A High Resolution Total Variation

Diminishing Scheme for Hyperbolic

Conservation Law and Related Problems

M. K. Kadalbajoo

Department of Mathematics, Indian Institute of Technology, Kanpur, India.

email: kadal@iitk.ac.in

Ritesh Kumar

Department of Mathematics, Indian Institute of Technology, Kanpur, India.

email: riteshkd@iitk.ac.in

Abstract

A high resolution, explicit second order accurate, Total Variation Diminishing (TVD)difference scheme using flux limiter approach for scalar hyperbolic conservation lawand closely related convectively dominated diffusion problem is presented. Boundsand TVD region is given for these limiters such that the resulting scheme is TVD. Aclass of such limiters is given which gives second order accurate TVD scheme.Numericalresults for one dimensional scalar conservation law and convectively dominated dif-fusion problem are presented. Comparison of numerical results with some of theclassical difference schemes are given.

Key words: Conservation Law, Convection, Finite Difference Schemes, HighResolution Schemes, Total Variation Diminishing Schemes.

1 Introduction

For simplicity we consider the scalar conservation law

∂u

∂t+

∂f(u)

∂x= 0

∂u

∂t+ a(u)

∂u

∂x= 0 (1)

and closely related one dimensional convectively dominated diffusion equation

Preprint submitted to Elsevier Science 29 November 2006

∂u

∂t+

∂f(u)

∂x=

∂

∂x(Γ

∂u

∂x)

∂u

∂t+ a(u)

∂u

∂x=

∂

∂x(Γ

∂u

∂x) (2)

together with appropriate initial and boundary conditions.

Here a(u) = ∂f(u)∂u

is the characteristic (convective) speed depending on thevariable u, Γ may be a turbulent diffusion coefficient and u may be the concen-tration of a passive tracer, the temperature, salinity or the velocity component.

It is easy to see that depending on the values of a and Γ, Eq.(2) changes itscharacter. For a = 0 and Γ 6= 0 it is parabolic; it is hyperbolic when a 6= 0and Γ = 0. If a, Γ are functions of x and t, the character changes locally withtime. The numerical treatment of such problems involves specific difficulties,which mainly originate from the different scales of convective and turbulentmotion.

It has been seen that for parabolic (pure diffusion) problems, Euler forwardin time and central in space difference scheme is suitable, but for convectivelydominated problems, the problems associated with hyperbolic conservationlaws comes into the play which makes difference schemes for convective termsquite sensitive to stability and accuracy. Although the classical first order finitedifference methods are monotonic and stable, but they suffer from numericaldissipation, and made the solution of the conservation law to become smearedout and are often grossly inaccurate. On the other hand, classical higher or-der finite difference methods (e. g. Lax-Wendroff, Warming-Beam, QUICK([1]),([2])) are less dissipative in nature but are susceptible to numerical insta-bilities and exhibit spurious oscillations around the point of discontinuity, i.e. at turning points.Infact it happens that turning points and steep gradientstend to appear together, and more repidly the gradient changes, the moreexaggerated the oscillations. This is why some authers write this phenomenaoccurs in the region of large gradients.([3]). For more details one can look into([3],[4]). Apart from the numerical instabilities, fundamental mechanical orthermo-dynamical principles can be violated. Fronts will not be resolved dueto numerical diffusion.

It has been obsevered that, even if the initial condition for conservation lawis very smooth, the wiggles get introduced into the solution by the differencescheme. One can avoid this by requiring that the total variation of the solu-tion does not increase from time step to time step and try to find a schemethat satisfy this requirement. In [5] it has been shown that non-conservativeschemes converge to wrong solution. In fact using conservative schemes tomodel physical problems help to avoid violation of findamental mechanical orthermo-dynamical principles. So an obvious choice for numerical treatement

2

of conservation laws is to use a TVD, conservative difference scheme.

In recent years, efforts has been made into obtaining a class of schemes forconservation laws which are conservative, generally (i.e. at most places) besecond order accurate (or higher) on smooth section of the solution and nonoscillatory in nature and capable of resolving discontinuities in the solution.This class of schemes are known as high resolution schemes. In this paper weare proposing a high resolution TVD scheme for scalar linear conservationlaw and convectively dominated diffusion equation. In section 2, we are givingsome useful existing results for system of hyperbolic conservation laws. wehave used them to prove some results in our work as they are true for scalarcase too. In section 3, we have drive the numerical scheme and proved someresults for our scheme. We have given some numerical results in 4.

2 Conservative difference scheme

Integrating Eq.(2) over the rectangle [xj− 1

2

, xj+ 1

2

] × [tn, tn+1] and introducingthe definitions of the spatial and temporal cell averages

unj =

1

∆x

xk+1

2∫

xk−

12

u(x, tn)dx,

hk+ 1

2

=h(U ; k +1

2) =

1

∆t

tn+1∫

tn

f(u(xk+ 1

2

, t))dt, (3)

Dk+ 1

2

=D(U ; k +1

2) =

1

∆t

tn+1∫

tn

(Γ∂u

∂x)(u(xk+ 1

2

, t))dt

one can obtain a difference scheme of the form

un+1k = un

k −∆t

∆x

{

hk+ 1

2

− hk− 1

2

}

+∆t

∆x

{

Dk+ 1

2

− Dk− 1

2

}

(4)

where the subscript k and superscript n denote the kth space step and nth

time step, hk± 1

2

and Dk± 1

2

are convection fluxes and diffusion fluxes on thecell boundaries at xk± 1

2

respectively.

Difference schemes of the form given by eq.(3),and eq. (4) are called the con-servative (flux form) difference schemes. The main advantage of using con-servative form to model the transport of the physical variable is that withconservative form it is simpler to assure that the total physical variable is

3

conserved. The other advantage is that it becomes easier to avoid non-linearinstabilities.

It has been pointed out [3] that for most flows of practical situations, theclassical, second order central scheme for diffusion terms is entirely adequate.Therefore, without any loss of generality, In this work we are considering onlythe convection term of the Eq. (1) and (2).

2.1 Some results and definitions [11]

Consider hyperbolic conservation law given by eq. (1)

∂u

∂t+

∂f(u)

∂x= 0

∂u

∂t+ a(u)

∂u

∂x= 0 (5)

where u is a K-vector and f(u) is a function that maps RK into R

K . Thecorresponding conservative difference scheme is given by

Un+1k = Un

k −∆t

∆x

{

hk+ 1

2

− hk− 1

2

}

(6)

where ∆t hk± 1

2

approximate the flux of material across the sides x = xk± 1

2

.

The approximate fluxes are written as

hnk+ 1

2

= h(unk−p, .........., u

nk+q)

hnk− 1

2

= h(unk−p−1, .........., u

nk+q−1) (7)

where numerical flux fuction hk± 1

2

depend on u at p + q + 1 points.

Proposition 1 If h(u, u, .....u) = f(u) then difference scheme (6) withnumerical flux function Eq.(7) will be consistent with conservation law Eq.(5)(cf. [11], page 157).

Proposition 2 The truncation error of a conservative scheme of the form

un+1k = Q(un

k−(p+1), ....... , unk+q) (8)

is given by

τnk = −∆t[q(u)ux]xu=un

k

+ O(∆t2 + ∆x2), (9)

4

where u = u(x, t) is a solution to conservation law eq.(5) and

q(u) =1

2

1

R2

q∑

j=−(p+1)

j2Qj(u, ....... , u) − [f′

(u)]2

. (10)

where R = ∆t/∆x and Qj represents the partial derivative of Q with respectto the (p + 2 + j)-th variable. (cf. [11], page 184).

Definition 3 A difference scheme is said to be total variation diminishingscheme (TVD) if the numerical solution obtained by the scheme satisfies

∞∑

k=−∞

|δ+un+1k | ≤

∞∑

k=−∞

|δ+unk | for all n ≥ 0 (11)

3 High resolution scheme

In this section, we are deriving a higher resolution TVD scheme using fluxlimiters in much the same way as described by Sweby [6]. The approach hefollowed was similar to the Flux Corrected Transport (FCT) of Boris and Book[7], the application of a low order scheme along with an additional “limited”(or Corrected) flux. This flux is a difference between the flux of a high orderscheme and that of the low order scheme which has been “corrected” in sucha way that the resulting scheme is TVD.

There are two main differences between the approach adopted by Sweby andof Boris and Book [7]. Firstly the FCT algorithm was essentially a two-stepprocedure, whereas Sweby [6] adopted a single-step procedure; and secondlythe FCT limiter was constricted by unity whilst the Sweby allowed a moregenerous upper limit. We are giving a single-step procedure for the hyperbolicscalar conservation law similar to Sweby [6]. Though the approach is similarbut the our proposed TVD scheme, TVD region and the class of limiters donot match with Sweby’s work.

Infact first order upstream and second order upstream scheme, both obey thehyperbolicity condition. Hence using their linear combination seems to givebetter accuracy. The numerical results of proposed scheme also support thisfact.

For clarity of approach we consider the linear scalar equation

ut + aux = 0, a > 0 (12)

5

The approach that we are using to construct high resolution scheme as follows.We define the numerical flux function of the new scheme in terms of numericalflux function of the first order upstream scheme and second order upstreamscheme as

hnk+ 1

2

= hnL

k+ 12

+ φnk [hn

Hk+1

2

− hnL

k+12

] (13)

where hL and hH are the numerical flux function of first order and secondorder upstream schemes respectively, and are given by

hnL

k+ 12

= aunk (14)

hnH

k+ 12

=a

2(3un

k − unk−1) (15)

and φ is the limiter which is defined in such a way that the resulting numericalflux function has the smoothing capabilities of the lower order scheme whenit is necessary (i.e near discontinuities) and the accuracy of the higher orderscheme (i.e. in the smooth sections of solution) when it is possible. Note that,since we formulate the scheme by defining the numerical flux function, theresulting scheme will automatically be conservative. Since the flux given byhL is considered to be too diffusive, the correction term φn

k [hnH

k+12

− hnL

k+12

] is

referred to as the anti-diffusive flux. We try to add as much as anti-diffusiveflux as possible without increasing the variation of the solution.

The resulting conservative difference scheme is

un+1k = un

k − c[

(unk − un

k−1) +1

2

{

φnk(un

k − unk−1) − φn

k−1(unk−1 − un

k−2)}

]

(16)

where c = a △t

△xis the courant number, and φ is taken to be nonnegative so as

to maintain the anti-diffusive flux.

Like Roe [8], Vanleer [9] and Sweby [6] and many others we take the limiter φto be a function of consecutive gradients (in the case of fixed mesh size it turnsout to be function of consecutive differences), i.e. φn

k = φ(θnk ). The smoothness

parameter θ is given by

θnk =

∆unk− 1

2

∆unk+ 1

2

(17)

where ∆−uk+1 = ∆+uk = ∆uk+ 1

2

= uk+1 − uk.

6

We now seek to choose the function φ(θ) in such a way that the limitedanti-diffusive flux is maximized in amplitude subject to the constraint of theresulting scheme being TVD.

Theorem 4 The scheme given by Eq.(16 ) is TVD under the constraint 0 ≤φ ≤ 2, 0 ≤ φ ≤ 2

θ.

Proof

Let us rewrite the given scheme as

un+1k = un

k −

[

c

(

1 +φn

k

2

)

−c

2φn

k−1

∆−unk−1

∆−unk

]

∆−unk (18)

Comparing scheme with the incremental form (I-form)

uk+1k = un

k + Ck+ 1

2

∆+unk − Dk− 1

2

∆−unk (19)

where Ck+ 1

2

and Dk− 1

2

depends on unk and its neighboring values, so we have

Ck+ 1

2

= 0, Dk− 1

2

=

[

c

(

1 +φn

k

2

)

−c

2φn

k−1

∆−unk−1

∆−unk

]

(20)

Harten [10] has proved that a sufficient condition for any scheme of the formEq.(19) for solving Eq.(1) to be TVD is that the coefficients Ck+ 1

2

, Dk+ 1

2

satisfy

Ck+ 1

2

≥ 0; Dk+ 1

2

≥ 0; 0 ≤ Ck+ 1

2

+ Dk+ 1

2

≤ 1 (21)

Hence scheme (18) will be TVD if Ck+ 1

2

and Dk+ 1

2

given by Eq.(20), satisfy

Conditions (21), i.e. if

0 ≤ Dk− 1

2

≤ 1

i.e.,

0 ≤

[

c

(

1 +φn

k

2

)

−c

2φn

k−1

∆−unk−1

∆−unk

]

≤ 1 (22)

using the definition of smootheness parameter θ, we see that Inequality (22)

7

satisfies if (dropping out the suffixes and prefixes)

−2 ≤ φ − φθ ≤ 2(1 − c)

c(23)

under the CFL condition 0 ≤ c ≤ 12, (which is the stability condition for

second order conservative scheme using numerical flux given by Eq.(15) ). Wesee that Inequality Eq.(23) satisfies for

0 ≤ |φ − φθ| ≤ 2 (24)

In addition to requiring φ(θ) to be nonnegative we must have φ(θ) = 0, θ < 0.

It is easily seen that Inequality (24) along with non-negativity of φ are satisfiedfor the bounds

0 ≤ (φ, φθ) ≤ 2 (25)

Hence for the scheme Eq.(16) to be TVD the limter function must satisfybounds given by (25). Geomatrically limiter function φ(θ) must lie in theshaded region of Fig. 1(a). Proposed TVD region is compared with TVDregion proposed by Sweby in fig 1.

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5 6 7 8 0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5

Proposed TVD region 1(a) Sweby’s TVD region 1(b).

Fig. 1.

We need to maximize the limiter φ in order to maximize the antidiffusive fluxthat we add to the first order scheme, subject to the TVD constraints. Anobvious choice is

φ(θ) = min(2, 2/θ), θ > 0

which is the upper boundary of the region shown in Fig. 1. The extra con-dition imposed on φ in order to make resulting scheme Eq.(16) be secondorder accurate whenever posssible is that φ(1) = 1. This is also required forLipschitz continuity of φ(θ). Note that the measure of smoothness θ breaksdown near extreme points of solution, where the denominator may be close to

8

zero and θ become arbitrary large, or negative even if the solution is smooth.Hence maintaining the second order accuracy at extreme points is impossiblewith TVD methods. Following theorem gives conditions on φ which garanteessecond order accuracy away from extrema of solution.

Theorem 5 If the flux limiter φ is bounded, then the difference scheme (16)is consistent with the linear scalar equation (12). If φ(1) = 1 and φ is Lipschitzcontinuous at θ = 1, then the difference scheme (16) is second order accurateon smooth solutions with ux bounded away from zero.

Proof

Consider the associated numerical flux function with difference scheme (16)

hk+ 1

2

= h(unk+1, u

nk , u

nk−1) = aun

k −a

2φn

k

[

unk − un

k−1

]

Obviously, since φ is bounded, for linear scalar equation (12)

h(u, u, u) = f(u)

First claim follows from Proposition 1.

We prove a slight variation of the second claim. So that we can apply Proposi-tion 2, we prove the second claim under the assumption that φ is differentiableat θ = 1. The proof using the assumption that φ is only Lipschitz continousat θ = 1 is a slight, technical variation of the proof given.

The difference scheme (16) can be written as,

un+1k = Q(un

k−2, ....., unk+1)

= unk − c

[

(unk − un

k−1) +1

2

{

φnk(un

k − unk−1) − φn

k−1(unk−1 − un

k−2)}

]

(26)

where p = q = 1, and note that φnk , φ

nk−1depend on un

k+1, unk , u

nk−1, u

nk , u

nk−1, u

nk−2

respectively. An easy calculation shows that

1∑

j=−2

j2Qj(u, u, u, u) = 0 (27)

by Eq. (10)

q(u) =−a2

2(28)

9

hence by Eq. (9)

τnk = −∆t

{(

−a2

2ux

)

x

}

u=un

k

+ O(∆t2 + ∆x2), (29)

Eq.(29 shows that truncation error is of O(∆t, ∆x2), i.e. the difference schemeis atleast second order accurate.

In the light of above results we are proposing a new class of limiters whichare Lipschitz continuous and lie within the proposed TVD region such thatφ(1) = 1. This class of limiters is given by

φ(θ) =

0, θ ≤ 0

min[

2, 2θ, 1+δ

δ+θ

]

, θ > 0, for any fixed δ ∈ [0,∞)

We see that for every value θ > 0 and any fixed δ ∈ [0,∞), the limitersare bounded, Lipschitz continuous and satisfies the condition φ(1) = 1, hencegives second order accuracy in the smooth region of solution. Also since thisclass of limiters lies in the proposed TVD region hence yields TVD scheme.

4 Numerical Results

In all the Numerical results we have used the limiters corresponding to δ =1 and δ = 9 for proposed scheme. We have found camparable results for limiterscorresponding to some of the other test values of δ ∈ [0,∞).

4.1 Pure Convection problem with discontinous Initial Condition [3]

One of the simplest model problems is the one-dimensional convection at con-stant velocity of an initial data with step function in the variable u withoutphysical diffusion.

ut + aux = 0, (30)

10

The analytical solution of the above equation is u = f(x − at), where f(x) isthe initial distribution of u given by

f (x) =

1, x ≤ 0.1

0, x > 0.1(31)

The solution describes a wave propagating in the positive x-direction (if a > 0)with velocity a.

Comparison of proposed scheme with some of the classical schemes and ex-isting high resolution schemes are presented. Numerical results for variousscheme are shown in fig. 2 and fig. 3 for dimensionless convection velocitya = 0.01, grid size ∆x = 0.005 for the domain x ∈ [0, 1], Courant numberc = 0.01, and c = 0.5 at dimensionless time t = 50, at this time the jump ismoving through x = 0.6 from its initial position x = 0.1. We are using theCourant number c = 0.01 for numerical comparison because our main interestis in various spatial differnce schemes for convection . Since all the camparedschemes are not second order accurate in time, so in order to avoid numericalerror due to discretization in time as far as possible, we have chosen such asmall Courant number. For Courant Number c = 0.5 comperison are givenonly with Sweby’s scheme in fig.4 as most of the higher order schemes becomeunstable and give too oscillatory results .

The classical first order Upstream scheme is diffusive and solution smeared outbecause of the inherent numerical diffusion of these schemes. For first orderschemes reducing the grid size ∆x (increasing grid number N) reduces numer-ical diffusion, but it makes using these schemes computationally expenssive.

On the other hand Standard second or higher order difference methods, e.g.,Central, Lax-Wendroff, Second order upstream, Beam-Warming, Fromm, QUICKschemes eliminate such diffusion but introduce dispersive effects that lead tounphysical oscillations in the numerical solution. The central and Lax-wendroffdifference schemes have leading odd order error term which corresponds topropagating numerical dispersion terms which corrupt large region of flowwith unphysical oscillations. Also, these are behind the advancing front andare damped with the distance to the front. The second order upstream andBeam-Warming schemes eliminate artificial diffusion, while minimizing nu-merical dispersion. Their leading truncation errors (potentially oscillatory)third order derivative terms. The damped oscillations before the advancingfronts are typical of these second order upstream difference methods. How-ever, the fourth order derivative (corresponds to dissipative error) is largeenough to dampen short-wavelength components of the dispersion to someextent. QUICK scheme has a leading fourth order derivative truncation errorterm which is dissipative, but higher order dispersion term still cause over-

11

shoots and a few oscillations when excited by nearly discontinuous behaviorof the advection variable. But, they are considerably smaller than the othersecond order schemes see fig. 2. Numerical results for Courant number C = 0.5are compared in Fig.4.

12

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I Order Upstream II-Order Upstream

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Fromm Central

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Lax-Wendroff Beam-Warming

exsol−−−−−

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Fig. 2. Numerical Results of some of the classical schemes for travelling shock prob-lem corresponding to Courant number C=0.01.

Numerical results of proposed scheme with limiter function corresponding toδ = 1 and δ = 9, are compared with Sweby’s [6] scheme

un+1k = un

k − c[

∆−unk +

1

2(1 − c)∆− (φn

k∆+unk)]

(32)

13

using Superbee, Vanleer and BW-LW(minmod) limiters.

Superbee φ(θ) = max [0, min(1, 2θ), min(θ, 2)] (33)

Vanleer φ(θ) =|θ| + θ

1 + |θ|(34)

BW-LW φ(θ) = max [0, min(θ, 1)] (35)

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Sweby’s scheme (superbee limiter) Sweby’s scheme (vanleer limiter)

exact sol .........

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Fig. 3. Numerical Results of the High Resolution schemes corresponding to Courantnumber C=0.01.

14

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Proposed scheme δ = 9

Fig. 4. Numerical Results of some of the classical and High resolution schemes forcorresponding to Courant number C=0.5.

Numerical results shows that proposed sceme gives remarkably sharp profileand better resolution near points of discontinuities, Fig. 3. and Fig. 4.

15

4.2 Pure Convection problem with smooth initial condition [14]

Consider the Eq.(30) along with the following initial condition

f (x) =

sin(10πx) 0 ≥ x ≤ 0.1

0 0.1 ≥ x ≤ 1.0(36)

with boundary conditions u(0, t) = 0 and u(1, t) = 0 The solution describesa sine wave propagating in the positive x-direction (if a > 0) with velocity a.The exact solution, upto t = 0.9/u; is

u (x, t)) =

0 0 ≤ x ≤ ut

sin[10π(x − ut)] ut < x ≤ ut + 0.1

0 ut + 0.1 < x ≤ 1.0

(37)

The sine wave propagates with no reduction in amplitude at a speed a = 0.1.The exact solution and approximate solution with C = 0.4, ∆t = 0.1, ∆x =0.025, and convection velocity a = 0.1 at time t = 8, are shown in fig. 5.

16

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2

’exact sol’

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2

’exact sol’

Lax-Wendroff I order Upwind

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2

’exact sol’

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2

’exact sol’

Sweby’s scheme (superbee limiter) Sweby’s scheme (vanleer limiter)

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2

’exact sol

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2

’exact sol’

Proposed scheme for δ = 1 Proposed scheme for δ = 9

Fig. 5. Comparision of numerical results travelling sine wave problem correspondingto Courant number C=0.4.

4.3 Propagating Temperature Front [14]

The simplest model to describe propagating temperature front is given by onedimensional linear convection dominated diffusion equation,

∂T

∂t+ a

∂T

∂x=

∂T

∂x(Γ

∂T

∂x) (38)

At t = 0 a sharp front is located at x = 0. For subsequent time the frontconvect to the right with speed a and its profile loses its sharpness under theinfluence of thermal diffusivity Γ. Consequently, for a given time t, the largerthe value of cell Reynolds number Rcell = a.∆x/Γ, the sharper the profile ofthe front.

17

The initial condition for this problem is shown in the Fig. 5.

0

0.5

1

1.5

2

2.5

3

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

Fig. 6. Initial condition for propagating temperature front

with the following boundary conditions.

T (−2, t) = 1.0, T (2, t) = 0.0 (39)

The series solution of this problem (cf.[[14], page 306]), by the separation ofvariables technique, is given by

T (x, T ) = 0.5 −2

π

N∑

k=1

sin

[

(2k − 1)π(x − at)

L

]

exp [−Γ(2k − 1)2πt/L2]

2k − 1

Numerical results for different schemes are given and compared with the exactsolution. It has been seen that for low cell Reynolds number, most of theclassical schemes give acceptable numerical solution Fig. 6. but for highercell Reynold number they exhibit oscillatory behaviour, and give unphysicalresults. Numerical results have shown that for high Reynolds number proposedscheme give comparable accuracy with Sweby’s scheme as shown in Fig. 7.

18

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

’exsol.txt’

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

’exsol.txt’

upstream Lax-Wendroff

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

’exsol.txt’

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

’exsol.txt’

Sweby’s scheme (Superbee limiter) Sweby’s scheme (Vanleer limiter)

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.2

0

0.2

0.4

0.6

0.8

1

1.2

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

’exsol’

Proposed scheme for δ = 1 Proposed scheme for δ = 9

Fig. 7. Numerical results when Rcell =3.33, ∆x = 0.1,c = 0.5, at timet = 1.0, a = 1.0,Γ = 0.03003

19

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

’exsol.txt’

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

’exsol.txt’

Upstream Lax-Wendroff

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

’exsol.txt’

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

’exsol.txt’

Sweby’s scheme (Superbee limiter) Sweby’s scheme (Vanleer limiter)

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

’exsol’

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

Proposed scheme for δ = 1 Proposed scheme for δ = 9

Fig. 8. Numerical results when Rcell =100, ∆x = 0.1,c = 0.5, at timet = 1.0, a = 1.0,Γ = 0.001

5 Concluding Remarks

The derivation of high resolution second order accurate scheme is investigatedby means of adding anti-diffusive flux to a general first order monotone scheme.Constraints on the limiters, as function of gradient ratios, have been obtainedso that the resulting scheme is TVD, and a class of limiters proposed whichsatisfy these constraints and yields a second order accurate TVD scheme forthe smooth section of solution. Numerical results for one dimensional linearcase are compared with some of the classical schemes along with the Sweby’sscheme using Superbee, Vanleer and BW-LW limiters. As the proposed schemeobeys the hyperbolicity condition hence for Hyperbolic cases it gives better ac-curacy. Numerical results have shown that in case of shocks and travelling sinewave, proposed scheme give better resolution near points of discontinuity. In

20

convectively dominated diffusion problems proposed scheme give comparableresults with other high resolution schemes.

References

[1] B. P. Leonard, A stable and accurate convective modelling procedure based onquadratic upstream interpolation, Comp. Mathods in App. Mech. and Eng., 19(1979), 59-98.

[2] B. P. Leonard, Order of accuracy of QUICK and related convection-diffusionschemes, Applied Mathematics Modelling, 1995, 19(11), 640-653.

[3] Yongqi Wang Kolumban Hutter, Comparisons of numerical methods withrespect to convectively dominated problems, Int. J. Numer. Meth. Fluids, 37(2001), 721-745.

[4] Richard E. Ewing and Hong Wang, A summary of numerical methods for time-depandent advection-dominated partial differential equations, J. of Comp. andApp. Math. 128(2001), 423-445.

[5] Thomas Y. Hou and Philippe G. Lefloch, Why Non conservative schemesconverge to wrong solutions: Error analysis, Math. Comp. Vol 62, No. 206,(1994), 497-530.

[6] P. K. Sweby, High resolution schemes using flux limiters for Hyperbolicconservation Laws, SIAM J. Numer. Anal, Vol 21, No. 5, 1984.

[7] J. P. Boris and D. L. Book, Flux corrected transport, I, SHASTA, A fluidtransport algorithm that works, J. Comp. Phys., 11 (1973), 38-69.

[8] P.L. Roe, Numerical algorithms for the linear wave equation, Royal AircraftEstablishment Technical Report 81047,1981.

[9] B. Vanleer, Towards the ultimate conservative scheme, II. Monotonicity andconservation combined in a second order scheme, J. Comp. Phys., 14 (1974),361-370.

[10] A. Harten, High resolution schemes for conservation law, J. Comp. Phys., 49(1983), 357-393.

[11] J. W. Thomas, Numerical Partial Differential Equations, conservation laws andelliptic equations, Text in Applied mathematics 33, Springer, 1999.

[12] R. J. Levaque, Numerical methods for conservation Laws, Lectures inMathematics, ETH Zurich, Birkhauser, 1999.

[13] E. F. Toro, Riemann Solvers and Numerical methods for Fluid Dynamics, Apractical introduction, 2nd edition, Springer, 1999.

[14] C. A. J. Fletcher, Computational Techniques for Fluid Dynamics, Part I,Springer Series in Computational Physics, 1988.

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[15] Serge Piperno and Sophie Depeyre, Criteria for the design of limiters yieldingefficient high resolution TVD schemes, Comp. and Fluids,vol. 27, No. 2, 1998,183-197.

[16] B. Jonathan and Randall J. Levaque, A Geometric Approach to High ResolutionSchemes, Siam J. Numer. Anal., vol. 25 No. 2, 1998, 268-284.

[17] C.W. Shu, An overview on high order numerical methods for convectiondominated PDEs, in Hyperbolic Problems: Theory, Numerics, Applications,T.Y. Hou and E. Tadmor, editors, Springer-Verlag, Berlin, 2003, pp.79-88.

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