High order finite difference WENO schemes with positivity-preserving. limiter for correlated random walk with density-dependent turning rates

Transcription

1 High order finite difference WENO schemes with positivity-preserving limiter for correlated random walk with density-dependent turning rates Yan Jiang, Chi-Wang Shu and Mengping Zhang 3 Abstract In this paper, we discuss high order finite difference weighted essentially non-oscillatory (WENO) schemes, coupled with total variation diminishing (TVD) Runge-Kutta (RK) temporal integration, for solving the semilinear hyperbolic system of a correlated random walk model describing movement of animals and cells in biology. Since the solutions to this system are non-negative, we discuss a positivity-preserving limiter without compromising accuracy. Analysis is performed to justify the maintanance of third order spatial / temporal accuracy when the limiters are applied to a third order finite difference scheme and third order TVD-RK time discretization for solving this model. Numerical results are also provided to demonstrate these methods up to fifth order accuracy. Key Words: weighted essentially non-oscillatory scheme, high order accuracy, positivitypreserving, correlated random walk Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 3, P.R. China. jiangy@mail.ustc.edu.cn. Research supported by The USTC Special Grant for Postgraduate Research, Innovation and Practice and by The CAS Special Grant for Postgraduate Research, Innovation and Practice. Division of Applied Mathematics, Brown University, Providence, RI 9, USA. shu@dam.brown.edu. Research supported by AFOSR grant F and NSF grant DMS Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 3, P.R. China. mpzhang@ustc.edu.cn. Research supported by NSFC grants 734, 93 and 945.

2 Introduction In this paper, we consider the random walk model in biology. The system is given as u t + γu = λ u + λ v, (, t) R [, T] v t γv = λ u λ v, (, t) R [, T] () u(, ) = u (), v(, ) = v (), R This model describes two kinds of particles moving in opposite directions on a line. u(, t) and v(, t) are the densities of left-moving and right-moving individuals. The particles move in a constant speed γ and change their directions with rates λ and λ. This model has been studied as the classical Goldstein-Kac theory for correlated random walk in [4, 7] when the turning rates are constants, λ = λ = µ. Since biological phenomena are complicated, the assumption of a constant speed and constant turning rates may not always be true. Often, individuals in a group change their directions when interacting with their neighbors locally or globally. These interactions can be direct through the neighbors density [,, 5, 3, ], or indirect through the chemicals produced by their neighbors []. Here, we will consider alignment, attraction and repulsion between individuals. Numerical results in [,, 3, ] demonstrate a variety of patterns by using first order upwind and second order La-Wendroff schemes. More recently, in [9], third-order positivity-preserving eplicit Runge-Kutta discontinue Galein (RKDG) methods are designed. Weighted essentially non-oscillation (WENO) scheme is another class of popular schemes for solving hyperbolic equations, which has the advantage of simplicity on uniform or smooth meshes as well as better control on spurious oscillations for discontinuous or sharp gradient solutions. In this paper, we will discuss positivity-preserving high order finite difference weighted essentially non-oscillation (WENO) schemes for the correlated random walk model with eplicit Runge-Kutta time discretization. WENO schemes are usually used to approimate hyperbolic conservation laws and the first derivative convection terms in the convection dominated partial differential equations, which give sharp, non-oscillatory discontinuity transitions and at the same time

3 provide high order accurate resolutions for the smooth part of the solution. The first WENO scheme was introduced in 994 by Liu, Osher and Chan in their pioneering paper [8], in which a third order accurate finite volume WENO scheme in one space dimension was constructed. In [], a general framewo is provided to design arbitrary order accurate finite difference WENO schemes, which are more efficient for multi-dimensional calculations. Very high order WENO schemes are documented in []. Details about the development and applications of WENO schemes can be found in [3]. Since the densities u(, t) and v(, t) in () should be positive, it is desirable to have numerical schemes also satisfy this property. Recently, Zhang et al. developed a framewo to obtain positivity-preserving finite volume and discontinuous Galein schemes which are proven to maintain the original high order accuracy of these schemes [9,,, 3]. The wo in [9] followed this approach to design positivity-preserving discontinuous Galein methods for the random walk model. Unfortunately, this framewo is not easy to be generalized to finite difference schemes. The wo in [] uses this framewo for designing positivity-preserving finite difference WENO schemes, however accuracy can be maintained only away from vacuum. On the other hand, in [5, ], Xiong et al. developed a parameter maimum principle preserving (MPP) flu limiter for finite difference WENO schemes with total variation diminishing (TVD) Runge-Kutta (RK) temporal integration, following the ideas in [7, 8]. The MPP properties of high order schemes are realized by limiting the high order flu towards a first order monotone flu, where the flu limiters are obtained by decoupling the linear, eplicit maimum principle constraints. Analysis on one-dimensional scalar conservation law was performed in [5], in which it is shown that the MPP limiter can maintain third order accuracy when applied to third order finite difference schemes with third order TVD Runge-Kutta method. In this paper, we will follow the idea in [5] to design and analyze positivitypreserving finite difference WENO schemes on the correlated random walk model, which contains global integral source terms and needs modifications to the algorithm as well 3

4 as its analysis. The rest of the paper is organized as follows. In Section, we will introduce our model. A first order upwind scheme is introduced to prove its positivity-preserving property under a suitable CFL condition. A short review of finite difference WENO schemes will be given in Section 3. In Section 4 we discuss the positivity-preserving limiter to guarantee positivity of the numerical solution. We provide analysis to verify that, when used to a third order finite difference scheme with third order TVD-RK time discretization, the limiter can keep third order accuracy under a suitable CFL condition, for both the source terms and the numerical flues. In Section 5 we present numerical results to demonstrate our numerical methods. Concluding remas are given in Section. The proof of some of the technical lemmas are given in Section 7, which serves as an appendi. The correlated random walk model In this paper, we consider the correlated random walk model in [, 9]. It is a nonlocal onedimensional hyperbolic system with a constant speed γ and density-dependent turning rate functions. The turning rate functions λ, λ are defined as follows λ = a + a f(y [u, v, ]) = a + a f() + a (f(y [u, v, ]) f()) () λ = a + a f(y [u, v, ]) = a + a f() + a (f(y [u, v, ]) f()) (3) where a, a are positive constants, a + a f() is the autonomous turning rate, and a (f(y [u, v, ]) f()) and a (f(y [u, v, ]) f()) are the bias turning rates. Here, we consider the cases with three social interactions: attraction (y,a, y,a ), repulsion 4

5 (y,r, y,r ) and alignment (y,al, y,al ). f(y) = tanh(y y ), p = u + v, y [u, v, ] = y,r [u, v, ] y,a [u, v, ] + y,al [u, v, ], y [u, v, ] = y,r [u, v, ] y,a [u, v, ] + y,al [u, v, ], y,r [u, v, ] = q r K r (s)(p( + s) p( s))ds, y,r [u, v, ] = q r K r (s)(p( s) p( + s))ds, y,a [u, v, ] = q a K a (s)(p( + s) p( s))ds, y,a [u, v, ] = q a K a (s)(p( s) p( + s))ds, y,al [u, v, ] = q al K al (s)(v( + s) u( s))ds, y,al [u, v, ] = q al K al (s)(u( s) v( + s))ds, K i (s) = πm i ep( (s s i ) /(m i)), i = r, a, al, s [, ) We will study the system () on the interval [, L] with periodic boundary conditions u(, t) = u(l, t), v(, t) = v(l, t) (4) with the solution u, v etended periodically on R with period L. We assume L > s i for i = r, al, al. Here the parameters are taken as in [3, 9], listed in Table. The following lemma is proved in [9], which shows not only the positivity-preserving property for the densities u and v of the first order upwind scheme but also the positivitypreserving property of the solution to the system () itself. Lemma [9]: If the initial conditions u (), v () are nonnegative, then the first order upwind scheme u n+ j v n+ j u n j t vj n t + γ un j un j γ vn j+ vn j = (λ ) n j u n j + (λ ) n j v n j (5) = (λ ) n j u n j (λ ) n j v n j () 5

6 Parameter Description Units Fied value γ Speed L/T No a Turning rate /T No a Turning rate /T No y Shift of the turning function q a Magnitude of attraction L/N No q al Magnitude of alignment L/N No q r Magnitude of repulsion L/N No s a Attraction range L s al Alignment range L.5 s r Repulsion range L.5 m a Width of attraction kernel L /8 m al Width of alignment kernel L.5/8 m r Width of repulsion kernel L.5/8 L Domain size L Table : List of the parameters in the model. can maintain positivity under the time step restriction t γ/ + a + a (7) where u n j and v n j are approimations to the solutions u( j, t n ) and v( j, t n ) at the grid point j = j and time level t n = n t. The turning rate functions (λ ) n j = λ (y [u n, v n, j ]) and (λ ) n j = λ (y [u n, v n, j ]) can be obtained by the rectangular rule. 3 Review of finite difference WENO schemes In this section, we briefly review finite difference WENO schemes for solving a onedimensional hyperbolic conservation law { ut + f(u) =, [a, b] u(, ) = u (8) with periodic boundary conditions. We denote the grid as with a = / < 3/ <... < N / < N+/ = b I i = [ i /, i+/ ], i = ( i / + i+/ ), = b a N.

7 On the uniform mesh, a semi-discrete conservative finite difference scheme has the following form d dt u i(t) + (Ĥi+/ Ĥi /) = (9) where u i (t) is an approimation to the point value u( i, t), and the numerical flu Ĥ i+/ = ˆf(u i p,, u i+q ) is consistent with the physical flu f(u) and is Lipschitz continuous with respect to all arguments. To achieve a high order accuracy (Ĥi+/ Ĥi /) = f(u) i + O( k ) () the scheme can use the following Lemma in [4]: Lemma [4]: If a function h() satisfies the following relationship then f(u()) = + h(ξ)dξ () ( h( + ) ) h( ) = f(u()). () Therefore, the numerical flu Ĥi+/ can be taken as h( i+/ ), which can be obtained by using a WENO reconstruction from neighboring cell averages of h(): h j = h(ξ)dξ = f(u( j, t)), I j j = i p,, i + q. For stability, it is important that upwinding is used in the construction of the flu. When f (u), a stencil with one more point from the left will be taken to reconstruct Ĥ i+/, i.e. p = q; otherwise, a stencil with one more point from the right will be used, p = q. When f (u) changes sign over the domain, a flu splitting can be applied. The simplest smooth splitting is the La-Friedrichs splitting. As an eample, we will list the procedure on the fifth order finite difference WENO scheme for (8): 7

8 . Split f(u) into two flues f + (u) and f (u) with the property f + (u)/ u and f (u)/ u. For eample, the La-Friedrichs splitting: f ± (u) = (f(u) ± αu) where α = ma u f (u) over the relevant range of u.. Identify v i = f + (u i ) and use the fifth WENO reconstruction to obtain the cell boundary values v + i+/ for all i. The upwind stencil is chosen as S = {I i,...,i i+ }, and the three small stencils are S () = {I i, I i+, I i+ }, S () = {I i, I i, I i+ }, S () = {I i, I i, I i }. On all small stencils and the big stencil we use standard reconstruction, obtaining and the linear weights which lead to v () i+/ = 3 v i + 5 v i+ v i+ v () i+/ = v i + 5 v i + 3 v i+ v () i+/ = 3 v i 7 v i + v i v big i+/ = 3 v i 3 v i + 47 v i + 9 v i+ v i+ The nonlinear weights are taken as with d = 3, d = 3 5, d = v big i+/ = d v () i+/ + d v () i+/ + d v () i+/. ω r = α r = α r s= α, r =,, s d r (β r + ǫ), r =,, Here, ǫ = is introduced to avoid the denominator to become. β r is the smooth indicators of the stencil S (r). For the fifth order WENO reconstruction, 8

9 we have β = 3 ( v i v i+ + v i+ ) + 4 (3 v i 4 v i+ + v i+ ) β = 3 ( v i v i + v i+ ) + 4 ( v i v i+ ) β = 3 ( v i v i + v i ) + 4 ( v i 4 v i + 3 v i ) Finally, the WENO reconstruction is v + i+/ = r= ω rv (r) i+/. 3. Take the positive numerical flu as ˆf + i+/ = v+ i+/. 4. Identify v i = f (u i ) and use the WENO reconstruction to obtain the cell boundary values v i+/ for all i. The upwind stencil is chosen as S = {I i,...,i i+3 } and the three small stencils are S () = {I i+, I i+, I i+3 }, S () = {I i, I i+, I i+ } and S () = {I i, I i, I i+ }. Following a mirror-symmetric (with respect to i+/) procedure we can obtain the WENO reconstruction v i+/, then we take the negative numerical flu as ˆf i+/ = v i+/. 5. Form the numerical flu as ˆf i+/ = ˆf + i+/ + ˆf i+/. For one-dimensional system of conservation laws, u(, t) = (u (, t),, u m (, t)) T is a vector, and f(u) = (f (u,, u m ),, f m (u,, u m )) T is also a vector. We could use the WENO reconstruction procedure on each component of u as in the scalar case. For our system which is diagonal, this is equivalent to the procedure of reconstruction in the local characteristic fields, which can effectively eliminate spurious oscillations when there are discontinuities in the solution. 9

10 4 Positivity-preserving limiter Here, we follow the idea of the positivity-preserving (PP) limiters in [5, ]. We describe the procedure and analysis only for the u-component of the system (). Similar results can be easily obtained for the v-component as well. We use a third order TVD Runge-Kutta time integration as an eample, u () = u n + tl(u n ) u () = 3 4 un + 4 (u() + tl(u () )) u n+ = 3 un + 3 (u() + tl(u () )). (3) Here, L(u n ) = (Ĥn j+/ Ĥn j / ) + Ĝn j, where Ĥn j+/ is the numerical flu from the WENO reconstruction based on u n, and Ĝn j is the approimation of the point value of the source term λ [u n, v n, j ]u n j + λ [u n, v n, j ]v n j. Similarly, let Ĥ() j+/ and Ĥ() j+/ be the numerical flues which are reconstructed based on u () and u (), and let Ĝ() j and Ĝ() j be the approimation of the source term point values λ [u (), v (), j ]u () j + λ [u (), v (), j ]v () j and λ [u (), v (), j ]u () j + λ [u (), v (), j ]v () j. Then the scheme (3) can be rewritten as u n+ j = u n j λ(ĥ j+/ Ĥ j / ) + tĝ j (4) where Ĥ j+/ = Ĥn j+/ + Ĥ() j+/ + 3Ĥ() j+/ Ĝ j = Ĝn j + Ĝ() j + 3Ĝ() j and λ = t/. Based on Eqn. (4), we propose to replace the numerical flu Ĥ j+/ and the source term Ĝ j by H j+/ and G j such that u n+ j = u n j λ( H j+/ H j /) + t G j (5) while attempting to maintain the original high order accuracy.

11 With the definitions of Ĥ j+/ Lemma 3: Using Taylor epansion, we can get Ĝ j and Ĝ j, we have the following results: =g[u n, v n, j ] + t ( λ,tu λ u t + λ,t v + λ v t ) n j + O( t + 3 ) () Ĥ j+/ =γun j + {( λγ)γu + λγg} n j + {γ( 3 λ γ λγ + )u + λγ( 3 λγ)( λ,u λ u + λ, v + λ v ) + 3 λ γ( λ,t u λ u t + λ,t v + λ v t )} n j + O( 3 + t 3 ) (7) where g[u, v, ] = λ [u, v, ]u(, t) + λ [u, v, ]v(, t). The proof of this lemma is given in the Appendi (Section 7). 4. PP-limiter for the source term First, we choose the time step t = CFL γ/ + a + a with CFL, such that the first order scheme u n+ j = u n j λ(γun j γun j ) + tgn j is positivity preserving by Lemma. We denote ˆf j+/ as the first order upwind numerical flu. Then we modify the source term by ũ n+ j = u n j λ( ˆf j+/ n ˆf j / n ) + t G j (8) such that ũ n+ j, where G j = r j (Ĝ j g n j ) + gn j. (9) Denote ũ j = u n j λ( ˆf j+/ n ˆf j / n ) + tĝ j, r j can be chosen to be { min( un j λ( ˆf j+/ n ˆf j / n )+ tgn j, ), if ũ r j = t(ĝ j gn j ) j <, otherwise ()

12 For this PP-limiter for the source term, we have the following theorem: Theorem : We use a third order finite difference spatial discretization and a third order RK time integration for the system. Assume the global error e n j = u( j, t n ) u n j = O( 3 + t 3 ), n, j. () Using the limiter on the source term (9), we can get with CFL. t G j Ĝ j = O( 3 + t 3 ), j () To prove Theorem, we need the following lemma as a tool, whose proof is given in the Appendi (Section 7). Lemma 4: We consider the characteristic line passing through the point (, t ) l : l = γ(t t ) + (3) We also define w (t;, t ) = u(γ(t t ) +, t) and w (t;, t ) = v(γ(t t ) +, t). Then we have the conclusion w (t;, t ) = u(, t ) + (t t ) g[u(, t ), v(, t ), ] + (t t ) { λ,t u λ u t + λ,t v + λ v t + γ( λ, u λ u + λ, v + λ v )} (,t ) + O((t t ) 3 ) (4) Proof of Theorem : If r j =, the limiter does not take effect. So we just need to consider the case This implies r j = un j λ( ˆf j+/ n ˆf j / n ) + tgn j t(ĝ j gj n) <. u n j λ( ˆf n j+/ ˆf n j / ) + tĝ j <

13 G j = un j λ( ˆf j+/ n ˆf j / n ) t t(ĝ j G j ) = u n j λ( ˆf n j+/ ˆf n j /) + tĝ j. Since u n j λ( ˆf n j+/ ˆf n j / ) + tĝ j <, to obtain (), it is sufficient to show that Lemma 3 tells us that Ĝ j u n j λ( ˆf n j+/ ˆf n j / ) + tĝ j = O( 3 + t 3 ). = g n j + t ( λ,tu λ u t + λ,t v + λ v t ) n j + O( t ) Since the first order upwind numerical flu is ˆf n j+/ = γun j, we have t(ĝ j G j ) = ( λγ)un j + λγun j + tgn j + t ( λ,tu λ u t + λ,t v + λ v t ) n j + O( t3 ) To simplify the notations, we use u j to denote u n j and u() to denote u(, tn ). From our assumption (), the difference between u( j, t n ) and u n j is of high order (third order). In our proof below, we use u( j, t n ) and u n j interchangeably when such high order difference allows. Denote I j to be the local minimum point in cell I j. We can epand t(ĝ j G j ) at. Denote u = u( ), and z = ( j )/. Thus t(ĝ j G j ) = u + {(z λγ)u + λg} + { (z λγz + λγ)u + λz( λ, u λ u + λ, v + λ v ) + λ ( λ,t u λ u t + λ,t v + λ v t )} + O( 3 + t 3 ) where all unspecified u, g and their derivatives are values at the location. Then We first consider the case ( j /, j+/ ), with u =, u and z (, ). t(ĝ j G j ) =u + tg + t { λ,t u λ u t + λ,t v + λ v t + γ( λ, u λ u + λ, v + λ v )} 3

14 + (z λγz + λγ)u + t (z λγ)( λ,u λ u + λ, v + λ v ) + O( 3 + t 3 ) =u( + (z λγ) ) + tg[u, v, + (z λγ) ] + t { λ,t u λ u t + λ,t v + λ v t + γ(λ, u λ u + λ, v + λ v )} +(z )λγ + ( λγz + λγ 4 λ γ )u + O( 3 + t 3 ) We have λγ = t γ = when CFL. Since z (, ), we can get CFL γ γ + (a + a ) λγz + λγ 4 λ γ λγ 4 λ γ i.e. ( λγz + λγ 4 λ γ )u. For the first three terms, they are an approimation of w (t n + t; + (z λγ), tn ) with third order accuracy O( 3 + t 3 ). In summary, t(ĝ j G j ) is equal to some non-negative term within O( 3 + t 3 ). In the case u() reaches its local minimum = j /, we have u and z = ( j )/ = /. Thus t(ĝ j G j ) =u j / + {( λγ)u + λg} + { 8 u + λ( λ,u λ u + λ, v + λ v ) + λ ( λ,t u λ u t + λ,t v + λ v t )} + O( 3 + t 3 ) =u( j / + ( λγ) ) + tg[u, v, j / + ( λγ) ] + t { λ,t u λ u t + λ,t v + λ v t + γ( λ, u λ u + λ, u + λ u )} j / + ( λγ) + ( λγ)u ( j / ) + 8 λγ( λγ)u ( j / ) 4

15 + O( 3 + t 3 ) Let s = λγ, then t(ĝ j G j ) = I + II + O( 3 + t 3 ) where I =u( j / + ( λγ) s ) + t { λ [u, v, j / + ( λγ) ] u( j / + ( λγ) s ) + λ [u, v, j / + ( λγ) ] v( j / + ( λγ) )} + t { λ,t [u, v, j / + ( λγ) ] u( j / + ( λγ) s ) λ [u, v, j / + ( λγ) ] u t( j / + ( λγ) s ) + λ,t [u, v, j / + ( λγ) ] v( j / + ( λγ) ) + λ [u, v, j / + ( λγ) ] v t( j / + ( λγ) ) + γ( λ, [u, v, j / + ( λγ) ] u( j / + ( λγ) s ) λ [u, v, j / + ( λγ) ] u ( j / + ( λγ) s ) + λ, [u, v, j / + ( λγ) ] v( j / + ( λγ) ) + λ [u, v, j / + ( λγ) ] v ( j / + ( λγ) ))} II = ( λγ)u ( j / ) + t( λγ)( λ [u, v, j / + ( λγ) ]) u ( j / ) We define ũ(, t) = u( s, t), ṽ(, t) = v(, t) g = λ [ũ( + s, t), ṽ(, t), ]ũ(, t) + λ [ũ( + s, t), ṽ(, t), ]ṽ(, t) For the new system ũ t + γũ = g, (, t) R [t n, T] ṽ t γṽ = g, (, t) R [t n, T] ũ(, t n ) = u( s, t n ), ṽ(, t n ) = v(, t n ) 5

16 using the similar idea of Lemma, we can prove that ũ and ṽ satisfy positive preserving principle. Considering the value along the characteristic line l : l = γ(t t n ) + j / + ( λγ) then we can see that I approimates ũ(γ t + j / + ( λγ), tn + t) with O( 3 + t 3 ). Since λγ + ( λγ) t ( λ [u, v, j / + ( λγ) ]) λγ + ( λγ) CFL ( a a ) γ/ + a + a = [( λγ) γ γ/ + a + a + (a + a )( λγ CFL( λγ))] [ γ γ/ + a + a λγ( γ + a + a )] γ = [ γ/ + a + a CFL ] when CFL, we have II. In summary, when loc = j /, we can also conclude that t(ĝ j to some non-negative term within O( 3 + t 3 ). G j ) is equal If = j+/, then u ( j+/ ) and z = ( j )/ = /. Thus t(ĝ j G j ) =u j / + (( λγ)u + λg) + { ( 4 + λγ)u λ( λ,u λ u + λ, v + λ v ) + λ ( λ,t u λ u t + λ,t v + λ v t )} + O( 3 + t 3 ) =u( j+/ ( + λγ) ) + tg[u, v, j+/ ( + λγ) ] + t { λ,t u λ u t + λ,t v + λ v t + γ( λ, u λ u + λ, v + λ v )} j+/ (+λγ) + ( λγ)u ( j+/ ) + 8 λγ( λγ)u ( j+/ ) + O( 3 + t 3 )

17 Denote s = +λγ+ +8λγ, then t(ĝ j G j ) = I + II + O( 3 + t 3 ) where I =u( j+/ ( + λγ) s ) + t{ λ [u, v, j+/ ( + λγ) ] u( j+/ ( + λγ) s ) + λ [u, v, j+/ ( + λγ) ] v( j+/ ( + λγ) )} + t { λ,t [u, v, j+/ ( + λγ) ] u( j+/ ( + λγ) s ) λ [u, v, j+/ ( + λγ) ] u t( j+/ ( + λγ) s ) + λ,t [u, v, j+/ ( + λγ) ] v( j+/ ( + λγ) ) + λ [u, v, j+/ ( + λγ) ] v t( j+/ ( + λγ) ) + γ( λ, [u, v, j+/ ( + λγ) ] u( j+/ ( + λγ) s ) λ [u, v, j+/ ( + λγ) ] u ( j+/ ( + λγ) s ) + λ, [u, v, j+/ ( + λγ) ] v( j+/ ( + λγ) ) + λ [u, v, j+/ ( + λγ) ] v ( j+/ ( + λγ) ))} II = ( λγ + s)u ( j+/ ) + ts ( λ [u, v, j+/ ( + λγ) ] )u ( j+/ ) Similar to the case of = j /, we can show that I equals to some non-negative term within O( 3 + t 3 ). For II, since λγ + s λ s t λγ + s( (a + a ) t) and (a + a ) t = γ/ + ( CFL)(a + a ) γ/ + a + a 7

18 if CFL, we can obtain that λγ + s λ s t Hence, we also can get II. For all the cases, we can see that t(ĝ j plus O( 3 + t 3 ). Since we know t(ĝ j G j ) equals to some non-negative term G j ), we have t(ĝ j G j ) = O( 3 + t 3 ). 4. PP-limiter for the numerical flu We would like to modify the numerical flu H j+/ = θ j+/(ĥ j+/ ˆf n j+/ ) + ˆf n j+/ (5) such that u n j λ( H j+/ We denote Γ j = u n j + λ( ˆf j+/ n ˆf j / n ) t G j inequality () can be rewritten as H j /) + t G j () and F j+/ = Ĥ j+/ ˆf n j+/. Thus, λθ j / ˆFj / λθ j+/ ˆFj+/ Γ j (7) with Γ j. We need to find a pair (Λ /,Ij, Λ +/,Ij ) such that any pair (θ j /, θ j+/ ) [, Λ /,Ij ] [, Λ +/,Ij ] would satisfy (7).. If F j / and F j+/, the pair would be (Λ /,Ij, Λ +/,Ij ) = (, );. If F j / and F j+/ >, the pair can be given as (Λ /,Ij, Λ +/,Ij ) = (, min(, Γ j λf j+/ )); 3. If F j / < and F j+/, the pair can be given as (Λ /,Ij, Λ +/,Ij ) = (min(, Γ j λf j / ), ); 8

19 4. If F j / < and F j+/ >, when (θ j /, θ j+/ ) = (, ) satisfies (7), the pair can be given as (Λ /,Ij, Λ +/,Ij ) = (, ). However, in the case that the pair (Λ /,Ij, Λ +/,Ij ) = (, ) does not satisfy (7), intersection is given as the pair (Λ /,Ij, Λ +/,Ij ) = ( Then the new flu will be defined as Γ j Γ λf j / λf j+/, j λf j / λf j+/ ). H j+/ = Λ j+/ (Ĥ j+/ ˆf n j+/) + ˆf n j+/ (8) Λ j+/ = min(λ +/,Ij, Λ /,Ij+ ) It is easy to check that the new flu defined as (8) can satisfy (). Theorem : We use a third order finite difference spatial discretization and a third order RK time discretization for the system. Assume the global error e n j = u( j, t n ) u n j = O( 3 + t 3 ), n, j. (9) Then using the limiter (8), we can get with CFL. Ĥ j+/ H j+/ = O( 3 + t 3 ), j (3) Proof: Let us look at the four cases for the choice of (Λ /,Ij, Λ +/,Ij ). For Case, the limiter does not take effect. For Case 4, we only need to consider the situation when Λ +/,Ij = i.e. Γ j > λf j / λf j+/. Γ j λf j / λf j+/ <, H j+/ Ĥ j+/ =Λ +/,I j (Ĥ j+/ ˆf n j+/ ) + ˆf n j+/ Ĥ j+/ =(Λ +/,Ij )(Ĥ j+/ ˆf n j+/) Γ j =( λf j / λf )(Ĥ j+/ ˆf j+/) n j+/ = λf j / λf j+/ Γ j F j+/ λf j+/ λf j / 9

20 It is sufficient to show that λf j / λf j+/ Γ j λf j+/ λf j / F j+/ = O( 3 + t 3 ). Because Γ j > λf j / λf j+/, which means λf j / λf j+/ Γ j = u n j λ(ĥ j+/ Ĥ j / ) + t G j < u( j, t n+ ), and u n j λ(ĥ j+/ Ĥ j / ) + t G j approimates u( j, t n+ ) within third order accuracy, we can get λf j / λf j+/ Γ j = O( 3 + t 3 ). Since < F j+/ λf j+/ λf j /, we have λ λf j / λf j+/ Γ j λf j+/ λf j / F j+/ = O( 3 + t 3 ) For Case, (similarly for Case 3), we only need to consider the situation when with Since Λ +,I j = Γ j λf j+/ < H j+/ Ĥ j+/ = un j λĥ j+/ + λ ˆf j / n + t G j λ Γ j λf j+/ <, which equals to u j λĥ j+/ + λ ˆf j / n + t G j <, it is sufficient to verify that u n j λĥ j+/ + λ ˆf j / n + t G j = O( 3 + t 3 ) We have showed that t prove that G j Ĝ j = O( 3 + t 3 ) in Theorem, so we can alternatively u n j λĥ j+/ + λ ˆf n j / + tĝ j = O( 3 + t 3 ) Using the results in Lemma 3 and ˆf j / n = γun j, we can get u n j λĥ j+/ + λ ˆf j / n + tĝ j =u + {(z 3 λγ + λ γ )u + λ( λγ)g} + {z + λγ(λγ 3)z 3 λ3 γ 3 + λ γ + 5 λγ}u + t {( λγ)z + λ γ 4 λγ}( λ,u λ u + λ, v + λ v )

21 + t ( 3 λγ)( λ,tu λ u t + λ,t v + λ v t ) + O( 3 + t 3 ) Similar to the procedure in the proof of Theorem, all unmaed values of u, g and their derivatives are located at (, t n ), where is the local minimum point of u(, t n ) in I j and z = ( j )/. If ( j /, j+/ ), then u =, u, u n j λĥ j+/ + λ ˆf j / n + tĝ j =u + t( λγ)g + t ( 3 λγ){ λ,tu λ u t + λ,t v + λ v t + γ( λ, u λ u + λ, v + λ v )} + {z + λγ(λγ 3)z 3 λ3 γ 3 + λ γ + 5 λγ} u + t {( λγ)z + 3 λ γ 3 4 λγ} ( λ,u λ u + λ, v + λ v ) + O( 3 + t 3 ) = ( λγ) λγ {u + λγ 3λγ tg + λγ ( 3λγ t) [λ,t u λ u t + λ,t v + λ v t 3 + γ( λ, u λ u + λ, v + λ v )]} + λγ( 8 + 3λγ) 4( 3 + λγ) u + {z + λγ(λγ 3)z 3 λ3 γ 3 + λ γ + 5 λγ} u + t {( λγ)z + 3 λ γ 3 4 λγ} ( λ,u λ u + λ, v + λ v ) + O( 3 + t 3 ) Denote s = z + 4λ γ 9λγ, then λγ u n j λĥ j+/ + λ ˆf j / n + tĝ j = ( λγ) 3 λγ {u( + s ) + λγ 3λγ tg[u, v, + s ] + λγ ( 3λγ t) [ λ,t u λ u t + λ,t v + λ v t + γ( λ, u λ u + λ, v + λ v )] +s } λγ( 8 + 3λγ) + 4( 3 + λγ) u

22 + u {z + λ γ z 3λγz 3 λ3 γ 3 + λ γ + 5 λγ ( λγ) 3 λγ s } + O( 3 + t 3 ) Using Lemma 3, the first term is a third order approimation of ( λγ) w (t n + λγ t; + 3 λγ 3λγ s, t n ). We can check that z + λ γ z 3λγz 3 λ3 γ 3 + λ γ + 5 λγ ( λγ) 3 λγ s when z (, ) and λγ [, ]. t 3 ). So, u n j λĥ j+/ + λ ˆf n j / + tĝ j approimates some non-negative term O( 3 + If = j /, then u, and z =. u n j λĥ j+/ + λ ˆf j / n + tĝ j =u j / + ( λγ) tg + ( 3 λγ) t { λ,t u λ u t + λ,t v + λ v t + γ( λ, u λ u + λ, v + λ v )} + ( 3 λγ + λ γ )u + t ( λγ + 3 λ γ )( λ, u λ u + λ, v + λ v ) Denote s = s 3 = + ( 3 λ3 γ 3 + λ γ 3 λγ + 4 )u + O( 3 + t 3 ) λγ( 8 + 3λγ) Then the equation is where s = λγ + 3 λ γ λγ 4( 3 + λγ) [ 3 λ3 γ 3 + λ γ 3 λγ + 4 ( λγ) λγ (s ) ] 3 4( 3 + λγ) λγ( 8 + 3λγ) [ 3 λ3 γ 3 + λ γ 3 λγ + 4 ( λγ) 3 λγ (s ) ] u n j λĥ j+/ + λ ˆf n j / + tĝ j = I + II + III + O( 3 + t 3 ) I = ( λγ) 3 λγ {u( j / + s )

23 + λγ 3 λγ t [ λ [u, v, j / + s ] u( j + s ) + λ [u, v, j / + s ] v( j + s )] + ( λγ 3 ) t [ λ,t [u, v, j / + s ] u( j + s ) λ [u, v, j / + s ] u t ( j + s ) + λ,t [u, v, j / + s ] v( j + s ) + λ [u, v, j / + s ] v t ( j + s ) + γ( λ, [u, v, j / + s ] u( j + s ) λ [u, v, j / + s ] u ( j + s ) + λ, [u, v, j / + s ] v( j + s ) + λ [u, v, j / + s ] v ( j + s ))]} λγ( 8 + 3λγ) II = 4( 3 + λγ) u( j / s 3 ) + ( λγ)( λ)λ [u, v, j / + s ]u ( j / ) III = { 3 λγ + λ γ ( λγ) 3 λγ (s ) + s }u ( j / ) We can check that 3 λγ + λ γ ( λγ) 3 λγ (s λγ) + s So III. Considering the value along characteristic lines, we can see that I + O( 3 + t 3 ). Since λγ( 8+3λγ) 4( 3+λγ), all we need to consider is u( j / s 3 t) + σu ( j / ), where σ = λλ ( 4( 3+λγ) λγ). It is sufficient to prove that u( λγ( 3+8λγ) j / s 3 t) + σu ( j / ) or u ( j / ) = O( ). If u is not monotone in [ j / s 3 3

24 , j / s 3 ], then there eists, [ j / s 3, j / s 3 ] such that u (, ) =, and u ( j / ) = O( ). If u is monotonically decreasing in [ j / s 3, j / s 3 ], then there eists, [ j / s 3, j / ] such that u (, ) =, and u ( j / ) = O( ). If u is monotonically increasing in [ j / s 3, j / s 3 ], assume u( j / s 3 ) C <, where C = σ. There eists,3 [ j / s 3, j / s 3 ] such that u( j / s 3 ) = u( j / s 3 ) + u (,3 ). where u (,3 ). Therefore u (,3 ) C u ( j / ), i.e. u (,3 ) = O( ). Thus we can get u ( j / ) = O( ), which implies now II + O( 3 + t 3 ). In the case of = j+/,we can get similar results. The proof is now complete. 5 Numerical eamples In this section, we will present some numerical results using the schemes discussed above. We will use the third order finite difference WENO scheme and the fifth order finite difference WENO scheme in space, denoted as WENO-3 and WENO-5. Both schemes are combined with the third order TVD Runge-Kutta temporal integration. Without special declaration, the time step is chosen as t = CFL γ/ + a + a (3) We take CFL =. in the numerical tests, such that the PP-limiter will not destroy accuracy when. For the infinite integrals, based on the fact that s i K i (s)ds 5, i = r, a, al, we can just compute the integrals on the compact interval [, s i ]. Here, the rectangular rule is used, which is the most accurate rule for compactly supported smooth functions. 4

25 To avoid the effect of rounding error, we change the condition (8) and () into u n j λ( ˆf j+/ n ˆf j / n ) + t G j ǫ (3) u n j λ( H j+/ H j / ) + t G j ǫ (33) where ǫ =. Eample. We first test the accuracy with constant coefficients λ = λ =.5. The initial condition is u(, ) = { ( ( 4) 4 ), 4, 4 <. (34) {, < v(, ) = ( ( )( ) ),. 4 (35) with -periodic boundary condition. Clearly u(, ), v(, ) C 5 (R). The final time is T = 5. Since the eact solution is not available in closed form, we obtain an accurate reference solution by using the spectral method with,8 grid points to serve as the eact reference solution. Table : Eample : Constant coefficients. u-component. WENO-3. Without the PP-limiter. n L error order L error order L error order min value 4.3E E-3.385E E-5 8.3E E E E-5.595E-3.5.9E E E E E E E E E E E-8 8.E E E E- We test the WENO-3 scheme with the time step (3) without the PP-limiter and with it. The WENO-5 scheme without the PP-limiter and with it are also tested, with time step t = CFL (γ/ ) 5/3 + a + a (3) 5

26 Table 3: Eample : Constant coefficients. u-component. WENO-3. With the PPlimiter. n L error order L error order L error order min value 4.3E E-3.385E-3.E- 8.3E E E E-.595E E E-4.94.E E E E E E E E E- 8.E E E E- Table 4: Eample : Constant coefficients. u-component. WENO-5. Without the PP-limiter. n L error order L error order L error order min value 4.E-3 5.4E-4.4E-4 -.9E E E E E- 7.43E E E E E E E E- 4.5E E E E E E E E-3 which is designed to match the temporal and spatial orders of accuracy. Errors and orders of accuracy of the u-component are shown in Tables -5. The minimum values of numerical solutions at the final time are also listed. It is clear that the PP-limiter can strictly keep the solutions non-negative without loss of accuracy. Eample. Net, we test the accuracy for system () with variable coefficients. We choose the parameters as γ =., a =., a =.9, Table 5: Eample : Constant coefficients. u-component. WENO-5. With the PPlimiter. n L error order L error order L error order min value 4.E-3 5.E-4.4E-4.E E E E-5 4..E- 7.43E E E E E E E E- 4.5E E E E E E E E-

27 q a =., q al =., q r =.5. The same initial conditions are used as in Eample (34)-(35). Table : Eample : Variable coefficients. u-component. WENO-3. Without the PPlimiter. n L error order L error order L error order min value E-.487E-.4E- -.9E E E E E-5 5.4E E E E-5 3.7E E E E E E E E-8 8.7E E E E-9 Table 7: Eample : Variable coefficients. u-component. WENO-3. With the PPlimiter. n L error order L error order L error order min value E-.48E-.4E-.E- 8.43E E E E- 5.4E E E-4..E- 3.7E E E-4.3.E E E E-5.83.E- 8.7E E E E- Table 8: Eample : Variable coefficients. u-component. WENO-5. Without the PPlimiter. n L error order L error order L error order min value E-.445E-.4E E- 8.93E E E E-.74E E E E E E E E E E E E E E E E- In Tables -9, we list the order of accuracy and minimum values of the u-component at the final time T = 5 of WENO-3 with the time step (3) and WENO-5 with the time step (3). They show that our schemes can achieve the designed order of accuracy and the PP-limiter can keep positivity without destroying accuracy. 7

28 Table 9: Eample : Variable coefficients. u-component. WENO-5. With the PPlimiter. n L error order L error order L error order min value E-.445E-.4E-.E- 8.93E E E E-.74E E E E E E E-7 9..E- 4.57E E E E E E E- 5..E- Eample 3. In this eample, we will test the eample of stationary pulses as in [9]. Suppose (u, v ) is the homogeneous steady state with u + v = A, where A is the total population density. When q al =, we have only one steady state (u, v = (A/, A/)). However, when q al, the system can have one, three or five solutions, and these solutions are obtained by the steady equation from (), u (a + a f(aq al u q al y)) + (A u )(a + a f( Aq al + u q al y ) =. We choose the parameters as γ =., a =., a =.9, q a =, q al =, q r =.4, also we choose A =, which means that the steady state is (u, v ) = (, ). The initial conditions are taken as a small perturbation on this steady state { u(, ) = +. sin(.π) v(, ) = +.5 cos(.π) (37) The solution evolves into stationary pulses. In Figure, we plot the numerical solutions of the total density p = u + v from t = 5 to t =, using the first order upwind scheme, the third order WENO-3 scheme and the fifth order WENO-5 scheme with n = 5 grid points. We also plot the solution obtained with the first order upwind scheme using grid points as a converged reference solution. The numerical solutions are stationary for all the schemes. 8

29 All schemes converge to the reference solution well, which can be seen in Figure for the cuts of p = u + v at the final time t =. In Figure (b), we can see that the higher order schemes produce results closer to the reference solution. Converged solutions in Figure 3(d) show that u and v almost overlap with each other, and numerical solutions u and v generated by the WENO-3 scheme and the WENO-5 scheme also overlap with each other, while u and v generated by the upwind scheme still show a slight translation. V3 V3 9 8 time time (a) Upwind scheme. 5 grid points (b) WENO-3. 5 grid points V3 V3 9 8 time time (c) WENO-5. 5 grid points (d) Converged solution Figure : Eample 3: Stationary pulses. u + v from t = 5 to t =. Eample 4. In this eample, we consider the traveling pulses problem. We choose the 9

30 4 Converged solution Upwind scheme WENO-3 WENO-5 4 Converged solution Upwind scheme WENO-3 WENO (a) (b) Figure : Eample 3: Stationary pulses. Cut of p = u + v at time T =. Figure (b) is the enlarged view of Figure (a) parameters as γ =., a =., a =.9, q a =., q al =, q r =.5. The initial condition is the same as in Eample 3. Here, we use grid points for the WENO-3 scheme, the WENO-5 scheme and the first order upwind scheme. Also, the numerical solution using the upwind scheme with grid point is taken as the converged reference solution. In Figure 4, we plot the total density p = u + v from time t = 5 to t =. The numerical solutions are traveling for all schemes. In Figure 5, we give a cut of p = u + v at the final time t =, and we can see that the higher order schemes produce results closer to the converged reference solution. Eample 5. In this eample, we test the system () with discontinuous initial conditions u(, ) = {, 4, 4 < <. (38) v(, ) = {, < <,. (39) 3

31 (a) Upwind scheme. 5 grid points (b) WENO-3. 5 grid points (c) WENO-5. 5 grid points (d) Converged solution Figure 3: Eample 3: Stationary pulses. u and v at the final time t =. The solid lines are the u-component and the dash lines are the v-component. The small figures are the enlarged view inside the rectangles. We choose the parameters as γ =., a =., a =.9, q a =., q al =., q r =.5. In Figure, we plot the numerical solutions u at time t =, with the first order upwind scheme, the WENO-3 scheme with and without the PP-limiter, and the WENO-5 scheme with and without the PP-limiter. We also test the third order finite difference (FD-3) scheme without the PP-limiter, which is the third order finite difference WENO 3

32 time 8 7 V time 9 V (a) Upwind scheme. grid points 8 (b) WENO-3. grid points time 8 7 V3 V time (c) WENO-5. grid points (d) Converged solution Figure 4: Eample 4: Traveling pulses. u + v from t = 5 to t =. scheme using the linear weights. Numerical solution obtained with the first order upwind scheme using grid points are used as a converged reference solution. Minimum values of the u-component at t = of the WENO-3 scheme and the WENO-5 scheme are shown in Table. We can see that without the nonlinear weights, the FD-3 scheme has oscillations near the interfaces. Comparing Figure (c), Figure (d), Figure (e) and Figure (f), we can see that the PP-limiter does not affect the non-oscillatory discontinuity transitions of WENO schemes when maintaining positivity. 3

33 8 Converged solution Upwind scheme WENO-3 WENO Converged solution WENO-3 WENO (a) (b) Figure 5: Eample 4: Traveling pulses. Cut of p = u + v at time T =. Fig.5(b) is the enlarged view inside the rectangle in fig.5(a) Table : Eample 5: Minimum values of the u-component at t =. Conclusion with the PP-limiter without the PP-limiter WENO-3.E E-8 WENO-5.E- -.34E-7 In this paper, we discuss high order finite difference WENO schemes coupled with total variation diminishing (TVD) Runge-Kutta (RK) temporal integration for a nonlocal hyperbolic system of a correlated random walk model. A positivity-preserving limiter is introduced to guarantee positivity of the solution. Analysis is given to show that when the limiter is applied to a third order finite difference scheme with third order TVD-RK time discretization solving this model, the scheme can maintain third order accuracy for both the source and the numerical flues, under the standard CFL condition. Numerical results are provided to demonstrate these methods up to fifth order accuracy. 7 Appendi We will give the proof of some of the technical lemmas in this section as an appendi. 33

34 (a) Upwind scheme. points grid (b) FD-3. grid points (c) WENO-3 with the PPlimiter. grid points (d) WENO-3 without the PPlimiter. grid points (e) WENO-5 with the PPlimiter. grid points (f) WENO-5 without the PPlimiter. grid points Figure : Eample 5: The u-component at t =. The solid lines are converged solutions and the symbols are numerical solutions. The small figures are the enlarged view inside the rectangles. 7. The proof of Lemma 3 Suppose u n (), v n () is the solution at time t n. With the third order TVD Runge-Kutta u () = u n + tl(u n ) we define the functions ũ () and ũ () as Since u () = 3 4 un + 4 u() + 4 tl(u() ) u n+ = 3 un + 3 u() + 3 tl(u() ) (4) ũ () () = u n () + t( γu n + g[un, v n, ]) ũ () () = 3 4 un () + 4ũ() () + 4 t( γũ() + g[ũ(), ṽ (), ]) (4) u () j = u n j + t( Ĥn j+/ Ĥn j / + g[u n, v n, j ]) 34

35 u () j = 3 4 un j + 4 u() j + t( Ĥ() j+/ Ĥ() j / + g[u (), u (), j ]) (4) 4 where Ĥn j+/ and Ĥ() j+/ and u (). Hence we can get are the numerical flues with third order accuracy w.r.t. un u () j = ũ () ( j ) + O( t 3 ) u () j = ũ () ( j ) + O( t 3 ) We also have similar definitions and results for the v-component. In the proof below, all unmaed quantities are evaluated at time level n. Since ũ () = u n + t( γu + g[u, v, ]) ṽ () = v n + t(γv g[u, v, ]) we have y [ũ (), ṽ (), ] =y [u + t( γu + g), v + t(γv g), ] =y [u, v, ] + ty [ γu + g, γv g, ] y [ũ (), ṽ (), ] =y [u + t( γu + g), v + t(γv g), ] =y [u, v, ] + ty [ γu + g, γv g, ] and λ [ũ (), ṽ (), ] =a + a f(y [ũ (), ṽ (), ]) =a + a f(y [u, v, ]) + a tf (y [u, v, ])y [ γu + g, γv g, ] + O( t ) =λ + a tf (y [u, v, ])y [ γu + g, γv g, ] + O( t ) λ [ũ (), ṽ (), ] =a + a f(y [ũ (), ṽ (), ]) =a + a f(y [u, v, ]) + a tf (y [u, v, ])y [ γu + g, γv g, ] + O( t ) =λ + a tf (y [u, v, ])y [ γu + g, γv g, ] + O( t ) 35

36 Hence, we can get g[ũ (), ṽ (), ] = λ [ũ (), ṽ (), ]ũ () + λ [ũ (), ṽ (), ]ṽ () (43) = λ u + λ v + t{ a f (y [u, v, ])y [ γu + g, γv g, ]u λ ( γu + g) + a f (y [u, v, ])y [ γu + g, γv g, ]v + λ (γv g)} + O( t ) (44) With the definitions, we obtain ũ () and ṽ () : ũ () = 3 4 u j + 4ũ() j + 4 t( γũ() =u n + t( γu + g) + g[ũ (), ṽ (), ]) + 4 t {γ u γ( λ, u λ u + λ, v + λ v ) a f (y [u, v, ])y [ γu + g, γv g, ]u λ ( γu + g) + a f (y [u, v, ])y [ γu + g, γv g, ]v + λ (γv g)} + O( t 3 ) ṽ () = 3 4 v j + 4ṽ() j =v + t(γv g) + 4 t(γṽ() g[ũ (), ṽ (), ]) + 4 t {γ v γ( λ, u λ u + λ, v + λ v ) + a f (y [u, v, ])y [ γu + g, γv g, ]u + λ ( γu + g) a f (y [u, v, ])y [ γu + g, γv g, ]v λ (γv g)} + O( t 3 ) And we repeat the procedure of g[ũ (), ṽ (), ] to get g[ũ (), ṽ (), ], therefore g[ũ (), ṽ (), ] = λ u + λ v + t{ a f (y [u, v, ]) y [ γu + g, γv g, ]u Since λ ( γu + g) + a f (y [u, v, ]) y [ γu + g, γv g, ]v + λ (γv g)} + O( t ) (45) y [ γu + g, γv g, ] = y [u t, v t, ] = y,t [u, v, ] 3

37 we write the above epressions as y [ γu + g, γv g, ] = y [u t, v t, ] = y,t [u, v, ] g[ũ (), ṽ (), ] = g + t( λ,t u λ u t + λ,t v + λ v t ) + O( t ) (4) g[ũ (), ṽ (), ] = g + t( λ,tu λ u t + λ,t v + λ v t ) + O( t ) (47) ũ () = u + t( γu + g) + 4 t {γ u γ( λ, u λ u + λ, v + λ v ) + ( λ,t u λ u t + λ,t v + λ v t )} + O( t 3 ) (48) ṽ () = v + t(γu g) + 4 t {γ v γ( λ, u λ u + λ, v + λ v ) ( λ,t u λ u t + λ,t v + λ v t )} + O( t 3 ) (49) Hence Ĝ j = g[un, v n, j ] + g[ũ(), ṽ (), j ] + 3 g[ũ(), ṽ (), j ] + O( 3 ) =g j + t( λ,tu λ u t + λ,t v + λ v t ) + O( t + 3 ) (5) If we use a third order finite difference scheme for the spatial discretization, we have Ĥ j+/ = 3 f j+/ 4 f j / 4 f j+3/ + O( 3 ) So if we epand H j+/ at j for the u-component with f(u) = γu, we can get Ĥ j+/ = (3 f(un ) j+/ 4 f(un ) j / 4 f(un ) j+3/ ) + (3 f(ũ() ) j+/ 4 f(ũ() ) j / 4 f(ũ() ) j+3/ ) + 3 (3 f(ũ() ) j+/ 4 f(ũ() ) j / 4 f(ũ() ) j+3/ ) + O( 3 ) =γu j + ( ( λγ)γu + λγg) + { ( 3 γ λ λγ + )u + ( 3 λγ + )( λ,u λ u + λ, v + λ v ) + λ γ( λ,t u λ u t + λ,t v + λ v t )} + O( 3 + t 3 ) (5) 37

38 7. The proof of Lemma 4 We will find the u-component values along the characteristic line l : l = γ(t t ) + We denote w l (t;, t ) as w l (t), l =,. Then we can get dw dt dw dt = γu + u t = g = γv + v t = γv g Hence, w (t) = w (t ) + t t g[u(, τ), v(, τ), γ(τ t ) + ]dτ Because w (τ) =w (t ) + (τ t ) dw dt (t ) + O((τ t ) ) =w (t ) + (τ t )(u t + γu ) (,t ) + O((τ t ) ) w (τ) =w (t ) + (τ t ) dw dt (t ) + O((τ t ) ) p(γ(τ t ) +, τ) =w (τ) + w (τ) =w (t ) + (τ t )(v t + γv ) (,t ) + O((τ t ) ) =w (t ) + w (t ) + (τ t )(u t + v t + γ(u + v )) (,t ) + O((τ t ) ) =p(, t ) + (τ t )(p t (, t ) + γp (, t )) + O((τ t ) we can get q r K r (s) [p(γ(τ t ) + + s, τ) p(γ(τ t ) + s, τ)]ds =q r K r (s) [p( + s, t ) + (τ t )(p t ( + s, t ) + γp ( + s, t )) + O((τ t ) ) p( s, t ) (τ t )(p t ( s, t ) + γp ( s, t )) + O((τ t ) )]ds =q r K r (s) [p( + s, t ) p( s, t )]ds 38

39 + (τ t )q r K r (s) [p t ( + s, t ) p t ( s, t )]ds + γ(τ t )q r K r (s) [p ( + s, t ) p ( s, t )]ds + O((τ t ) ) Thus y [u(, τ), v(, τ), γ(τ t ) + ] =q r K r (s)[p(γ(τ t ) + + s, τ) p(γ(τ t ) + s, τ)]ds q a K a (s)[p(γ(τ t ) + + s, τ) p(γ(τ t ) + s, τ)]ds + q al K al (s)[v(γ(τ t ) + + s, τ) u(γ(τ t ) + s, τ)]ds =y [u(, t ), v(, t ), ] + (τ t ) y [u t (, t ), v t (, t ), ] + γ(τ t ) y [u (, t ), v (, t ), ] + O((τ t ) ) =y [u(, t ), v(, t ), ] + (τ t ) y,t [u(, t ), v(, t ), ] + γ(τ t ) y, [u(, t ), v(, t ), ] + O((τ t ) ) and λ [u(, τ), v(, τ), γ(τ t ) + ] =a + a f(y [u(, τ), v(, τ), γ(τ t ) + ]) =a + a f(y [u(, t ), v(, t ), ] + a (τ t )f (y [u(, t ), v(, t ), ]) y,t [u(, t ), v(, t ), ] + a γ(τ t )f (y [u(, t ), v(, t ), ]) y, [u(, t ), v(, t ), ] + O((τ t ) ) =λ [u(, t ), v(, t ), ] + (τ t )λ,t [u(, t ), v(, t ), ] + γ(τ t )λ, [u(, t ), v(, t ), ] + O((τ t ) ) 39

40 Similarly, λ [u(, τ), v(, τ), γ(τ t ) + ] =λ [u(, t ), v(, t ), ] + (τ t )λ,t [u(, t ), v(, t ), ] + γ(τ t )λ, [u(, t ), v(, t ), ] + O((τ t ) ) Hence g[u(, τ), v(, τ), γ(τ t ) + ] = λ [u(, t ), v(, t ), γ(τ t ) + ] w (τ) + λ [u(, t ), v(, t ), γ(τ t ) + ] w (τ) = λ [u(, t ), v(, t ), ] w (t ) + λ [u(, t ), v(, t ), ] w (t ) + (τ t ) [ λ,t u λ u t + λ,t v + λ v t + γ( λ, u λ u + λ, v + λ v )] (,t ) + O((τ t ) ) and Thus t g[u(, τ), v(, τ), γ(τ t ) + ]dt t =(t t ) g[u(, t ), v(, t ), ] + (t t ) { λ,t u λ u t + λ,t v + λ v t + γ( λ, u λ u + λ, v + λ v )} (,t ) + O((τ t ) 3 ) u(γ(t t ) +, t ) =u(, t ) + (t t ) g[u(, t ), v(, t ), ] + (t t ) { λ,t u λ u t + λ,t v + λ v t + γ( λ, u λ u + λ, v + λ v )} (,t ) + O((t t ) 3 ) References [] D. Balsara and C.-W. Shu, Monotonicity preserving weighted essentially nonoscillatory schemes with increasingly high order of accuracy, Journal of Computational Physics, (),

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