

i=1

∂xivi = 0,

∂tvi +d∑

j=1

∂xj(PΛiΛjf) =

1

ǫ(gi(u) − vi) . (2.12)

Consider the following expansion of f as a perturbation series

f = F + ǫf (1) + O(ǫ2). (2.13)

Then

vi = gi(u) − ǫ

(

∂tvi +d∑

j=1

∂xj(PΛiΛjf)

)

+ O(ǫ2)

= gi(u) − ǫ

(

∂tvi +d∑

j=1

∂xj(PΛiΛjF )

)

+ O(ǫ2).

Substituting this in (2.12) yields

∂tu +

d∑

i=1

∂xigi(u) = ǫ

d∑

i=1

∂xi

(

∂tvi +

d∑

j=1

∂xj(PΛiΛjF )

)

+ O(ǫ2). (2.14)

Again, we have

∂tvi = Ai(u)∂tu = −

d∑

j=1

Ai(u)Aj(u)∂xju + O(ǫ).

Then we get from (2.14) upto the first order terms

∂tu +d∑

i=1

∂xigi(u) = ǫ

d∑

i=1

∂xi

(

d∑

j=1

Bij(u)∂xju

)

, (2.15)

where

Bij = PΛiΛjDF (u) − Ai(u)Aj(u).

Thus B = (Bij) is the viscosity matrix. For B to be admissible we must have the usualpositive definiteness condition

d∑

i,j=1

〈Bijξi, ξj〉 ≥ 0 (2.16)

for all ξ1 ∈ Rm, . . . , ξd ∈ R

m. Under this condition (2.15) is parabolic and hence (2.16) isthe required stability criterion.

4

2.2 A new relaxation system

In this section we give a precise description of the construction of relaxation systems. Forany system of m conservation laws in d space dimensions of the form (2.1) we constructrelaxation models of the type given in (2.3). The most important part is the constructionof the diagonal matrices Λi, the Maxwellians F and the matrix P for taking the moments.Following Aregba-Driollet and Natalini [1] we use a block structure for obtaining Λi. Wechoose L = Nm and P = (Im, Im, . . . , Im) with N blocks of the identity matrix Im ofR

m. Each matrix Λi is constructed using N diagonal blocks of size m × m. The numberN depends on the space dimension. In our two-dimensional computations we have chosenN = 4. Thus,

Λi = diag(

C(i)1 , C

(i)2 , . . . , C

(i)N

)

, (2.17)

whereC

(i)j = λ

(i)j Im, λ

(i)j ∈ R. (2.18)

Now (2.3) can be written as

∂tf i +d∑

j=1

λ(j)i ∂xj

f i =1

ǫ(F i − f i) for i = 1, 2, . . . , N (2.19)

with

u =N∑

i=1

f i. (2.20)

In [4] Bochut has characterised the space of Maxwellians so that the kinetic system (2.3)admits a kinetic entropy and the entropy inequality is satisfied in the equilibrium limitas ǫ → 0. Folllowing [4] we choose the Maxwellians as the linear combinations of theconserved variable and the fluxes

F i(u) = ai0u +

d∑

j=1

aijgj(u). (2.21)

Here the coefficients aij are to be chosen according to the consistency conditions (2.5).For the construction of the matrices Λi we use the orthogonal velocity method given in [1].

We choose Λi in such a way that the coefficients λ(i)j satisfy the following two conditions

N∑

j=1

λ(i)j = 0, for i = 1, 2, . . . , d (2.22)

andN∑

k=1

λ(i)k λ

(j)k = 0, for i, j = 1, 2, . . . , d, i 6= j. (2.23)

With the above choice of the discrete velocities, λ(i)j , the Maxwellian F = (F 1, F 2, . . . , F N)

can be obtained to be

F i(u) =u

N+

d∑

j=1

gj(u)

a2j

λ(i)j for i = 1, 2, . . . , N, (2.24)

5

where a2j =

∑Nk=1 λ

(j)k . For better understanding we give now an example of the relax-

ation system for a single conservation law in two space dimensions. Consider a singleconservation law

∂tu + ∂xg1(u) + ∂yg2(u) = 0 (2.25)

We propose the relaxation system

∂tf + Λ1∂xf + Λ2∂yf =1

ǫ(F − f ), (2.26)

where

f =

f1

f2

f3

f4

, F =

F1

F2

F3

F4

. (2.27)

The diagonal matrices Λ1 and Λ2 are given as

Λ1 =

−λ 0 0 00 λ 0 00 0 λ 00 0 0 −λ

and Λ2 =

−λ 0 0 00 −λ 0 00 0 λ 00 0 0 λ

. (2.28)

The Maxwellians can be obtained to be

F1

F2

F3

F4

=

u4 −

g1(u)4λ

−g2(u)4λ

u4 +

g1(u)4λ

−g2(u)4λ

u4 +

g1(u)4λ

+g2(u)4λ

u4 −

g1(u)4λ

+g2(u)4λ

. (2.29)

The relaxation system (2.26) with this choice of discrete velocities and Maxwellians isfirst introduced by Manisha et.al [16] and has been used by Arun et.al [2] to developa genuinely multi-dimensional relaxation scheme. Note that this choice of the discretevelocities admits the following nice feature, which we call isotropy. The bicharactericscurves of (2.3) through any point P (x, y, t + ∆t) falls evenly in all the four quadrantsaround the point Q(x, y, t); see Figure 1.

P(x,y,t+∆ t)

Q1

Q2

Q3 Q4

Q(x,y,t)

t

y

x

Figure 1: Feet of the bicharacteristics through P

The isotropy is a special property of our relaxation system (2.26). This feature enablesus to design a genuinely multi-dimensional finite volume scheme for (2.3). For the twodimensional relaxation system (2.3) the stability condition (2.16) reads

λ2 ≥ (∂ug1(u))2 + (∂ug2(u))2 . (2.30)

6

Note that in the case of two dimensions our stability criterion is very simple and it differssignificantly from that in [1]. The condition (2.30) is a realistic extension of the sub-characteristic condition [13].

3 A numerical scheme based on bicharacteristics

In the literature numerous approaches based on the relaxation systems can be found. Forexample, in [11] Jin and Xin classify their schemes into two categories, namely, relaxingschemes and relaxed schemes. Relaxing schemes depend on ǫ and the artificial variables v

and w. The zero relaxation limit of the relaxing schemes are called relaxed schemes. Therelaxed schemes are stable descretizations of the original conservation law and thus areindependent of ǫ and the artificial variables v and w. Jin and Xin construct upwindingbased on characteristic variables. To achieve second order accuracy they use van Leer’sMUSCL descretization. Time discretization is realized by means of a second order TVDRunge-Kutta method. New discrete kinetic schemes have been introduced by Aregba-Driollet and Natalini in [1]. The authors start with a set of decoupled equations, inwhich the dependent variables are the characteristic variables. The main advantage inthis approach is the fact that upwinding becomes very simple. In order to treat multi-dimensional systems the dimensional splitting has been used.Our goal is to use the discrete Boltzmann equation (2.3) and approximate it in a genuinelymulti-dimensional way using the isotropy of the system. The discrete Boltzmann equation(2.3) is solved by splitting method as

∂tf + Λ1∂xf + Λ2∂yf = 0 (convection step), (3.1)

df

dt=

1

ǫ(F − f ) (relaxation step). (3.2)

The relaxation step is further simplified by taking ǫ = 0, leading to f = F . Hence at anystage we need to solve only the set of linear convection equations in (3.1).

Let Ω be our two-dimensional computational domain. We subdivide Ω by finite partitionsΩij := [xi−1/2, xi+1/2] × [yj−1/2, yj+1/2]. If we integrate (3.1) over a mesh cell Ωij and overthe time interval from n∆t to (n + 1)∆t, application of Gauss formula gives,

fn+1 = fn − ∆t

∫ ∆t

0

[

Λ1δxfn+t/∆t + Λ2δyf

n+t/∆t]

dt. (3.3)

In this formula fn+1 and fn represents the cell averages, while δxfn+t/∆t involves averages

along the cell edges to the right and left and δyfn+t/∆t along the edges to the top and

bottom at an intermediate time step n+ t/∆t. We approximate the time integrals in (3.3)by midpoint rule at t = ∆t/2. Then the cell boundary flux fn+1/2 is evolved using theexact evolution operator E∆t/2 for (3.1) and suitable recovery operators Rh. For exampleon vertical edges,

fn+1/2 =1

h

∫ h

0

E∆t/2Rhfndsy, (3.4)

where dsy is an element of arc length along the vertical edges. A similar expression holdsfor horizontal edges too. We have used the Simpson rule for the cell interface integrals in(3.4) and evaluated the fluxes at the midpoints and corners of the edges, see figure 2 forthe 9-point stencil used in the computation. Note that the Simpson quadrature allows usto take into account the multi-dimensional corner effects.

7

i,j i+1,j

i+1,j+1

i+1,j−1i,j−1

i,j+1i−1,j+1

i−1,j

i−1,j−1

Figure 2: 9-point stencil and quadrature points for flux evaluation

For betterreadability we shall briefly explain our multi-dimensional flux evaluation. Asgiven in (3.4) we compute the fluxes at half time step tn+∆t/2. This auxiliary informationof the solution at half time step is computed using the exact solution of the linear equationsin (3.1). For example, the exact solution of the first convection equation in (3.1) gives

f1(x, y, t + ∆t) = f1(x + λ∆t, y + λ∆t, t). (3.5)

The flux in the x − direction at a right hand corner corresponding to the variable f1 canbe computed as

−λfn+1/21,i+1/2,j = −λf1(xi+1/2, yj, tn + ∆t/2)

= f1(xi+1/2 + λ∆t/2, yj + λ∆t/2, tn). (3.6)

Analogous expressions hold for other quadrature nodes and also for the remaining vari-ables. We now employ the recovery functions Rh to compute these values at the feet ofthe bicharacteristics. The piecewise linear recovery functions is given as

f(x, y, tn) = fnij + (x − xi)(fx)

nij + (y − yj)(f y)

nij in Ωij , (3.7)

where the gradients (fx, fy) are calculated using just finite differences with standardlimiting procedures. Note that we limit the overshoots using the minmod limiter.

4 Numerical results

The new multi-dimensional relaxation scheme is tested on some standard test problemsfor inviscid Burgers equation and the Euler equations in two space dimensions. In all theproblems the computations were carried out on uniform Cartesian grids.

Burgers equation test cases: These test cases are taken from Spekreijse [22]. It modelsa shock and a smooth variation representing an expansion fan in 2D domain. The inviscidBurgers equation considered here is given by

∂tu + ∂x

(

u2

2

)

+ ∂yu = 0.

8

The computational domain is [0, 1]× [0, 1]. The following two test cases has been consid-ered.

Test case 1: The boundary conditions are given by

u(0, y) =1 0 < y < 1,

u(1, y) = − 1 0 < y < 1,

u(x, 0) =1 − 2x 0 < x < 1.

The solution is computed for time T = 1.5 on a 128 × 128 grid with a CFL number0.7. The solution contains a shock originating at the point (0.5, 0.5). The isolines of thesolution is given in Figure 3.

Test case 2: The boundary conditions are

u(0, y) =1.5 0 < y < 1,

u(1, y) = − 0.5 0 < y < 1,

u(x, 0) =1.5 − 2x 0 < x < 1.

The solution is computed for time T = 1.5 on a 128 × 128 grid with a CFL number 0.7.The solution contains an oblique shock originating at the point (0.75, 0.5). See Figure 3.

Euler equations test cases:

Test case 3: The first test case is the cylindrical explosion problem. The computationaldomain is the square [−1, 1] × [−1, 1]. The initial data read,

ρ = 1, u = 0, v = 0, p = 1, |x| < 0.4,

ρ = 0.125, u = 0, v = 0, p = 0.1, else.

The solution is computed at time T = 0.2 with a CFL number 0.7. The solution exhibitsa circular shock and circular contact discontinuity moving away from the centre of thecircle and circular rarefaction wave moving in the opposite direction. The isolines of thedensity, x−component of the velocity, y−component of the velocity and pressure are givenin Figure 4.Next two test cases are two-dimensional Riemann problems. The computational domain[−1, 1]× [−1, 1] is divided into four quadrants. The initial data consist of single constantstates in each of these four quadrants. These constant values are chosen in such a waythat each pair of quadrants defines a one-dimensional Riemann problem.

Test case 4: In this test problem we choose the initial data in such a way that two forwardmoving shocks and two standing slip lines are produced. The initial data read,

ρ = 0.5313, u = 0.0, v = 0.0, p = 0.4, if x > 0, y > 0,

ρ = 1.0, u = 0.0, v = 0.7276, p = 1.0, if x > 0, y < 0,

ρ = 1.0, u = 0.7276, v = 0.0, p = 1.0, if x < 0, y > 0,

ρ = 0.8, u = 0.0, v = 0.0, p = 1.0, if x < 0, y < 0.

9

The solution is computed at time T = 0.52 with a CFL number 0.7. The isolines of thedensity and pressure are given in Figure 5.

Test case 5: This test case is a two dimensional Riemann problem that produces twoforward moving shocks and two backward moving shocks. Initial data are taken as

ρ = 1.1, u = 0.0, v = 0.0, p = 1.1, if x > 0, y > 0,

ρ = 0.5065, u = 0.0, v = 0.8939, p = 0.35, if x > 0, y < 0,

ρ = 0.5065, u = 0.8939, v = 0.0, p = 0.35, if x < 0, y > 0,

ρ = 1.1, u = 0.8939, v = 0.8939, p = 1.1, if x < 0, y < 0.

Computation are done untill time T = 0.25 with a CFL number 0.7. The density andpressure isolines are given in Figure 6.

Test case 6: Mach 3 wind tunnel with a step (Forward facing step problem) The setting ofthis problem is the following. A right going Mach 3 uniform flow enters a wind tunnel of1 unit wide and 3 units long. The step is 0.2 units high and is located 0.6 units from theleft end of the tunnel. The initial conditions correspond to a uniform right going Mach 3flow. Reflecting boundary conditions are applied along the walls of the tunnel and inflowand outflow boundary conditions are applied respectively at the entrance and exit of thetunnel. The solution is computed at time T = 4.0. The isolines of the density, pressureand Mach number are given in Figure 7. We can notice the triple point structure due tothe interaction of the shocks with the expansion wave.

5 Conclusions

A novel genuinely multi-dimensional relaxation scheme is presented, based on a multi-dimensional relaxation system in which the foot of the characteristics traverses all quad-rants in an isotropic way. This scheme is tested on some bench-mark problems for scalarand vector conservation laws in two dimensions and the results demonstrate its efficiencyin capturing the flow features accurately.

Acknowledgement. The authors sincerely thank the Department of Science and Tech-nology (DST), Goverment of India and the German Academic Exchange Service (DAAD)for providing the financial support for collaborative research under the DST-DAAD ProjectBased Personnel Exchange Programme. K.R. Arun would like to express his gratitude tothe Council of Scientific & Industrial Research (CSIR) for supporting his research at theIndian Institute of Science under the grant-09/079(2084)/2006-EMR-1.

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Mechanics and Analysis 2004; 172:247-266.

test case 1

0.2 0.4 0.6 0.8

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

test case 2

0.2 0.4 0.6 0.8

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure 3: Burgers equation test cases on a 128 × 128 mesh

12

rho

−0.5 0 0.5

−0.5

0

0.5

u

−0.5 0 0.5

−0.5

0

0.5

v

−0.5 0 0.5

−0.5

0

0.5

p

−0.5 0 0.5

−0.5

0

0.5

Figure 4: Cylidrical explosion problem: Isolines of the solution calculated on a 400× 400mesh at time T = 0.2

rho

−0.5 0 0.5

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

p

−0.5 0 0.5

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Figure 5: Two dimensional Riemann problem containing two shocks/two slip lines. Iso-lines of density and pressure calculated on a 400 × 400 mesh at time T = 0.52

13

rho

−0.5 0 0.5

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

p

−0.5 0 0.5

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Figure 6: Two dimensional Riemann problem with four shocks. Isolines of density andpressure calculated on a 400 × 400 mesh at time T = 0.25

density

0.5 1 1.5 2 2.5

0.2

0.4

0.6

0.8

pressure

0.5 1 1.5 2 2.5

0.2

0.4

0.6

0.8

Mach number

0.5 1 1.5 2 2.5

0.2

0.4

0.6

0.8

Figure 7: Isolines of density, pressure and Mach number for the Mach 3 wind tunnelproblem calculated on a 240 × 160 mesh at time T = 4.0

14