30 crete maximum principle, which all imply the bound-preserving property. But most

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1 A HIGH ORDER ACCURATE BOUND-PRESERVING COMPACT FINITE DIFFERENCE SCHEME FOR SCALAR CONVECTION DIFFUSION EQUATIONS HAO LI, SHUSEN XIE, AND XIANGXIONG ZHANG Abstract We show that the classical fourth order accurate compact finite difference scheme with high order strong stability preserving time discretizations for convection diffusion problems satisfies a weak monotonicity property, which implies that a simple limiter can enforce the boundpreserving property without losing conservation and high order accuracy Higher order accurate compact finite difference schemes satisfying the weak monotonicity will also be discussed Key words finite difference method, compact finite difference, high order accuracy, convection diffusion equations, bound-preserving, maximum principle AMS subject classifications M, M Introduction The bound-preserving property Consider the initial value problem for a scalar convection diffusion equation u t + f(u) x = a(u) xx, u(x, ) = u (x), where a (u) Assume f(u) and a(u) are well-defined smooth functions for any u [m, M] where m = min x u (x) and M = max x u (x) Its exact solution satisfies: () min u (x) = m u(x, t) M = max u x (x), t x In this paper, we are interested in constructing a high order accurate finite difference scheme satisfying the bound-preserving property () For a scalar problem, it is desired to achieve () in numerical solutions mainly for the physical meaning For instance, if u denotes density and m =, then negative numerical solutions are meaningless In practice, in addition to enforcing (), it is also critical to strictly enforce the global conservation of numerical solutions for a time-dependent convection dominated problem Moreover, the computational cost for enforcing () should not be significant if it is needed for each time step 7 Popular methods for convection problems For the convection prob- 8 lems, ie, a(u), a straightforward way to achieve the above goals is to require 9 a scheme to be monotone, total-variational-diminishing (TVD), or satisfying a dis- 3 crete maximum principle, which all imply the bound-preserving property But most 3 schemes satisfying these stronger properties are at most second order accurate For 3 instance, a monotone scheme and traditional TVD finite difference and finite volume 33 schemes are at most first order accurate [7] Even though it is possible to have high 34 order TVD finite volume schemes in the sense of measuring the total variation of 3 reconstruction polynomials [, ], such schemes can be constructed only for the 3 one-dimensional problems The second order central scheme satisfies a discrete max- 37 imum principle min j u n j un+ j max j u n j where un j denotes the numerical solution 38 at n-th time step and j-th grid point [8] Any finite difference scheme satisfying H Li and X Zhang were supported by the NSF grant DMS-93 S Xie was supported by NSFC grant and Fundamental Research Funds for the Central Universities Department of Mathematics, Purdue University, N University Street, West Lafayette, IN (li497@purdueedu, zhan9@purdueedu) School of Mathematical Sciences, Ocean University of China, 38 Songling Road, Qingdao, PR China (shusenxie@ouceducn) This manuscript is for review purposes only

2 H LI, S XIE AND X ZHANG such a maximum principle can be at most second order accurate, see Harten s example in [4] By measuring the extrema of reconstruction polynomials, third order maximum-principle-satisfying schemes can be constructed [9] but extensions to multidimensional nonlinear problems are very difficult For constructing high order accurate schemes, one can enforce only the boundpreserving property for fixed known bounds, eg, m = and M = if u denotes the density ratio Even though high order linear schemes cannot be monotone, high order finite volume type spatial discretizations including the discontinuous Galerkin (DG) method satisfy a weak monotonicity property [3, 4, ] Namely, in a scheme consisting of any high order finite volume spatial discretization and forward Euler time discretization, the cell average is a monotone function of the point values of the reconstruction or approximation polynomial at Gauss-Lobatto quadrature points Thus if these point values are in the desired range [m, M], so are the cell averages in the next time step A simple and efficient local bound-preserving limiter can be designed to control these point values without destroying conservation Moreover, this simple limiter is high order accurate, see [3] and the appendix in [] With strong stability preserving (SSP) Runge-Kutta or multistep methods [4], which are convex combinations of several formal forward Euler steps, a high order accurate finite volume or DG scheme can be rendered bound-preserving with this limiter These results can be easily extended to multiple dimensions on cells of general shapes However, for a general finite difference scheme, the weak monotonicity does not hold For enforcing only the bound-preserving property in high order schemes, efficient alternatives include a flux limiter [9, 8] and a sweeping limiter in [] These methods are designed to directly enforce the bounds without destroying conservation thus can be used on any conservative schemes Even though they work well in practice, it is nontrivial to analyze and rigorously justify the accuracy of these methods especially for multi-dimensional nonlinear problems 3 The weak monotonicity in compact finite difference schemes Even though the weak monotonicity does not hold for a general finite difference scheme, in this paper we will show that some high order compact finite difference schemes satisfy such a property, which implies a simple limiting procedure can be used to enforce bounds without destroying accuracy and conservation To demonstrate the main idea, we first consider a fourth order accurate compact finite difference approximation to the first derivative on the interval [, ]: (f i+ + 4f i + f i ) = f i+ f i + O( x 4 ), x where f i and f i are point values of a function f(x) and its derivative f (x) at uniform grid points x i (i =,, N) respectively For periodic boundary conditions, the following tridiagonal linear system needs to be solved to obtain the implicitly defined approximation to the first order derivative: 4 f f 4 f () = f x 4 f N f N 4 f N f N We refer to the tridiagonal (, 4, ) matrix as a weighting matrix For the one- This manuscript is for review purposes only

3 A BOUND-PRESERVING COMPACT FINITE DIFFERENCE SCHEME dimensional scalar conservation laws with periodic boundary conditions on [, ]: (3) u t + f(u) x =, u(x, ) = u (x), the semi-discrete fourth order compact finite difference scheme can be written as 83 (4) dū i dt = x [f(u i+) f(u i )], 84 where ū i is defined as ū i = (u i + 4u i + u i+ ) Let λ = t x, then (4) with the 8 forward Euler time discretization becomes 8 87 () ū n+ i = ū n i λ[f(un i+) f(u n i )] The following weak monotonicity holds under the CFL λ max u f (u) 3 : ū n+ i = (un i + 4u n i + u n i+) + λ[f(un i+) f(u n i )] = [u i 3λf(u n i )] + [un i+ + 3λf(u n i+)] + 4 un i = H(u n i, u n i, u n i+) = H(,, ), where denotes that the partial derivative with respect to the corresponding argument is non-negative Therefore m u n i M implies m = H(m, m, m) ū n+ i H(M, M, M) = M, thus () m (un+ i + 4un+ i + u n+ i+ ) M If there is any overshoot or undershoot, ie, u n+ i > M or u n+ i < m for some i, then () implies that a local limiting process can eliminate the overshoot or undershoot Here we consider the special case m = to demonstrate the basic idea of this limiter, and for simplicity we ignore the time step index n + In Section we will show that (u i + 4u i + u i+ ), i implies the following two facts: max{u i, u i, u i+ } ; If u i <, then (u i ) + + (u i+) + u i >, where (u) + = max{u, } By the two facts above, when u i <, then the following three-point stencil limiting process can enforce positivity without changing i u i: (u i ) + (u i+ ) + v i = u i + u i ; v i+ = u i+ + u i, (u i ) + + (u i+ ) + (u i ) + + (u i+ ) + replace u i, u i, u i+ by v i,, v i+ respectively In Section, we will show that such a simple limiter can enforce the bounds of u i without destroying accuracy and conservation Thus with SSP high order time discretizations, the fourth order compact finite difference scheme solving (3) can be rendered bound-preserving by this limiter Moreover, in this paper we will show that such a weak monotonicity and the limiter can be easily extended to more general and practical cases including two-dimensional problems, convection diffusion problems, inflow-outflow boundary conditions, higher order accurate compact finite difference approximations, compact finite difference schemes with a total-variation-bounded (TVB) limiter [3] However, the extension to non-uniform grids is highly nontrivial thus will not be discussed In this paper, we only focus on uniform grids This manuscript is for review purposes only

4 H LI, S XIE AND X ZHANG 4 The weak monotonicity for diffusion problems Although the weak monotonicity holds for arbitrarily high order finite volume type schemes solving the convection equation (3), it no longer holds for a conventional high order linear finite volume scheme or DG scheme even for the simplest heat equation, see the appendix in [] Toward satisfying the weak monotonicity for the diffusion operator, an unconventional high order finite volume scheme was constructed in [] Second order accurate DG schemes usually satisfies the weak monotonicity for the diffusion operator on general meshes [] The only previously known high order linear scheme in the literature satisfying the weak monotonicity for scalar diffusion problems is the third order direct DG (DDG) method with special parameters [], which is a generalized version of interior penalty DG method On the other hand, arbitrarily high order nonlinear positivity-preserving DG schemes for diffusion problems were constructed in [,, 4] In this paper we will show that the fourth order accurate compact finite difference and a few higher order accurate ones are also weakly monotone, which is another class of linear high order schemes satisfying the weak monotonicity for diffusion problems It is straightforward to verify that the backward Euler or Crank-Nicolson method with the fourth order compact finite difference methods satisfies a maximum principle for the heat equation but it can be used be as a bound-preserving scheme only for linear problems The method is this paper is explicit thus can be easily applied to nonlinear problems It is difficult to generalize the maximum principle to an implicit scheme Regarding positivity-preserving implicit schemes, see [] for a study on weak monotonicity in implicit schemes solving convection equations See also [] for a second order accurate implicit and explicit time discretization for the BGK equation 4 Contributions and organization of the paper Although high order 4 compact finite difference methods have been extensively studied in the literature, eg, 4 [,, 3,, 3, 7], this is the first time that the weak monotonicity in compact finite 43 difference approximations is discussed This is also the first time a weak monotonicity 44 property is established for a high order accurate finite difference type scheme The 4 weak monotonicity property suggests it is possible to locally post process the numerical 4 solution without losing conservation by a simple limiter to enforce global bounds 47 Moreover, this approach allows an easy justification of high order accuracy of the 48 constructed bound-preserving scheme 49 For extensions to two-dimensional problems, convection diffusion problems, and sixth order and eighth order accurate schemes, the discussion about the weak mono- tonicity in general becomes more complicated since the weighting matrix may become a five-diagonal matrix instead of the tridiagonal (, 4, ) matrix in () Nonethe- 3 less, we demonstrate that the same simple three-point stencil limiter can still be used 4 to enforce bounds because we can factor the more complicated weighting matrix as a product of a few of tridiagonal c+ (, c, ) matrices with c The paper is organized as follows: in Section we demonstrate the main idea 7 for the fourth order accurate scheme solving one-dimensional problems with periodic 8 boundary conditions Two-dimensional extensions are discussed in in Section 3 Sec- 9 tion 4 is the extension to higher order accurate schemes Inflow-outflow boundary conditions and Dirichlet boundary conditions are considered in Section Numerical tests are given in Section Section 7 consists of concluding remarks 3 4 A fourth order accurate scheme for one-dimensional problems In this section we first show the fourth order compact finite difference with forward Euler time discretization satisfies the weak monotonicity Then we discuss how to design This manuscript is for review purposes only

5 7 8 9 A BOUND-PRESERVING COMPACT FINITE DIFFERENCE SCHEME a simple limiter to enforce the bounds of point values To eliminate the oscillations, a total variation bounded (TVB) limiter can be used We also show that the TVB limiter does not affect the bound-preserving property of ū i, thus it can be combined with the bound-preserving limiter to ensure the bound-preserving and non-oscillatory solutions for shocks High order time discretizations will be discussed in Section 7 7 One-dimensional convection problems Consider a periodic function f(x) on the interval [, ] Let x i = i N (i =,, N) be the uniform grid points on 7 the interval [, ] Let f be a column vector with numbers f, f,, f N as entries, 73 where f i = f(x i ) Let W, W, D x and D xx denote four linear operators as follows: 4 f f W f = 4 f, D x f = f 74, 4 f N f N 4 f N f N W f = f f f N f N, D xx f = f f f N f N The fourth order compact finite difference approximation to the first order derivative () with periodic assumption for f(x) can be denoted as W f = x D xf The fourth order compact finite difference approximation to f (x) is W f = x D xx f The fourth compact finite difference approximations can be explicitly written as f = x W D x f, f = x W D xx f, 8 where W and W are the inverse operators For convenience, by abusing notations 83 we let W f i denote the i-th entry of the vector W f 84 Then the scheme (4) solving the scalar conservation laws (3) with periodic d 8 boundary conditions on the interval [, ] can be written as W dt u i = x [f(u i+) 8 f(u i )], and the scheme () is equivalent to W u n+ i = W u n i λ[f(un i+ ) 87 f(u n i )] As shown in Section 3, the scheme () satisfies the weak monotonicity 88 Theorem Under the CFL constraint t x max u f (u) 3,if un i [m, M], 89 then u n+ computed by the scheme () satisfies () A three-point stencil bound-preserving limiter In this subsection, we consider a more general constraint than () and we will design a simple limiter to enforce bounds of point values based on it Assume we are given a sequence of periodic point values u i (i =,, N) satisfying () m c + (u i + cu i + u i+ ) M, i =,, N, c, where u := u N, u N+ := u and c is a constant We have the following results: Lemma The constraint () implies the following for stencil {i, i, i + }: This manuscript is for review purposes only

6 H LI, S XIE AND X ZHANG 97 () min{u i, u i, u i+ } M, max{u i, u i, u i+ } m 98 (u () If u i > M, then i M) + (M u i ) ++(M u i+) + c 99 (m u If u i < m, then i) + (u i m) ++(u i+ m) + c Here the subscript + denotes the positive part, ie, (a) + = max{a, } Remark 3 The first statement in Lemma states that there do not exist three consecutive overshoot points or three consecutive undershoot points But it does not necessarily imply that at least one of three consecutive point values is in the bounds [m, M] For instance, consider the case for c = 4 and N is even, define u i for all odd i and u i for all even i, then c+ (u i + cu i + u i+ ) [, ] for all i but none of the point values u i is in [, ] Remark 4 Lemma implies that if u i is out of the range [m, M], then we can set u i m for undershoot (or u i M for overshoot) without changing the local sum u i + u i + u i+ by decreasing (or increasing) its neighbors u i± Proof We only discuss the upper bound The inequalities for the lower bound can be similarly proved First, if u i, u i, u i+ > M then c+ (u i + cu i + u i+ ) > M which is a contradiction to () Second, () implies u i + cu i + u i+ (c + )M, thus c(u i M) (M u i ) + (M u i+ ) (M u i ) + + (M u i+ ) + If u i > M, we get (M u i ) + + (M u i+ ) + > Moreover, (u i M) + (M u i ) ++(M u i+) + = u i M (M u i ) ++(M u i+) + c For simplicity, we first consider a limiter to enforce only the lower bound without 7 destroying global conservation For m =, this is a positivity-preserving limiter Algorithm A limiter for periodic data u i to enforce the lower bound Require: The input u i satisfies ū i = c+ (u i + cu i + u i+ ) m, i =,, N, with c Let u, u N+ denote u N, u respectively Ensure: The output satisfies v i m, i =,, n and N i= v i = N i= u i First set v i = u i, i =,, N Let v, v N+ denote v N, v respectively for i =,, N do if u i < m then (u v i v i i m) + (u i m) ++(u i+ m) + (m u i ) + v i+ v i+ v i m end if end for (u i+ m) + (u i m) ++(u i+ m) + (m u i ) Remark Even though a for loop is used, Algorithm is a local operation to an undershoot point since only information of two immediate neighboring points of the undershoot point are needed Thus it is not a sweeping limiter Theorem The output of Algorithm satisfies N v i = N u i and v i m Proof First of all, notice that the algorithm only modifies the undershoot points and their immediate neighbors Next we will show the output satisfies v i m case by case: If u i < m, the i-th step in for loops sets v i = m After the (i + )-th step in for loops, we still have v i = m because (u i m) + = i= i= This manuscript is for review purposes only

7 A BOUND-PRESERVING COMPACT FINITE DIFFERENCE SCHEME If u i = m, then v i = m in the final output because (u i m) + = If u i > m, then limiter may decrease it if at least one of its neighbors u i and u i+ is below m: v i = u i (u i m) + (m u i ) + (u i m) + + (u i m) + (u i m) + (m u i+ ) + (u i m) + + (u i+ m) + u i c (u i m) + c (u i m) + > m, where the inequalities are implied by Lemma and the fact c Finally, we need to show the local sum v i + v i + v i+ is not changed during the i-th step if u i < m If u i < m, then after (i )-th step we still have v i = u i because (u i m) + = Thus in the i-th step of for loops, the point value at x i is increased by the amount m u i, and the point values at x i and x i+ are decreased (u by i m) + (u (u i m) ++(u i+ m) + (m u i ) + + i+ m) + (u i m) ++(u i+ m) + (m u i ) + = m u i So v i + v i + v i+ is not changed during the i-th step Therefore the limiter ensures the output v i m without changing the global sum The limiter described by Algorithm is a local three-point stencil limiter in the sense that only undershoots and their neighbors will be modified, which means the limiter has no influence on point values that are neither undershoots nor neighbors to undershoots Obviously a similar procedure can be used to enforce only the upper bound However, to enforce both the lower bound and the upper bound, the discussion for this three-point stencil limiter is complicated for a saw-tooth profile in which both neighbors of an overshoot point are undershoot points Instead, we will use a different limiter for the saw-tooth profile To this end, we need to separate the point values {u i, i =,, N} into two classes of subsets consisting of consecutive point values In the following discussion, a set refers to a set of consecutive point values u l, u l+, u l+,, u m, u m For any set S = {u l, u l+,, u m, u m }, we call the first point value u l and the last point value u m as boundary points, and call the other point values u l+,, u m as interior points A set of class I is defined as a set satisfying the following: It contains at least four point values Both boundary points are in [m, M] and all interior points are out of range 3 It contains both undershoot and overshoot points Notice that in a set of class I, at least one undershoot point is next to an overshoot point For given point values u i, i =,, N, suppose all the sets of class I are S = {u m, u m+,, u n }, S = {u m,, u n },, S K = {u mk,, u nk }, where m < m < < u mk A set of class II consists of point values between S i and S i+ and two boundary points u ni and u mi+ Namely they are T = {u, u,, u m }, T = {u n,, u m }, T = {u n,, u m3 },, T K = {u nk,, u N } For periodic data u i, we can combine T K and T to define T K = {u nk,, u N, u,, u m } In the sets of class I, the undershoot and the overshoot are neighbors In the sets of class II, the undershoot and the overshoot are separated, ie, an overshoot is not next to any undershoot We remark that the sets of class I are hardly encountered in the numerical tests but we include them in the discussion for the sake of completeness When there are no sets of class I, all point values form a single set of class II We will use the same procedure as in Algorithm for T i and a different limiter for S i to enforce both the lower bound and the upper bound This manuscript is for review purposes only

8 8 H LI, S XIE AND X ZHANG Algorithm A bound-preserving limiter for periodic data u i satisfying ū i [m, M] Require: the input u i satisfies ū i = c+ (u i + cu i + u i+ ) [m, M], c Let u, u N+ denote u N, u respectively Ensure: the output satisfies v i [m, M], i =,, N and N i= v i = N i= u i : Step : First set v i = u i, i =,, N Let v, v N+ denote v N, v respectively : Step I: Find all the sets of class I S,, S K (all local saw-tooth profiles) and all the sets of class II T,, T K 3: Step II: For each T j (j =,, K), the same limiter as in Algorithm (but for both upper bound and lower bound) is used: 4: for all index i in T j do : if u i < m then (u : v i v i i m) + (u i m) ++(u i+ m) + (m u i ) + 7: v i+ v i+ 8: v i m 9: end if : if u i > M then : v i v i + (u i+ m) + (u i m) ++(u i+ m) + (m u i ) + (M u i ) + (M u i ) ++(M u i+) + (u i M) + (M u : v i+ v i+ + i+) + (M u i ) ++(M u i+) + (u i M) + 3: v i M 4: end if : end for : Step III: for each saw-tooth profile S j = {u mj,, u nj } (j =,, K), let N and N be the numbers of undershoot and overshoot points in S j respectively 7: Set U j = n j i=m j v i 8: for i = m j +,, n j do 9: if u i > M then : v i M : end if : if u i < m then 3: v i m 4: end if : end for : Set V j = N M + N m + v mj + v nj 7: Set A j = v mj + v nj + N M (N + )m, B j = (N + )M v mj v nj N m 8: if V j U j > then 9: for i = m j,, n j do 3: v i v i vi m A j (V j U j ) 3: end for 3: else 33: for i = m j,, n j do 34: v i v i + M vi B j (U j V j ) 3: end for 3: end if This manuscript is for review purposes only

9 A BOUND-PRESERVING COMPACT FINITE DIFFERENCE SCHEME Theorem 7 Assume periodic data u i (i =,, N) satisfies ū i = c+ (u i + cu i + u i+ ) [m, M], c for all i =,, N with u := u N and u N+ := u, then the output of Algorithm satisfies N i= v i = N i= u i and v i [m, M], i Proof First we show the output v i [m, M] Consider Step II, which only modifies the undershoot and overshoot points and their immediate neighbors Notice that the operation described by lines -8 will not increase the point value of neighbors to an undershoot point thus it will not create new overshoots Similarly, the operation described by lines -3 will not create new undershoots In other words, no new undershoots (or overshoots) will be created when eliminating overshoots (or undershoots) in Step II Each interior point u i in any T j belongs to one of the following four cases: u i m or u i M m < u i < M and u i, u i+ M 3 m < u i < M and u i, u i+ m 4 m < u i < M and u i > M, u i+ < m (or u i+ > M, u i < m) We want to show v i [m, M] after Step II For the first three cases, by the same arguments as in the proof of Theorem, we can easily show that the output point values are in the range [m, M] For case (), after Step II, if u i m then v i = m; if u i M then v i = M For case (), v i u i only if at least one of u i and u i+ is an undershoot If so, then v i = u i (u i m) + (m u i ) + (u i m) + + (u i m) + (u i m) + (m u i+ ) + (u i m) + + (u i+ m) + u i c (u i m) + c (u i m) + > m Similarly, for case (3), v i u i only if at least one of u i and u i+ is an overshoot, and we can show v i < M Notice that case () and case (3) are not exclusive to each other, which however does not affect the discussion here When case () and case (3) overlap, we have u i, u i, u i+ [m, M] thus v i = u i [m, M] after Step II For case (4), without loss of generality, we consider the case when u i+ > M, u i [m, M], u i < m, and we need to show that the output v i [m, M] By Lemma, we know that Algorithm will decrease the value at x i by at most c (u i m) to eliminate the undershoot at x i then increase the point value at x i by at most c (M u i) to eliminate the overshoot at x i+ So after Step II, v i u i + c (M u i) M (because c, u i < M); v i u i c (u i m) m (because c, u i > m) Thus we have v i [m, M] after Step II By the same arguments as in the proof of Theorem, we can also easily show the boundary points are in the range [m, M] after Step II It is straightforward to verify that N i= v i = N i= u i after Step II because the operations described by lines -8 and lines -3 do not change the local sum v i + v i + v i+ Next we discuss Step III in Algorithm Let N = + N + N = n j m j + be the cardinality of S j = {u mj,, u nj } We need to show that the average value in each saw-tooth profile S j is in the range [m, M] after Step II before Step III Otherwise it is impossible to enforce This manuscript is for review purposes only

10 H LI, S XIE AND X ZHANG the bounds in S j without changing the sum in S j In other words, we need to show Nm U j = v i S j v i NM We will prove the claim by conceptually applying the upper or lower bound limiter Algorithm to S j Consider a boundary point of S j, eg, u mj [m, M], then during Step II the point value at x mj can be unchanged, moved down at most c (u m j m) or moved up at most c (M u m j ) We first show the average value in S j after Step II is not below m: (a) Assume both boundary point values of S j are unchanged during Step II If applying Algorithm to S j after Step II, by the proof of Theorem, we know that the output values would be greater than or equal to m with the same sum, which implies that v i S j v i Nm (b) If a boundary point value of S j is increased during Step II, the same discussion as in (a) still holds because an increased boundary value does not affect the discussion for the lower bound (c) If a boundary point value v mj of S j is decreased during Step II, then with the fact that it is decreased by at most the amount c (u m j m), the same discussion as in (a) still holds Similarly if applying the upper bound limiter similar to Algorithm to S j after Step II, then by the similar arguments as above, the output values would be less than or equal to M with the same sum, which implies v i S j v i NM Now we can show the output v i [m, M] for each S j after Step III: Assume V j = N M + N m + v mj + v nj > U j before the for loops in Step III Then after Step III: if u i < m we get v i = m; if u i m we have M v i v i m (V j U j ) A j v i m =v i v mj + v nj + N M (N + )m (v m j + v nj + N M + N m U j ) v i m v i v mj + v nj + N M (N + )m (v m j + v nj + N M + N m Nm) =v i (v i m) = m Assume V j = N M + N m + v mj + v nj U j before the for loops in Step III Then after Step III: if u i > M we get v i = M; if u i M we have m v i + M v i (U j V j ) B j M v i =v i + (N + )M v mj v nj N m (U j v mj v nj N M N m) M v i v i + (N + )M v mj v nj N m ( NM v mj v nj N M N m) =v i +(M v i ) = M Thus we have shown all the final output values are in the range [m, M] Finally it is straightforward to verify that N i= v i = N i= u i The limiters described in Algorithm and Algorithm are high order accurate limiters in the following sense Assume u i (i =,, N) are high order accurate approximations to point values of a very smooth function u(x) [m, M], ie, u i u(x i ) = O( x k ) For fine enough uniform mesh, the global maximum points are well separated from the global minimum points in {u i, i =,, N} In other words, This manuscript is for review purposes only

11 A BOUND-PRESERVING COMPACT FINITE DIFFERENCE SCHEME there is no saw-tooth profile in {u i, i =,, N} Thus Algorithm reduces to the three-point stencil limiter for smooth profiles on fine resolved meshes Under these assumptions, the amount which limiter increases/decreases each point value is at most (u i M) + and (m u i ) + If (u i M) + >, which means u i > M u(x i ), we have (u i M) + = O( x k ) because (u i M) + < u i u(x i ) = O( x k ) Similarly, we get (m u i ) + = O( x k ) Therefore, for point values u i approximating a smooth function, the limiter changes u i by O( x k ) 3 A TVB limiter The scheme () can be written into a conservation form: () ū n+ i = ū n i t x ( ˆf i+ ˆf i ), which is suitable for shock calculations and involves a numerical flux (3) ˆfi+ = (f(un i+) + f(u n i )) To achieve nonlinear stability and eliminate oscillations for shocks, a TVB (total variation bounded in the means) limiter was introduced for the scheme () in [3] In this subsection we will show that the bound-preserving property of ū i () still holds for the scheme () with the TVB limiter in [3] Thus we can use both the TVB limiter and the bound-preserving limiter in Algorithm () at the same time The compact finite difference scheme with the limiter in [3] is (4) where the numerical flux ū n+ i ˆf (m) i+ = ū n i t (m) ( ˆf x i+ (m) ˆf ), i is the modified flux approximating (3) First we write f(u) = f + (u) + f (u) with the requirement that f + (u) u, and f (u) u The simplest such splitting is the Lax-Friedrichs splitting f ± (u) = (f(u)±αu), α = max f (u) Then we write the flux ˆf i+ as ˆf u [m,m] i+ = ˆf + + ˆf, i+ i+ where ˆf ± are obtained by adding superscripts ± in (3) Next we define i+ 377 d ˆf + i+ = ˆf + f + (ū i+ i ), d ˆf i+ = f (ū i+ ) ˆf i Here d ˆf ± are the differences between the numerical fluxes ˆf ± i+ i+ upwind fluxes f + (ū i ) and f (ū i+ ) The limiting is defined by and the first-order, (m) d ˆf = m(d ˆf +, + f + (ū i+ i+ i ), + f + (m) (ū i )), d ˆf = m(d ˆf, + f (ū i+ i+ i ), + f (ū i+ )), where + v i v i+ v i is the usual forward difference operator, and the modified minmod function m is defined by { a, if a () m(a,, a k ) = p x, m(a,, a k ), otherwise, where p is a positive constant independent of x and m is the minmod function { s min i k a m(a,, a k ) = i, if sign(a ) = = sign(a k ) = s,, otherwise This manuscript is for review purposes only

12 H LI, S XIE AND X ZHANG The limited numerical flux is then defined by f (ū i+ ) d ˆf +(m) i+ = f + (ū i ) + d +(m) (m) ˆf, ˆf = i+ i+ (m) (m) +(m) ˆf,and ˆf = ˆf + i+ i+ i+ Lemma 8 For any n and t such that n t T, scheme (4) is TVBM (m) ˆf The following result was proved in [3]: i+ 388 (total variation bounded in the means): T V (ū n ) = i ūn i+ ūn i C, where C is 39 independent of t, under the CFL condition max u ( u f + (u) u f (u)) t x 39 Next we show that the TVB scheme still satisfies () 39 Theorem 9 If u n i [m, M], then under a suitable CFL condition, the TVB 393 scheme (4) satisfies m (un+ i + 4un+ i + u n+ i+ ) M 394 Proof Let λ = t x, then we have ū n+ i = ū n (m) i λ( ˆf i+ = 4 (ūn i 4λ We will show ū n+ i under the CFL condition (m) ˆf ) i +(m) ˆf ) + i+ 4 (ūn i 4λ (m) ˆf ) + i+ 4 (ūn i + 4λ [m, M] by proving that the four terms satisfy ū n +(m) i 4λ ˆf [m 4λf + (m), M 4λf + (M)], i+ (m) ū i 4λ ˆf [m 4λf (m), M 4λf (M)], i+ ū n +(m) i + 4λ ˆf [m + 4λf + (m), M + 4λf + (M)], i (m) ū i + 4λ ˆf [m + 4λf (m), M + 4λf (M)], i () λ max u f (±) (u) +(m) ˆf ) + i 4 (ūn i + 4λ (m) ˆf ) i We only discuss the first term since the proof for the rest is similar We notice that u 4λf + (u) and u λf + (u) are monotonically increasing functions of u under the CFL constraint (), thus u [m, M] implies u 4λf + (u) [m 4λf + (m), M 4λf + (M)] and u λf + (u) [m λf + (m), M λf + (M)] For convenience, we drop the time step n, then we have +(m) ū i 4λ ˆf = ū i+ i 4λ(f + (ū i ) + d +(m) where the value of d ˆf has four possibilities: i+ +(m) If d ˆf =, then i+ +(m) ˆf ), i+ +(m) ū i 4λ ˆf = ū i+ i 4λf + (ū i ) [m 4λf + (m), M 4λf + (M)] (m) If d ˆf = d ˆf +, then we get i+ i+ +(m) ū i 4λ ˆf = i+ (u i + 4u i + u i+ ) 4λ f + (u i ) + f + (u i+ ) = u i + 3 (u i 3λf + (u i )) + (u i+ λf + (u i+ )) This manuscript is for review purposes only

13 A BOUND-PRESERVING COMPACT FINITE DIFFERENCE SCHEME By the monotonicity of the function u λf + (u) and u 3λf + (u), we have u i 3λf + (u i ) [m 3λf + (m), M 3λf + (M)], u i+ λf + (u i+ ) [m λf + (m), M λf + (M)], +(m) which imply ū i 4λ ˆf [m 4λf + (m), M 4λf + (M)] i+ = + f + (ū i ), ū i 4λ +(m) 3 If d ˆf i+ ˆf +(m) i+ = ū i 4λf + (ū i+ ) If + f + (ū i ) >, ū i 4λf + (ū i+ ) < ū i 4λf + (ū i ) M 4λf + (M), which implies the upper bound holds Due to the definition of the minmod function, we can get < + f + (ū i ) < d ˆf + + Thus, ˆf = f + (u i)+f + (u i+) i+ i+ = f + (ū i ) + d ˆf + > f + (ū i+ i ) + + f + (ū i ) = f + (ū i+ ) Then, ū i 4λf + (ū i+ ) > ū i 4λ f + (u i)+f + (u i+) m 4λf + (m), which gives the lower bound For the case + f + (ū i ) <, the proof is similar 4 If d = + f + (ū i ), the proof is the same as the previous case ˆf +(m) i One-dimensional convection diffusion problems We consider the one- 48 dimensional convection diffusion problems with periodic boundary conditions: u t + 49 f(u) x = a(u) xx, u(x, ) = u (x), where a (u) Let f n denote the column vector 43 with entries f(u n ),, f(u n N ) By notations introduced in Section, the fourth- 43 order compact finite difference with forward Euler can be denoted as: 43 (7) u n+ = u n t x W D x f n + t x W D xx a n Recall that we have abused the notation by using W fi n to denote the i-th entry of the vector W f n and we have defined ū i = W u i We now define ũ i = W u i Notice that W and W are both circulant thus they both can be diagonalized by the discrete Fourier matrix, so W and W commute Thus we have ū i = (W W u) i = (W W u) i = ũ i Let fi n = f(u n i ) and an i = a(un i ), then the scheme (7) can be written as ũ n+ i = ũ n i t x W D x f n i + t x W D xx a n i Theorem Under the CFL constraint t x max u f (u), t x max u a (u) 4, if un i [m, M], then the scheme (7) satisfies that m ũ n+ i M Proof Let λ = t x and µ = t x We can rewrite the scheme (7) as u n+ = (un λw D x f n ) + (un + µw D xx a n ), W W u n+ = W (W u n λd x f n ) + W (W u n + µd xx a n ), ũ n+ i = W (ū n i λd x f n i ) + W (ũ n i + µd xx a n i ) This manuscript is for review purposes only

14 H LI, S XIE AND X ZHANG By Theorem, we have ū n i λd xf n i [m, M] We also have ũ n i + µd xx a n i = (un + u n i + u n i+) + µ(a n i a n i + a n i+) ( ) ( ) ( ) = un i 4µa n i + un i + µa n i + un i+ + µa n i+ 44 Due to monotonicity under the CFL constraint and the assumption a (u), we get 44 ũ n i + µd xxa n i [m, M] Thus we get ũ n+ i [m, M] since it is a convex combination 447 of ū n i λd xfi n and ũ n i + µd xxa n i 448 Given point values u i satisfying ũ i [m, M] for any i, Lemma no longer 449 holds since ũ i has a five-point stencil However, the same three-point stencil limiter 4 in Algorithm can still be used to enforce the lower and upper bounds Given 4 ũ i = W W u i i =,, N, conceptually we can obtain the point values u i by first 4 computing ū i = W ũ i then computing u i = W ū i Thus we can apply the limiter 43 in Algorithm twice to enforce u i [m, M]: 44 Given ũ i [m, M], compute ū i = W ũ i which are not necessarily in the 4 range [m, M] Then apply the limiter in Algorithm to ū i, i =,, N 4 Let v i denote the output of the limiter Since we have ũ i = ū i = c + (ū i + cū i + ū i+ ), c =, all discussions in Section are still valid, thus we have v i [m, M] Compute u i = W v i Apply the limiter in Algorithm to u i, i =,, N Let v i denote the output of the limiter Then we have v i [m, M] High order time discretizations For high order time discretizations, we can use strong stability preserving (SSP) Runge-Kutta and multistep methods, which are convex combinations of formal forward Euler steps Thus if using the limiter in Algorithm for fourth order compact finite difference schemes considered in this section on each stage in a SSP Runge-Kutta method or each time step in a SSP multistep method, the bound-preserving property still holds In the numerical tests, we will use a fourth order SSP multistep method and a fourth order SSP Runge-Kutta method [4] Now consider solving u t = F (u) The SSP coefficient C for a SSP time discretization is a constant so that the high order SSP time discretization is stable in a norm or a semi-norm under the time step restriction t C t, if under the time step restriction t t the forward Euler is stable in the same norm or semi-norm The fourth order SSP Multistep method (with SSP coefficient C ms = 48) and the fourth order SSP Runge-Kutta method (with SSP coefficient C rk = 8) will be used in the numerical tests See [4] for their definitions In Section we have shown that the limiters in Algorithm and Algorithm are high order accurate provided u i are high order accurate approximations to a smooth function u(x) [m, M] This assumption holds for the numerical solution in a multistep method in each time step, but it is no longer true for inner stages in the Runge-Kutta method So only SSP multistep methods with the limiter Algorithm are genuinely high order accurate schemes For SSP Runge-Kutta methods, using the bound-preserving limiter for compact finite difference schemes might result in an order reduction The order reduction for bound-preserving limiters for finite volume and DG schemes with Runge-Kutta methods was pointed out in [3] due to the same reason However, such an order reduction in compact finite difference schemes is more prominent, as we will see in the numerical tests This manuscript is for review purposes only

15 A BOUND-PRESERVING COMPACT FINITE DIFFERENCE SCHEME Extensions to two-dimensional problems In this section we consider initial value problems on a square [, ] [, ] with periodic boundary conditions Let (x i, y j ) = ( i j N x, N y ) (i =,, N x, j =,, N y ) be the uniform grid points on the domain [, ] [, ] For a periodic function f(x, y) on [, ] [, ], let f be a matrix of size N x N y with entries f ij representing point values f(u ij ) We first define two linear operators W x and W y from R Nx Ny to R Nx Ny : 4 W x f = N x N x f f f,ny f f f,ny W y f = f Nx, f Nx, f Nx,N y f Nx, f Nx, f Nx,N y f f f,ny f f f,ny, f Nx, f Nx, f Nx,N y f Nx, f Nx, f Nx,N y N y N y We can define W x, W y, D x, D y, W x and W y similarly such that the subscript x denotes the multiplication of the corresponding matrix from the left for the x-index and the subscript y denotes the multiplication of the corresponding matrix from the right for the y-index We abuse the notations by using W x f ij to denote the (i, j) entry of W x f We only discuss the forward Euler from now on since the discussion for high order SSP time discretizations are the same as in Section 3 Two-dimensional convection equations Consider solving the two-dimensional convection equation: u t + f(u) x + g(u) y =, u(x, y, ) = u (x, y) By the our notations, the fourth order compact scheme with the forward Euler time discretization can be denoted as: (3) u n+ ij = u n ij t x W x D xf n ij t y W y D yg n ij We define ū n = W x W y u n, then by applying W y W x to both sides, (3) becomes (3) ū n+ ij (33) = ū n ij t x W yd x f n ij t y W xd y g n ij Theorem 3 Under the CFL constraint t x max f (u) + t u y max u g (u) 3, if u n ij [m, M], then the scheme (3) satisfies ūn+ ij [m, M] Proof For convenience, we drop the time step n in u n ij, f ij n, and introduce: U = u i,j+ u i,j+ u i+,j+ u i,j u i,j u i+,j, F = u i,j u i,j u i+,j f i,j+ f i,j+ f i+,j+ f i,j f i,j f i+,j f i,j f i,j f i+,j This manuscript is for review purposes only

16 H LI, S XIE AND X ZHANG Let λ = t x and λ = t y, then the scheme (3) can be written as ū n+ ij = W y W x u n ij λ W y D x fij n λ W x D y gij, n = : U λ 4 4 : F λ 4 : G, where : denotes the sum of all entrywise products in two matrices of the same size Obviously the right hand side above is a monotonically increasing function with respect to u lm for i l i +, j m j + under the CFL constraint (33) The monotonicity implies the bound-preserving result of ū n+ ij Given ū ij, we can recover point values u ij by obtaining first v ij = Wx ūij then u ij = Wy v ij Thus similar to the discussions in Section 4, given point values u ij satisfying ū ij [m, M] for any i and j, we can use the limiter in Algorithm in a dimension by dimension fashion to enforce u ij [m, M]: Given ū ij [m, M], compute v ij = Wx ūij which are not necessarily in the range [m, M] Then apply the limiter in Algorithm to v ij (i =,, N x ) for each fixed j Since we have ū ij = c + (v i,j + cv i,j + v i+,j ), c = 4, all discussions in Section are still valid Let v ij denote the output of the limiter, thus we have v ij [m, M] Compute u ij = Wy v ij Then we have v ij = c + (u i,j + cu i,j + u i,j+ ), c = 4 Apply the limiter in Algorithm to u ij (j =,, N y ) for each fixed i Then the output values are in the range [m, M] 3 Two-dimensional convection diffusion equations Consider the twodimensional convection diffusion problem: u t + f(u) x + g(u) y = a(u) xx + b(u) xx, u(x, y, ) = u (x, y), where a (u) and b (u) A fourth-order accurate compact finite difference scheme can be written as du dt = x W x D xf y W y D yg + x W x D xxa + y W y D yyb Let λ = t x, λ = t y, µ = t x and µ = t discretization, the scheme becomes (34) u n+ ij y With the forward Euler time = u n ij λ W x D xf n ij λ W y D yg n ij + µ W x D xxa n ij + µ W y D yyb n ij We first define ū = W x W y u and ũ = W x W y u, where W = W x W y and W = W x W y Due to the fact W W = W W, we have ū = W x W y (W x W y u) = W x W y (W x W y u) = ũ The scheme (34) is equivalent to the following form: ū n+ ij = ū n ij λ W y W x W y D x fij n λ W x W x W y D y gij n + µ W x W y W y D xx a n ij + µ W x W y W x D yy b n ij This manuscript is for review purposes only

17 A BOUND-PRESERVING COMPACT FINITE DIFFERENCE SCHEME 7 3 (3) Theorem 3 Under the CFL constraint t x max f (u) + t u y max u g (u), t x max u a (u) + t y max u b (u) 4, if u n ij [m, M], then the scheme (34) satisfies ū n+ ij Proof By using ū n ij = ū n ij + ũ n ij, we obtain [m, M] 4 ū n+ ij = W xw y [ū n ij λ W y D x f n ij λ W x D y g n ij] + W xw y [ũ n ij + µ W y D xx a n ij + µ W x D yy b n ij] Let v ij = ū n ij λ W y D x fij n λ W x D y gij n, w ij = ũ n ij +µ W y D xx a n ij +µ W x D yy b n ij 7 Then by the same discussion as in the proof of Theorem 3, we can show v ij [m, M] 8 For w ij, it can be written as 9 w ij = 44 : U + µ : A + µ : B, A = a i,j+ a i,j+ a i+,j+ b i,j+ b i,j+ b i+,j+ a i,j a i,j a i+,j, B = b i,j b i,j b i+,j a i,j a i,j a i+,j b i,j b i,j b i+,j Under the CFL constraint (3), w ij is a monotonically increasing function of u n ij involved thus w ij [m, M] Therefore, ū n+ ij [m, M] Given ū ij, we can recover point values u ij by obtaining first ũ ij = Wx W y ū ij then u ij = Wx W y ũij Thus similar to the discussions in the previous subsection, given point values u ij satisfying ū ij [m, M] for any i and j, we can use the limiter in Algorithm dimension by dimension several times to enforce u ij [m, M]: Given ū ij [m, M], compute ũ ij = Wx W y ū ij and apply the limiting algorithm in the previous subsection to ensure ũ ij [m, M] Compute v ij = Wx ũij which are not necessarily in the range [m, M] Then apply the limiter in Algorithm to v ij for each fixed j Since we have ũ ij = c + (v i,j + cv i,j + v i+,j ), c =, all discussions in Section are still valid Let ṽ ij denote the output of the limiter, thus we have ṽ ij [m, M] 3 Compute u ij = Wy ṽij Then we have ṽ ij = c+ (u i,j +cu i,j +u i,j+ ), c = Apply the limiter in Algorithm to u ij for each fixed i Then the output values are in the range [m, M] 4 Higher order extensions The weak monotonicity may not hold for a generic compact finite difference operator See [] for a general discussion of compact finite difference schemes In this section we demonstrate how to construct a higher order accurate compact finite difference scheme satisfying the weak monotonicity Following Section and Section 3, we can use these compact finite difference operators to construct higher order accurate bound-preserving schemes This manuscript is for review purposes only

18 8 H LI, S XIE AND X ZHANG Higher order compact finite difference operators Consider a compact finite difference approximation to the first order derivative in the following form: (4) β f i + α f i + f i + α f i+ + β f i+ = b f i+ f i 4 x f i+ f i + a, x where α, β, a, b are constants to be determined To obtain a sixth order accurate approximation, there are many choices for α, β, a, b To ensure the approximation in (4) satisfies the weak monotonicity for solving scalar conservation laws under some CFL condition, we need α >, β > By requirements above, we obtain (4) β = ( + 3α ), a = 9 (8 3α ), b = 8 ( 7 + 7α ), α > 3 With (4), the approximation (4) is sixth order accurate and satisfies the weak monotonicity as discussed in Section The truncation error of the approximation (4) and (4) is 4 7! (9α 4) x f (7) + O( x 8 ), so if setting (43) α = 4 9, β = 3, a = 4 7, b = 4, we have an eighth order accurate approximation satisfying the weak monotonicity Now consider the fourth order compact finite difference approximations to the second derivative in the following form: β f i + α f i + f i + α f i+ + β f i+ f i+ f i + f i f i+ f i + f i = b 4 x + a x, a = 3 (4 4α 4β ), b = 3 ( + α + 4β ) with the truncation error 4! ( + α 4β ) x 4 f () The fourth order scheme discussed in Section is the special case with α =, β =, a =, b = If β = α 4, we get a family of sixth-order schemes satisfying the weak monotonicity: (44) a = 78α , b = 9α 3, α > 4 The truncation error of the sixth order approximation is 3 8! (79α 344) x f (8) Thus we obtain an eighth order approximation satisfying the weak monotonicity if (4) α = , β = 3 38, a = 3 393, b = 3 393, with truncation error x8 f () 4 Convection problems For the rest of this section, we will mostly focus on the family of sixth order schemes since the eighth order accurate scheme is a special case of this family For u t + f(u) x = with periodic boundary conditions on the interval [, ], we get the following semi-discrete scheme: d dt u = W D x f, x This manuscript is for review purposes only

19 A BOUND-PRESERVING COMPACT FINITE DIFFERENCE SCHEME D x f = α β β α β α α β β β α α β β β β W u = + α + β α α β β β α α β β β α β α β β 4( + α + β ) u u u 3 u N u N u N, a b b a a a b b b a a b b a a b b b a a a b b a f f f 3 f N f N f N where f i and u i are point values of functions f(u(x)) and u(x) at uniform grid points x i (i =,, N) respectively We have a family of sixth-order compact schemes with forward Euler time discretization: (4) u n+ = u n t W D x f x, Define ū = W u and λ = t x, then scheme (4) can be written as ū n+ i = ū n i λ 4( + α + β ) (b f n i+ + a f n i+ a f n i b f n i ) Following the lines in Section, we can easily conclude that the scheme (4) satisfies ū n+ i [m, M] if u n i [m, M], under the CFL constraint t x f 9 (u) min{, (3α ) 8 3α 7α 7 } Given ū i [m, M], we also need a limiter to enforce u i [m, M] Notice that ū i has a five-point stencil instead of a three-point stencil in Section Thus in general the extensions of Section for sixth order schemes are more complicated However, we can still use the same limiter as in Section because the five-diagonal matrix W can be represented as a product of two tridiagonal matrices Plugging in β = ( + 3α ), we have W = W (), where W () = c () + c () c () c () c () W (), c () = α 7 3α 4α + 7α, α + 9α This manuscript is for review purposes only

20 H LI, S XIE AND X ZHANG 33 W () = c () + c () c () c () c (), c () = α 7 3α + 4α + 7α α + 9α In other words, ū = W () 34 u = W W () u Thus following the limiting procedure 3 in Section 4, we can still use the same limiter in Section twice to enforce the 3 bounds of point values if c (), c(), which implies 3 < α 9 In this case we have 37 9 min{ 8 3α, (3α ) 7α } = (3α ) 7 7α 7, thus the CFL for the weak monotonicity becomes 38 λ f (u) (3α ) 7α 7 We summarize the results in the following theorem Theorem 4 Consider a family of sixth order accurate schemes (4) with β = ( + 3α ), a = 9 (8 3α ), b = 8 ( 7 + 7α ), 3 < α 9, 39 which includes the eighth order scheme (43) as a special case If u n i [m, M] for all 4 i, under the CFL constraint t x max u f (u) (3α ) 7α 7, we have ūn+ i [m, M] () Given point values u i satisfying W W () u i = W u i = ū i [m, M] for any i, we can apply the limiter in Algorithm twice to enforce u i [m, M]: Given ū i [m, M], compute v i = [ W () ] ū i which are not necessarily in the range [m, M] Then apply the limiter in Algorithm to v i, i =,, N Let v i denote the output of the limiter Since we have ū i = c () +(v i + c () v i + v i+ ), c () >, all discussions in Section are still valid, thus we have v i [m, M] Compute u i = [ W () ] v i Apply the limiter in Algorithm to u i, i =,, N Since we have v i = c () +(u i + c () u i + u i+ ), c () >, all discus- sions in Section are still valid, thus the output are in [m, M] Diffusion problems For simplicity we only consider the diffusion problems and the extension to convection diffusion problems can be easily discussed following Section 4 For the one-dimensional scalar diffusion equation u t = g(u) xx with g (u) and periodic boundary conditions on an interval [, ], we get the sixth d order semi-discrete scheme: dt u = x W D xx g, where α β β α β α α β β β α α β β β β W u = + α + β α α β β β α α β β β α β α β β u u u 3 u N u N u N, This manuscript is for review purposes only

21 A BOUND-PRESERVING COMPACT FINITE DIFFERENCE SCHEME a b 4a b b 4a g 4a 8a b 4a b b g b 4a 8a b 4a b g 3 D xxg = 4(+α +β ), b 4a 8a b 4a b g N b b 4a 8a b 4a g N 4a b b 4a 8a b g N where g i and u i are values of functions g(u(x)) and u(x) atx i respectively As in the previous subsection, we prefer to factor W as a product of two tridiagonal matrices Plugging in β = α 4, we have: W () = W W (), where c () c () W () = c (), c () = α 8 + c () α 7α + 43α, 4 44α + α c () c () c () W () = c (), c () = α 8 + c () α + 7α + 43α 4 44α + α c () To have c (), c(), we need < α 3 The forward Euler gives (47) u n+ = u n + t W x D xx g Define ũ i = W u i and µ = t x, then the scheme (47) can be written as i = ũ n µ i + 4( + α + β ) ũ n+ [ b gi n + 4a gi n + ( 8a b )gi n + 4a gi+ n + b gi+ n ] Theorem 4 Consider a family of sixth order accurate schemes (47) with β = α 4, a = 78α , b = 9α 3, < α 3, 8 which includes the eighth order scheme (4) as a special case If u n i [m, M] for all i, under the CFL t x g 4 9 (u) < 3( α, the scheme satisfies ) ũn+ [m, M] () W W () As in the previous subsection, given point values u i satisfying u i = W u i = ũ i [m, M] for any i, we can apply the limiter in Algorithm twice to enforce u i [m, M] The matrices W and W commute because they are both circulant matrices thus diagonalizable by the discrete Fourier matrix The discussion for the sixth order scheme solving convection diffusion problems is also straightforward Extensions to general boundary conditions Since the compact finite difference operator is implicitly defined thus any extension to other type boundary conditions is not straightforward In order to maintain the weak monotonicity, the This manuscript is for review purposes only

22 H LI, S XIE AND X ZHANG boundary conditions must be properly treated In this section we demonstrate a high order accurate boundary treatment preserving the weak monotonicity for inflow and outflow boundary conditions For convection problems, we can easily construct a fourth order accurate boundary scheme For convection diffusion problems, it is much more complicated to achieve weak monotonicity near the boundary thus a straightforward discussion gives us a third order accurate boundary scheme 84 Inflow-outflow boundary conditions for convection problems For 8 simplicity, we consider the following initial boundary value problem on the interval 8 [, ] as an example: u t + f(u) x =, u(x, ) = u (x), u(, t) = L(t), where we 87 assume f (u) > so that the inflow boundary condition at the left cell end is a well- 88 posed boundary condition The boundary condition at x = is not specified thus 89 understood as an outflow boundary condition We further assume u (x) [m, M] 9 and L(t) [m, M] so that the exact solution is in [m, M] 9 Consider a uniform grid with x i = i x for i =,,, N, N + and x = N+ 9 Then a fourth order semi-discrete compact finite difference scheme is given by 93 d dt 4 4 u u N+ = x f f N With forward Euler time discretization, the scheme is equivalent to () ū n+ i = ū n i λ(f n i+ f n i ), i =,, N 9 Here u n = L(t n ) is given as boundary condition for any n Given u n i for i = 97,,, N +, the scheme () gives ū n+ i for i =,, N, from which we still 98 need u n+ N+ to recover interior point values un+ i for i =,, N 99 Since the boundary condition at x N+ = can be implemented as outflow, we 7 can use ū n+ i for i =,, N to obtain a reconstructed u n+ N+ If there is a cu- 7 bic polynomial p i (x) so that u i, u i, u i+ are its point values at x i, x i, x i+, then xi+ 7 x x i p i (x) dx = u i + 4 u i + u i+ = ū i, due to the exactness of the Simpson s 73 quadrature rule for cubic polynomials To this end, we can consider a unique cu- xj+ 74 bic polynomial p(x) satisfying four equations: x x j p(x) dx = ū n+ j, j = N 7 3, N, N, N If ū n+ j are fourth order accurate approximations to u(x j, t n+ )+ 4 7 u(x j, t n+ ) + u(x j+, t n+ ), then p(x) is a fourth order accurate approximation to 77 u(x, t n+ ) on the interval [x N 4, x N+ ] So we get a fourth order accurate u n+ N+ by () p(x N+ ) = 3ūN ūn 4 3 ūn + 7 ūn Since () is not a convex linear combination, p(x N+ ) may not lie in the bound [m, M] Thus to ensure u n+ N+ [m, M] we can define (3) u n+ N+ := max{min{p(x N+), M}, m} Obviously Theorem still holds for the scheme () For the forward Euler time discretization, we can implement the bound-preserving scheme as follows: Given u n i for all i, compute ū n+ i for i =,, N by () Obtain boundary values u n+ = L(t n+ ) and u n+ N+ by () and (3) This manuscript is for review purposes only

23 A BOUND-PRESERVING COMPACT FINITE DIFFERENCE SCHEME Given ū n+ i for i =,, N and two boundary values u n+ and u n+ N+, recover 77 point values u n+ i for i =,, N by solving the tridiagonal linear system 78 (the superscript n + is omitted): 4 u ū 4 u u ū 79 = 4 u N ū N 4 u N ū N u N Apply the limiter in Algorithm to the point values u n+ i for i =,, N Dirichlet boundary conditions for one-dimensional convection diffusion equations Consider the initial boundary value problem for a one-dimensional scalar convection diffusion equation on the interval [, ]: (4) u t + f(u) x = g(u) xx, u(x, t) = u (x), u(, t) = L(t), u(, t) = R(t), 7 where g (u) We further assume u (x) [m, M] and L(t), R(t) [m, M] so that 7 the exact solution is in [m, M] 77 We demonstrate how to treat the boundary approximations so that the scheme 78 still satisfies some weak monotonicity such that a certain convex combination of point 79 values is in the range [m, M] at the next time step Consider a uniform grid with 73 x i = i x for i =,,, N, N + where x = N+ The fourth order compact 73 finite difference approximations at the interior points can be written as: f x, f fx, f x f x, f W f x,n f x,n = x D x f N f N + f x,n+ + f N+ x, 4 W = 4, D x =, 4 4 W g xx, g xx, g xx,n g xx,n W = = x D xx g g g N g N +, D xx = gxx, + g x g xx,n+ + g N+ x,, This manuscript is for review purposes only

24 4 H LI, S XIE AND X ZHANG where f x,i and g xx,i denotes the values of f(u) x and g(u) xx at x i respectively Let F = fx, f x f x,n+ + f N+ x, G = gxx, + g x g xx,n+ + g N+ x Define W := W W = W W Here W and W commute because they have the same eigenvectors, which is due to the fact that W W is the identity matrix Let u = ( ) T ( u u u N, f = f(u ) f(u ) f(u N ) ) T ( and g = g(u ) g(u ) g(u N ) ) T 74 Then a fourth order compact finite difference approximation to (4) at the interior grid points is d dt u + W ( x D xf + F ) = W ( x D xx g + G) which is equivalent to d (W u) + dt x W D x f x W D xx g = W F + W G If u i (t) = u(x i, t) where u(x, t) is the exact solution to the problem, then it satisfies () u t,i + f x,i = g xx,i, where u t,i = d dt u i(t), f x,i = f(u i ) x and g xx,i = g(u i ) xx If we use () to simplify W F + W G, then the scheme is still fourth order accurate In other words, setting f x,i +g xx,i = u t,i does not affect the accuracy Plugging () in the original W F + W G, we can redefine W F + W G as 8 u t, + f x, + x f + 3 x g 7 u t, + 4 f + x g W F + W G := 7 u t,n+ 4 f N+ + x g N+ 8 u t,n+ + f x,n+ x f N+ + 3 x g N+ So we now consider the following fourth order accurate scheme: () 8 u t, + f x, + x f + 3 x g d (W u)+ dt x W D x f x W D xx g = The first equation in () is 7 u t, + 4 f + x g 7 u t,n+ 4 f N+ + x g N+ 8 u t,n+ + f x,n+ x f N+ + 3 x g N+ d dt (4u + 4u + 4u + u 3 ) = 7 4 x (f + f f f 3 ) + x (4g 7g + g + g 3 ) + f x, 77 After multiplying 7 d (7) = to both sides, it becomes dt (4u + 4u + 4u + u 3 ) = x (f + f f f 3 ) + x (4g 7g + g + g 3 ) + f x, This manuscript is for review purposes only

25 A BOUND-PRESERVING COMPACT FINITE DIFFERENCE SCHEME In order for the scheme (7) to satisfy a weak monotonicity in the sense that 4u n+ +4u n+ +4u n+ +u n+ 3 in (7) with forward Euler can be written as a monotonically increasing function of u n i under some CFL constraint, we still need to find an approximation to f(u) x, using only u, u, u, u 3, with which we have a straightforward third order approximation to f(u) x, : (8) f x, = x ( f + 3f 3 f + 3 f 3) + O( x 3 ) Then (7) becomes (9) d dt (4u + 4u + 4u + u 3 ) = x (9f + f 39f f 3 ) + x (4g 7g + g + g 3 ) The second to second last equations of () can be written as d dt (u i + 4u i + 4u i + 4u i+ + u i+ () ) = 7 4 x (f i + f i f i+ f i+ ) + x (g i + g i g i + g i+ + g i+ ), i N, which satisfies a straightforward weak monotonicity under some CFL constraint The last equation in () is d dt (4u N+ + 4u N + 4u N + u N ) = 7 4 x (f N + f N f N f N+ ) + x (g N + g N 7g N + 4g N+ ) + f x,n+ After multiplying 7 = to both sides, it becomes d dt (u N + 4u N + 4u N + 4u N+ ) = x (f N + f N f N f N+ ) + x (g N + g N 7g N + 4g N+ ) + f x,n+ Similar to the boundary scheme at x, we should use a third-order approximation: () f x,n+ = x ( 3 f N + 3 f N 3f N + f N+) + O( x 3 ) Then the boundary scheme at x N+ becomes () d dt (u N + 4u N + 4u N + 4u N+ ) = x (f N + 39f N f N 9f N+ ) + x (g N + g N 7g N + 4g N+ ) To summarize the full semi-discrete scheme, we can represent the third order scheme (9), () and (), for the Dirichlet boundary conditions as: d dt W ũ = x D x f(ũ) + x D xx g(ũ), This manuscript is for review purposes only

26 H LI, S XIE AND X ZHANG where D x = 4 W = Let ū = W ũ, λ = t x N (N+) N (N+), ũ = u u u N u N+, D xx = and µ = t x With forward Euler, it becomes (N+) 4 4, 4 4 N (N+) 793 (3) ū n+ i = ū n i λ D x fi + µ D xx g i, i =,, N We state the weak monotonicity without proof Theorem Under the CFL constraint t x max u f (u) 4 9, t 9 9, if un i We notice that [m, M], then the scheme (3) satisfies ūn+ i ū n+ = (4un+ + 4u n+ + 4u n+ + u3 n+ ) = un+ + 4u n+ + u n+ ū n+ N = (un+ N + 4un+ N + 4un+ N + 4un+ N+ ) = u n+ x max u g (u) [m, M] N + 4un+ N + un+ N + u n+ 8 Recall that the boundary values are given: u n+ = L(t n+ ) [m, M] and u n+ 8 R(t n+ ) [m, M], so we have u n+ + 4u n+ + u n+ u n+ + 4u n+ + u n+ u n+ N + 4un+ N + un+ N u n+ N + 4un+ N + un+ N + u n+ + + u n u n+ + u n+ 3 u n+ + 4u n+ + u n+ 3 N + 4un+ N u n+ N + 4un+ N + un+ N+ + un+ N+ M + M = M, m + m = m, M + M = M, + 4u n+ + u n+ 3 un+, + un+ N + 4un+ N + un+ N+ un+ N+ N+ = m + m = m Thus define w n+ = ( w n+, w n+, w3 n+,, w n+ N, ) T wn+ N as follows and we have: m w n+ i : = ū n+ i M, i =,, N, m w n+ : = u n+ + 4u n+ + u n+ + u n+ + 4u n+ + u n+ 3 M, m w n+ N : = u n+ N 3 + 4un+ N + un+ N + u n+ N + 4un+ N + un+ N M This manuscript is for review purposes only

27 A BOUND-PRESERVING COMPACT FINITE DIFFERENCE SCHEME 7 8 By the notations above, we get (4) K = N N w n+ = Kū n+ + u n+ bc = W ũ,, u bc = u u N+ N, W = N (N+) We notice that W can be factored as a product of two tridiagonal matrices: = N N 4 which can be denoted as W = W W Fortunately, all the diagonal entries of W and c W are in the form of c+, c > So given ū i = W u i [m, M], we construct w n+ i [m, M] We can apply the limiter in Algorithm twice to enforce u i [m, M]: Given u n i for all i, use the scheme (3) to obtain ū n+ i [m, M] for i =,, N Then construct w n+ i [m, M] for i =,, N by (4) Notice that W is a matrix of size N N Compute v = W w n+ Apply the limiter in Algorithm to v i and let v i denote the output values Since we have W v i [m, M], ie, m m v + v M, v + v + v 3 M, m v N + v N + v N M, m v N + v N M Following the discussions in Section, it implies v i [m, M] 3 Obtain values of u n+ i, i =,, N by solving a N N system: 4 u n+ v 4 u n+ v = un+ bc 4 u n+ N v N 4 v N u n+ N 4 Apply the limiter in Algorithm to u n+ i to ensure u n+ i [m, M] 4 4 N (N+), This manuscript is for review purposes only

28 8 H LI, S XIE AND X ZHANG 834 Numerical tests 83 One-dimensional problems with periodic boundary conditions In 83 this subsection, we test the fourth order and eighth order accurate compact finite 837 difference schemes with the bound-preserving limiter The time step is taken to 838 satisfy both the CFL condition required for weak monotonicity in Theorem and 839 Theorem and the SSP coefficient for high order SSP time discretizations 84 Example One-dimensional linear convection equation Consider u t + u x = 84 with and initial condition u (x) and periodic boundary conditions on the interval 84 [, π] The L and L errors for the fourth order scheme with a smooth initial 843 condition at time T = are listed in Table where x = π N, the time step is taken as t = C ms 3 x for the multistep method, and t = C ms 3 x for the Runge-Kutta 84 method so that the number of spatial discretization operators computed is the same as 84 in the one for the multistep method We can observe the fourth order accuracy for 847 the multistep method and obvious order reductions for the Runge-Kutta method 848 The errors for smooth initial conditions at time T = for the eighth order accu- 849 rate scheme are listed in Table For the eighth order accurate scheme, the time step 8 to achieve the weak monotonicity is t = C ms x for the fourth-order SSP multi- 8 step method On the other hand, we need to set t = x in fourth order accurate 8 time discretizations to verify the eighth order spatial accuracy To this end, the time step is taken as t = C ms x 83 for the multistep method, and t = C ms x for 84 the Runge-Kutta method We can observe the eighth order accuracy for the multistep 8 method and the order reduction for N = is due to the roundoff errors We can 8 also see an obvious order reduction for the Runge-Kutta method Table The fourth order accurate compact finite difference scheme with the bound-preserving limiter on a uniform N-point grid for the linear convection with initial data u (x) = + sin4 (x) Fourth order SSP multistep Fourth order SSP Runge-Kutta N L error order L error order L error order L error order 344E- - 49E- - 34E- - E E E E E E-4 4 9E E E-4 4 E- 4 8E- 4 9E E E-7 4 E- 4 4E- 38 3E- 3 Table The eighth order accurate compact finite difference scheme with the bound-preserving limiter on a uniform N-point grid for the linear convection with initial data u (x) = + sin4 (x) Fourth order SSP multistep Fourth order SSP Runge-Kutta N L error order L error order L error order L error order 3E- - E- - 44E- - 98E- - 33E- 7 9E E E E E- 8 E E E-9 8 E E E E- 74 E- 77 4E E This manuscript is for review purposes only

29 A BOUND-PRESERVING COMPACT FINITE DIFFERENCE SCHEME Next, we consider the following discontinuous initial data: {, if < x π, () u (x) =, if π < x π See Figure for the performance of the bound-preserving limiter and the TVB limiter on the fourth order scheme We observe that the TVB limiter can reduce oscillations but cannot remove the overshoot/undershoot When both limiters are used, we can obtain a non-oscillatory bound-preserving numerical solution See Figure for the performance of the bound-preserving limiter on the eighth order scheme Numerical Exact Numerical Exact (a) without any limiter (b) with only the bound-preserving limiter Numerical Exact Numerical Exact (c) with only the TVB limiter (d) with both limiters Fig Linear convection at T = Fourth order compact finite difference and fourth order SSP multistep with t = Cms x and grid points The TVB parameter in () is p = 3 84 Example One dimensional Burgers equation 8 Consider the Burgers equation u t + ( u ) x = with a periodic boundary condition 8 on [ π, π] For the initial data u (x) = sin(x)+, the exact solution is smooth up to 87 T =, then it develops a moving shock We list the errors of the fourth order scheme 88 at T = in Table 3 where the time step is t = 3 C ms x for SSP multistep and t = 3 C ms x for SSP Runge-Kutta with x = π N We observe the expected fourth 87 order accuracy for the multistep time discretization At T =, the exact solution 87 contains a shock near x = The errors on the smooth region [, π] at T = 87 are listed in Table 4 where high order accuracy is lost Some high order schemes 873 can still be high order accurate on a smooth region away from the shock in this test, 874 see [] We emphasize that in all our numerical tests, Step III in Algorithm was This manuscript is for review purposes only

30 3 H LI, S XIE AND X ZHANG Numerical Exact Numerical Exact (a) Without any limiter (b) With the bound-preserving limiter Fig Linear convection at T = Eighth order compact finite difference and the fourth order SSP multistep method with t = C ms x and grid points never triggered In other words, set of Class I is rarely encountered in practice So the limiter Algorithm is a local three-point stencil limiter for this particular example rather than a global one The loss of accuracy in smooth regions is possibly due to the fact that compact finite difference operator is defined globally thus the error near discontinuities will pollute the whole domain The solutions of the fourth order compact finite difference and the fourth order SSP multistep with the bound-preserving limiter and the TVB limiter at time T = are shown in Figure 3, for which the exact solution is in the range [, ] The TVB limiter alone does not eliminate the overshoot or undershoot When both the bound-preserving and the TVB limiters are used, we can obtain a non-oscillatory bound-preserving numerical solution Table 3 The fourth order scheme with limiter for the Burgers equation Smooth solutions 88 Fourth order SSP multistep Fourth SSP Runge-Kutta N L error order L error order L error order L error order 9E-4-4E-3-779E-4 - E E- 44 3E E E E- 4 E E- 3 9E- 4 E- 44 4E E E E E E E- Table 4 Burgers equation The errors are measured in the smooth region away from the shock Fourth order SSP multistep Fourth SSP Runge-Kutta N L error order L error order L error order L error order 9E- - E- - E- - 39E- - 4 E E- 93 E E E-4 73 E E-4 7 7E E-4 E-3 99 E-4 E E- E E- 3E-4 44 This manuscript is for review purposes only

31 A BOUND-PRESERVING COMPACT FINITE DIFFERENCE SCHEME 3 Numerical Exact Numerical Exact (a) without any limiter (b) with both limiters Fig 3 Burgers equation at T = Fourth order compact finite difference with t = Cms x and grid points The TVB parameter in () is set as p = 3 max x u (x) 9 88 Example 3 One dimensional convection diffusion equation 887 Consider the linear convection diffusion equation u t + cu x = du xx with a periodic 888 boundary condition on [, π] For the initial u (x) = sin(x), the exact solution is 889 u(x, t) = exp( dt)sin(x ct) which is in the range [, ] We set c = and d = 89 The errors of the fourth order scheme at T = are listed in the Table in which 89 t = C ms min{ x c, x 4 d } for SSP multistep and t = C msmin{ x c, x 4 d } for SSP Runge-Kutta with x = π N We observe the expected fourth order accuracy 893 for the SSP multistep method Even though the bound-preserving limiter is triggered, 894 the order reduction for the Runge-Kutta method is not observed for the convection 89 diffusion equation One possible explanation is that the source of such an order reduc- 89 tion is due to the lower order accuracy of inner stages in the Runge-Kutta method, 897 which is proportional to the time step Compared to t = O( x) for a pure con- 898 vection, the time step is t = O( x ) in a convection diffusion problem thus the 899 order reduction is much less prominent See the Table for the errors at T = of the eighth order scheme with t = C ms min{ 3 x c, 3 x 3 d } for SSP multistep and 9 t = C ms min{ 3 x c, 3 x 3 d } for SSP Runge-Kutta where x = π N Table The fourth order compact finite difference with limiter for linear convection diffusion Fourth order SSP multistep Fourth order SSP Runge-Kutta N L error order L error order L error order L error order 33E- - 9E- - 3E- - 9E- - 4 E E E- 4 3E E E E-7 44 E E E E E E- 4 83E- 4 9E- 4 83E- 4 9 Example 4 Nonlinear degenerate diffusion equations A representative test for validating the positivity-preserving property of a scheme solving nonlinear diffusion equations is the porous medium equation, u t = (u m ) xx, m > This manuscript is for review purposes only

32 3 H LI, S XIE AND X ZHANG Table The eighth order compact finite difference with limiter for linear convection diffusion SSP multistep SSP Runge-Kutta N L error order L error order L error order L error order 38E-7-9E-7-38E-7-9E-7-4E-9 8 E E E E- 8 8E- 8 48E- 8 89E E- 3 4E- 4 E E- 4 We consider the Barenblatt analytical solution given by B m (x, t) = t k [( k(m ) x m t k ) +] /(m ), where u + = max{u, } and k = (m + ) The initial data is the Barenblatt solution at T = with periodic boundary conditions on [, ] The solution is computed till time T = High order schemes without any particular positivity treatment will generate negative solutions [,, 4] See Figure 4 for solutions of the fourth order scheme and the SSP multistep method with t = 3m C ms x and grid points Numerical solutions are strictly nonnegative Without the bound-preserving limiter, negative values emerge near the sharp gradients 4th order Compact FD with limiter Exact solution of u t =(u ) xx 4th order Compact FD with limiter Exact solution of u t =(u 8 ) xx (a) m = (b) m = 8 Fig 4 The fourth order compact finite difference with limiter for the porous medium equation 9 One-dimensional problems with non-periodic boundary conditions 9 Example One-dimensional Burgers equation with inflow-outflow boundary 9 condition Consider u t + ( u ) x = on interval [, π] with inflow-outflow boundary condition and smooth initial condition u(x, ) = u (x) Let u (x) = sin(x) +, 94 we can set the left boundary condition as inflow u(, t) = L(t) and right boundary as 9 outflow, where L(t) is obtained from the exact solution of initial-boundary value prob- 9 lem for the same initial data and a periodic boundary condition We test the fourth 97 order compact finite difference and fourth order SSP multistep method with the bound- 98 preserving limiter The errors at T = are listed in Table 7 where t = C ms x and 99 x = π N See Figure for the shock at T = 3 on a -point grid with t = C ms x 9 9 Example One-dimensional convection diffusion equation with Dirichlet boundary conditions We consider equation u t + cu x = du xx on [, π] with boundary con- This manuscript is for review purposes only

33 A BOUND-PRESERVING COMPACT FINITE DIFFERENCE SCHEME 33 Table 7 Burgers equation The fourth order scheme Inflow and outflow boundary conditions N L error order L error order E-4-78E-4-4 4E- 48 E E E-7 44 E E E- 4 87E (a) Without any limiter (b) With the bound-preserving limiter Fig Burgers equation The fourth order scheme Inflow and outflow boundary conditions ditions u(, t) = cos( ct)e dt and u(π, t) = cos(π ct)e dt The exact solution is u(x, y, t) = cos(x ct)e dt We set c = and d = We test the third order boundary scheme proposed in Section and the fourth order interior compact finite difference with the fourth order SSP multistep time discretization The errors at T = are listed in Table 8 where t = C ms min{ 4 x 9 c, 9 x 9 d }, x = π N Table 8 A linear convection diffusion equation with Dirichlet boundary conditions N L error order L error order 8E-3-87E-3-47E-4 3 7E E E E E- 43 3E-8 47 E Two-dimensional problems with periodic boundary conditions In 98 this subsection we test the fourth order compact finite difference scheme solving two- 99 dimensional problems with periodic boundary conditions 93 Example 7 Two-dimensional linear convection equation Consider u t + u x + 93 u y = on the domain [, π] [, π] with a periodic boundary condition The scheme 93 is tested with a smooth initial condition u (x, y) = + sin4 (x + y) to verify the accuracy The errors at time T = are listed in Table 9 where t = C ms x for the SSP multistep method and t = C ms x for the SSP Runge-Kutta method with x = y = π N We can observe the fourth order accuracy for the multistep method 93 on resolved meshes and obvious order reductions for the Runge-Kutta method This manuscript is for review purposes only

34 34 H LI, S XIE AND X ZHANG Table 9 Fourth order accurate compact finite difference with limiter for the D linear equation Fourth order SSP multistep Fourth order SSP Runge-Kutta N N Mesh L error order L error order L error order L error order 47E- - 7E- - 84E- - 7E- - 47E E-3 37 E E E E E E E E- 49 9E E- 38 9E- 43 8E- 4 E- 88 E (a) Without any limiter (b) With bound-preserving limiter Fig Fourth order compact finite difference for the D linear convection We also test the following discontinuous initial data: {, if (x, y) [, ] [, ], u (x, y) =, otherwise The numerical solutions on a 8 8 mesh at T = are shown in Figure with t = C ms x and x = y = π N Fourth order SSP multistep method is used 939 Example 8 Two-dimensional Burgers equation Consider u t +( u ) x+( u ) y = 94 with u (x, y) = + sin(x + y) and periodic boundary conditions on [ π, π] [ π, π] 94 At time T =, the solution is smooth and the errors at T = on a N N mesh 94 x are shown in the Table in which t = C ms max x u (x) for multistep and t = 943 x C ms max x u (x) for Runge-Kutta with x = y = π N At time T =, the exact 944 solution contains a shock The numerical solutions of the fourth order SSP multistep 94 method on a mesh are shown in Figure 7 where t = max x u C (x) ms x 94 The bound-preserving limiter ensures the solution to be in the range [, ] 947 Example 9 Two-dimensional convection diffusion equation 948 Consider the equation u t + c(u x + u y ) = d(u xx + u yy ) with u (x, y) = sin(x + y) 949 and a periodic boundary condition on [, π] [, π] The errors at time T = 9 for c = and d = are listed in Table, in which t = C ms min{ x c, x 48d } 9 for the fourth-order SSP multistep method, and t = C ms min{ x c, x 48d } for the fourth-order SSP Runge-Kutta method, where x = y = π N 93 Example Two-dimensional porous medium equation This manuscript is for review purposes only

35 A BOUND-PRESERVING COMPACT FINITE DIFFERENCE SCHEME 3 Table Fourth order compact finite difference scheme with the bound-preserving limiter for the D Burgers equation SSP multistep SSP Runge-Kutta N N Mesh L error order L error order L error order L error order 8E E-3-9E-3-373E E E E E E- 44 4E E- 383 E E E- 49 3E- 3 4E- 8 87E E E-7 7 3E (a) Without the boundpreserving limiter (b) With the boundpreserving limiter 3 - (c) The exact solution Fig 7 The fourth order scheme D Burgers equation 94 9 We consider the equation u t = (u m ) with the following initial data {, if (x, y) [, ] [, ], u (x, y) =, if (x, y) [, ] [, ]/[, ] [, ], 9 and a periodic boundary condition on domain [, ] [, ] See Figure 8 for the 97 solutions at time T = for SSP multistep method with t = 48 max x u C (x) ms x 98 and x = y = The numerical solutions are strictly non-negative, which is 99 nontrivial for high order accurate schemes High order schemes without any positivity 9 treatment will generate negative solutions in this test, see [,, 4] (a) m = (b) m = (c) m = Fig 8 The fourth order scheme with limiter for D porous medium equations u t = (u m ) Concluding remarks In this paper we have demonstrated that fourth order accurate compact finite difference schemes for convection diffusion problems with periodic boundary conditions satisfy a weak monotonicity property, and a simple This manuscript is for review purposes only