A parametrized maximum principle preserving flux limiter for finite difference RK-WENO schemes with applications in incompressible flows.

Transcription

1 A parametrized maximum principle preserving flux limiter for finite difference RK-WENO schemes with applications in incompressible flows Tao Xiong Jing-ei Qiu Zhengfu Xu 3 Abstract In Xu [] a class of parametrized flux limiters is developed for high order finite difference/volume essentially non-oscillatory (ENO) and Weighted ENO (WENO) schemes coupled with total variation diminishing (TVD) Runge-Kutta (RK) temporal integration for solving scalar hyperbolic conservation laws to achieve strict maximum principle preserving (PP). In this paper we continue along this line of research but propose to apply the parametrized PP flux limiter only to the final stage of any RK method. Compared with the original work [] the proposed new approach has several advantages: First the PP property is preserved with high order accuracy without as much time step restriction; Second the implementation of the parametrized flux limiters is significantly simplified. Local truncation error analysis is performed to justify the maintenance of high order spatial/temporal accuracy when the PP flux limiters are applied to high order finite difference schemes solving linear problems. We further apply the limiting procedure to the simulation of the incompressible flow: we first design a first order PP scheme for the incompressible flow then apply an PP flux limiter to the flux of high order scheme toward that of a first order PP scheme. The PP property is guaranteed without affecting the designed order of spatial and temporal accuracy for the incompressible flow computation. The efficiency and effectiveness of the proposed scheme is demonstrated via several test examples. Keywords: hyperbolic conservation laws high order scheme WENO parametrized flux limiter maximum principle preserving Runge-Kutta method incompressible flow Department of athematics University of Houston Houston txiong@math.uh.edu Department of athematics University of Houston Houston jingqiu@math.uh.edu. The first and second authors are supported by Air Force Office of Scientific Computing YIP grant FA NSF grant DS and DS-7008 University of Houston. 3 Department of athematical Science ichigan Technological University Houghton zhengfux@mtu.edu

2 Introduction In this paper we consider the following scalar hyperbolic conservation laws u t + F(u) = 0 u(x 0) = u 0 (x). (.) Popular methods for solving (.) include finite difference/volume schemes based on high order essentially non-oscillatory (ENO) and Weighted ENO (WENO) reconstructions [ 5] and finite element discontinuous Galerkin (DG) methods coupled with total variation diminishing (TVD) Runge-Kutta (RK) time discretization [7]. Our focus of this paper is the finite difference RK-WENO scheme. The close relationship between the finite difference and finite volume scheme was first explained by Shu and Osher [7 8] by introducing a sliding function h(x). Compared with finite volume schemes high order finite difference schemes are more computationally efficient for high dimensional implementations. Compared with DG finite difference schemes with WENO reconstruction are more robust in capturing shocks without oscillations. An important property of the solution for hyperbolic conservation laws (.) is the strict maximum principle [3] namely u m u(x t) u if u m u 0 (x) u. The TVD schemes satisfy the strict maximum principle but it is only first order at the smooth extrema. ENO and WENO schemes are essentially non-oscillatory around discontinuities; however the numerical solutions do not necessarily preserve the strict maximum principle. A genuinely high order conservative scheme to preserve the global maximum principle has recently been developed by Zhang and Shu in [3 5]. PP limiters are applied to the reconstructed high order polynomials in the finite volume/dg framework around the cell averages in order for the updated cell-average values of the numerical solutions to satisfy the maximum principle. The techniques have recently been applied to a number of problems including the compressible/incompressible Euler equations shallow water equations among many others [ ]. However it was also pointed out in [6] that it is not trivial to apply the PP limiters to the finite difference schemes without destroying the designed order of accuracy since the non-physical sliding average function h(x) does not necessarily satisfy the maximum principle for u. oreover the CFL number required to preserve the PP property while maintaining high order spatial and temporal accuracy is much smaller than the one for the original scheme i.e the CFL must be less than for a third order finite volume scheme 6 with PP limiters [3]. In [] Xu developed a parametrized PP flux limiting technique to maintain the PP property of numerical solutions of the one-dimensional scalar hyperbolic conservation laws.

3 Compared with limiting the cell-wise reconstructed polynomials in [3 5] a general framework is proposed in [] to apply the PP flux limiter to high order numerical fluxes of conservative high order schemes toward first order monotone fluxes. The PP requirement of u m u h u is described by a group of explicit inequalities. By decoupling these inequalities the numerical fluxes are locally redefined leading to a consistent conservative maximum principle preserving high order scheme. When coupled with the TVD RK scheme a successive parametrized limiting approach with some relaxed upper and lower bound is proposed for each stage of the RK method. The method was later generalized to the high order methods for solving multi-dimensional scalar hyperbolic conservation laws []. The PP property is guaranteed under the same CFL time step restriction of the first order monotone scheme. However the scheme suffers from a mild time step restriction for the preservation of high order accuracy. In this paper following the idea in [] we focus on developing the PP flux limiter for conservative high order schemes exemplified by the finite difference WENO scheme coupled with TVD RK time discretization. There are two new ingredients in this paper. First we propose to implement the parametrized PP flux limiters only at the final stage of the multi-stage RK time discretization. It was commented in [] that if the PP flux limiter is applied at each of the intermediate stage of RK method due to the influence of the special cancelation of RK high order temporal accuracy could be lost. With the flux limiter applied only at the final RK stage high order temporal accuracy is preserved without as much time step restriction as []. Compared with the successive PP flux limiters proposed in [ ] the implementation complexity is significantly reduced. Local truncation error analysis is performed to confirm the maintenance of high order spatial and temporal accuracy. Secondly we design a high order finite difference scheme for incompressible flow with the PP property. The schemes are applied to advection in incompressible flows with stream-vortex formulation. It is numerically checked that the proposed schemes enjoy high order accuracy and maximum principle preserving without additional time step restriction. It is also numerically observed that a linear scheme (without the nonlinear WENO reconstruction) with the PP flux limiter could sometimes resolve discontinuities very well without oscillations. The rest of the paper is organized as follows. In Section we will first review the high order finite difference schemes [6] and the parametrized PP flux limiters in Xu []. In Section 3 we describe on how to apply the PP limiter on the final stage of a multi-stage RK method in one- and two-dimensional cases. Local truncation error analysis is provided to show that the newly proposed limiter preserves high order accuracy in both space and time 3

4 without excessive CFL restriction. In Section 4 a high order finite difference scheme with the PP property is proposed for incompressible flow. In Section 5 numerical examples are presented to show the efficiency and effectiveness of the proposed approach. Review of finite difference WENO scheme [6] and parametrized PP flux limiter []. Finite difference WENO scheme. We first briefly review the finite difference WENO scheme [6] for a simple one-dimensional hyperbolic conservation equation u t + f(u) x = 0 x [0 ] (.) with initial condition u(x 0) = u 0 (x). Without loss of generality we assume periodic boundary condition. We adopt the following spatial discretization for the spatial domain [0 ] where I j = [x j point x j = (x j 0 = x < x 3 < < x N+ x j+ ] has the mesh size x =. Let u N j(t) denote the solution at grid ) at continuous time t. The finite difference scheme evolves the + x j+ point values of the solution by approximating the differential form of the equation (.) directly in a conservative form. = d dt u j(t) + x (Ĥj+/ Ĥj /) = 0. (.) Ĥ j+/ = ˆf(u i p u i+q ) is a numerical flux consistent with the physical flux f(u) and is Lipschitz continuous with respect to all arguments. The stencil {u i p u i+q } is chosen to be upwind biased. Especially when f (u) 0 one more point from the left (p = q) will be taken to reconstruct ˆf i+ ; otherwise one more point from the right (p = q ) will be taken. When f (u) changes sign over the domain then a flux splitting e.g. the Lax-Friedrichs flux splitting can be applied. The spatial accuracy of the scheme is determined by how well x (Ĥi+ Ĥi ) approximates f(u) x. To obtain a high order approximation Shu and Osher [7] introduced a sliding average function h(x) such that x+ x h(ξ)dξ = f(u(x t)). (.3) x x x 4

5 Taking the x derivative of the above equation gives ( h(x + x ) x ) h(x x ) Therefore the numerical flux Ĥi+ = f(u) x. (.4) in equation (.5) can be taken as h(x i+ ) which can be reconstructed from neighboring cell averages of h(x) h j = x I j h(ξ)dξ (.3) = f(u(x j t)) j = i p i + q by WENO reconstruction. By adaptively assigning nonlinear weights to neighboring candidate stencils the WENO reconstruction preserves high order accuracy of the linear scheme around smooth regions of the solution while producing a sharp and essentially non-oscillatory capture of discontinuities. Equation (.) can be further discretized in time by a high order time integrator. For example the scheme with the first order forward Euler time discretization is where u n j step size. u n+ j = u n j λ(ĥj+/ Ĥj /) (.5) denotes the numerical solution at x j and at time t n λ = t x and t is the time The finite difference WENO schemes have been extended to the adaptive mesh refinement (AR) framework [4] and have been applied to a wide range of problems including computational fluid dynamics astrophysics plasma physics semi-conductor device simulations among many others [6]. It is well-known that the schemes enjoy the mass conservation high order spatial accuracy and the flexibility in implementation for high dimensional problems when compared with the finite volume framework. However the maximum principle preserving (PP) property of the solution for hyperbolic equation (.) is not preserved on the numerical level. For more details please see [6] and the references therein.. A parametrized PP flux limiter [] Below we illustrate the idea of parametrized flux limiters for preserving the maximum principle of the high order finite difference scheme. For simplicity we first consider the one step forward Euler time discretization. The idea will be generalized to a high order TVD Runge-Kutta time discretization in the next section. Let To preserve the PP property u m = min(u(x 0)) u = max(u(x 0)). x x u m u n j u j n (.6) 5

6 it is proposed in [] to modify the numerical fluxes Ĥj+/ as H j+/ = θ j+/ (Ĥj+/ ĥj+/) + ĥj+/ (.7) where ĥj+/ = ˆf(u j u j+ ) is the first order monotone flux. Notice that with the first order monotone flux in equation (.5) it is well-known that the PP property (.6) will be preserved. θ j+/ [0 ] is designed by taking advantage of the PP property with the use of the first order monotone flux ĥj+/ and the high order accuracy of Ĥj+/ reconstructed from WENO procedure. Below is a detailed procedure of designing θ j+/. For each θ j+/ limiting the numerical flux H j+/ we are looking for upper bounds Λ /Ij and Λ +/Ij from the need of keeping u n+ j within [u m u ]. Consequently θ j+/ [0 Λ + I ] [0 Λ j I ] j (.8) j+ provides a sufficient condition for the scheme to preserve the maximum principle. Let Γ j = u u j + λ(ĥj+/ ĥj /) Γ m j = u m u j + λ(ĥj+/ ĥj /) then from the PP property of a first order monotone scheme To ensure u n+ j Γ j 0 Γ m j 0. [u m u ] with H j+/ as in equation (.7) it is sufficient to have λθ j / (Ĥj / ĥj /) λθ j+/ (Ĥj+/ ĥj+/) Γ j 0 (.9) λθ j / (Ĥj / ĥj /) λθ j+/ (Ĥj+/ ĥj+/) Γ m j 0. (.0) The discussion is case by case based on the sign of F j±/. = Ĥ j±/ ĥj±/.. Assume θ j / [0 Λ /I j ] θ j+/ [0 Λ +/I j ] where Λ /I j and Λ +/I j are designed to preserve the upper bound by equation (.9) (a) If F j / 0 and F j+/ 0 (Λ /I j Λ +/I j ) = ( ). 6

7 (b) If F j / 0 and F j+/ < 0 (c) If F j / > 0 and F j+/ 0 (Λ /I j Λ +/I j ) = ( min( Γ j λf j+/ )). (d) If F j / > 0 and F j+/ < 0 (Λ /I j Λ +/I j ) = (min( Γ j λf j / ) ). If equation (.9) is satisfied with (θ j / θ j+/ ) = ( ) then (Λ /I j Λ +/I j ) = ( ). If equation (.9) is not satisfied with (θ j / θ j+/ ) = ( ) then Γ (Λ /I j Λ j Γ j +/I j ) = ( ). λf j / λf j+/ λf j / λf j+/. Similarly assume θ j / [0 Λ m /I j ] θ j+/ [0 Λ m +/I j ] where Λ m /I j and Λ m +/I j are designed to preserve the lower bound by equation (.0). (a) If F j / 0 and F j+/ 0 (Λ m /I j Λ m +/I j ) = ( ). (b) If F j / 0 and F j+/ > 0 (c) If F j / < 0 and F j+/ 0 (d) If F j / < 0 and F j+/ > 0 (Λ m /I j Λ m +/I j ) = ( min( (Λ m /I j Λ m +/I j ) = (min( Γ m j λf j+/ )). Γ m j λf j / ) ). If equation (.0) is satisfied with (θ j / θ j+/ ) = ( ) then (Λ m /I j Λ m +/I j ) = ( ). 7

8 If equation (.0) is not satisfied with (θ j / θ j+/ ) = ( ) then Γ m j Γ m (Λ m /I j Λ m j +/I j ) = ( ). λf j / λf j+/ λf j / λf j+/ Notice that the range of θ j+/ (.8) is determined by the need to ensure both the upper bound (.9) and the lower bound (.0) of numerical solutions in both cell I j and I j+. Thus the locally defined limiting parameter is given as θ j+/ = min(λ +/Ij Λ /Ij+ ) (.) with Λ +/Ij = min(λ +/I j Λ m +/I j ) Λ /Ij+ = min(λ /I j+ Λ m /I j+ ). The modified flux in equation (.7) with the θ j+/ designed above ensures the maximum principle. Such modified flux is consistent and monotone since it is a convex combination (θ j+/ [0 ]) of a high order flux Ĥj+/ with the first order flux ĥj+/. Since the scheme is in the flux difference form (.5) the mass conservation property is preserved. It is proven in [] that if the time step size t is small enough the high order spatial accuracy of the scheme with regular WENO flux Ĥj+/ is maintained with the modified flux H j+/. Remark.. In the parametrized flux limiter approach we start from looking for a range rather than a single value for the θ j+/. We let θ j+/ to be the upper bound of such ranges determined from the need to bound both u n+ j and u n+ j+ with PP property. 3 A new parametrized PP limiter for the RK-WENO 3. One-dimensional problem A successive PP limiting procedure was proposed in [] for limiting the upper and lower bounds of solutions at internal stages of a third order TVD RK method []. In this section we propose to apply the PP flux limiting procedure at the final stage of RK time discretization only. The newly proposed limiting procedure is very general in the sense that it can be applied to any high order explicit RK method. oreover the time step restriction to ensure both PP property and high order accuracy in both space and time is relieved compared with that proposed in [] see Theorem 3. below. To illustrate the idea without loss of generality we use a third order TVD RK time discretization below as an example. With the method-of-lines approach the third order 8

9 TVD RK time discretization [] can be written as u () = u n + tl(u n ) u () = u n + t( 4 L(un ) + 4 L(u() )) u n+ = u n + t( 6 L(un ) + 3 L(u() ) + 6 L(u() )). (3.) We note that the second term in equation (3.) is the fourth order Simpson s rule in approximating t n+ L(u(τ))dτ. Here L(u n ) =. t n x (Ĥn j+/ Ĥn j / ) where Ĥn j+/ is the numerical flux from WENO reconstruction based on u n. Similarly let Ĥ() j+/ and Ĥ() j+/ be the numerical fluxes reconstructed based on u () and u (). The equation (3.) can be rewritten as u n+ j = u n j λ(ĥrk j+/ Ĥrk j /) (3.) where Ĥ rk j+/. = 6Ĥn j+/ + 3Ĥ() j+/ + 6Ĥ() j+/. Based on equation (3.) we propose to replace the numerical flux Ĥrk j+/ one by the modified H rk j+/ = θ j+/ (Ĥrk j+/ ĥj+/) + ĥj+/ (3.3) where ĥj+/ is the first order monotone flux and θ j+/ is designed in the same way as that in the review section. to preserve the PP property. It shall be pointed out that most of the explicit RK temporal approximation method can be written in the form of (3.). Therefore the newly proposed PP flux limiting procedure can be directly applied to most of the high order conservative schemes equipped with an explicit RK temporal integration. It is proven in Theorem 3. below that the proposed flux limiters maintain high order spatial accuracy of WENO reconstruction and temporal accuracy of RK discretization. Remark 3.. The proposed flux limiting procedure is very simple and computationally efficient compared with that in []. Notice that numerical solutions at intermediate RK stages i.e. u () and u () are low order in time approximations to the exact solutions at the corresponding stages. If the PP properties are enforced at these intermediate stages the special cancelation of RK method may be affected. As a result very restrictive time step size would be needed to maintain high order accuracy in time see []. 9

10 In the following we will show that the high order accuracy in both space and time is maintained with the newly proposed PP limiter when the solution is smooth enough. As there is no rigorous error analysis for finite difference scheme with high order reconstruction by introducing the sliding function h(x) and high order RK method for hyperbolic equation (.) we make some assumptions on the error of the original high order finite difference scheme. Notice that these assumptions have been numerically verified extensively but haven t been rigorously proven. For the linear problem f(u) = u in (.) with ĥj+/ = u j we have Theorem 3.. Consider using a third order finite difference spatial discretization to reconstruct the numerical flux Ĥj+ in equation (.) and using a high order TVD RK method discretize the time derivative in equation (.). Assume that e n j = u n j u(x j t n ) = O( x 3 + t r ) n j. (3.4) Consider applying the limiting procedure to the numerical flux in equation (3.) as proposed in equation (3.3) then Ĥrk j+/ H rk j+/ = O( x 3 + t r ) j (3.5) with CFL subject to the linear stability constraint of the scheme without the proposed PP flux limiter. r is the order of accuracy of the chosen RK integration. Proof. We only consider the limiters for the maximum value case it is similar for the minimum value case. The statement is proved via discussing four cases discussed in Section.. holds. Case (a): No limiters are introduced in case (a) decoupling therefore equation (3.5) Case (d): This is the case of F j / > 0 and F j+/ < 0. From (3.3) it is sufficient to show that Γ j (λf j / λf j+/ ) λf j / λf j+/ F j+/ = O( x 3 + t r ) (3.6) when Γ j < λf j / λf j+/. Since F j / > 0 and F j+/ < 0 we have 0 < /λ. Recalling F j+/ λf j / λf j+/ Γ j (λf j / λf j+/ ) = u {u j λ(ĥrk j+/ Ĥrk j /)} < 0 (3.7) it suffices to show u {u j λ(ĥrk j+/ Ĥrk j /)} O( x 3 + t r ) (3.8) 0

11 which can be justified by using equation (3.4). with Case (b): Similar to case (d) we only need to consider the case when Λ +/Ij = Γ j λf j+/ <. (3.9) H rk j+/ Ĥrk j+/ = Γ j To prove (3.5) it suffices to prove + λf j+/ λ = u {u j λ(ĥrk λ j+/ ĥj /)}. (3.0) For the high order RK flux we have Ĥ rk j+ = t t n+ u (u j λ(ĥrk j+ u j )) O( x 3 + t r ) (3.) h(x j+ τ)dτ + O( t r ) = t n t t 0 h(x j+ τ t n )dτ + O( t r ) (3.) where the sliding average function h can be given in the following expanded form [8] s h(x j+ t) = u(x j+ t) + k= a k x k ( k ) u(x t) + O( x s+ ) (3.3) xk x=x j+ with properly defined {a k } to ensure h(x j+/ t) h(x j / t) = u x (x j t) + O( x s+ ) for arbitrary s. In the following we assume h is a global smooth function of (x t). For the advection problem h(x t) as a function of u and its derivatives in (3.3) follow the same characteristics as u which gives the second equality in (3.). The same argument will also be used in the 4th order proof in the appendix. For the third order case we take the first two terms in (3.3) and approximate h(x t n ) with the following polynomial P h (x) = u j + u j u j (x x j ) + u j+ u j + u j ((x x x 6 x j )(x x j+ ) +(x x j )(x x j+ 3 ) + (x x j+ )(x x j+ 3 )) (3.4) which is reconstructed based on the point values u j u j u j+. Thus t t 0 h(x j+ τ t n )dτ = xj+ h(ξ t n )dξ λ x x j+ λ x = 6 (5u j + u j+ u j ) + λ (u j u j+ ) + λ 6 (u j+ u j + u j ) + O( x 3 )

12 and u j λ ( t t 0 ) h(x j+ τ t n )dτ u j = ( λ 6 (5 λ)( + λ))u j + λ 6 (7 λ )u j λ 6 ( λ)( λ)u j+ + O( x 3 ). (3.5) We first prove (3.) in the case that x I j with u = u(x ) u = 0 and u 0. Other cases will be considered separately. We perform the Taylor expansions u j = u + u (x j x ) + u (x j x ) + O( x 3 ) u j = u + u (x j x x) + u (x j x x) + O( x 3 ) u j+ = u + u (x j x + x) + u (x j x + x) + O( x 3 ). Since u = 0 (3.5) can be rewritten as ( t ) u j λ h(x t j+ τ t n )dτ u j = u + u x f(z λ) + O( x 3 ) (3.6) with 0 f(z λ) = λ(5 λ)( + λ) + 6λ(λ 3)z + 6z (3.7) and z = (x j x )/ x. The minimum value for function f with respect to z is f min = f(z λ) = λ (λ )(λ )(5 3λ) 0 z= λ(λ 3) with λ [0 ]. Hence f(z λ) 0. From (3.) and (3.6) we have u (u j λ(ĥrk u j+ j )) = u x f(z λ) + O( x 3 + t r ) O( x 3 + t r ) (3.8) Now if u(x) reaches its maximum or local maximum at x j that is x = x j we have u = u 0 following the previous calculation and take z = (x j j x )/ x = / we have u j λ(ĥrk u j+ j ) = u + ( 3λ + λ + z)u x + u x f(z λ) + O( x 3 + t r ) = u + ( 3λ + λ )u x + (3 8λ + λ 4λ 3 )u x +O( x 3 + t r ) = u(x s x) + s u x + O( x 3 + t r ) (3.9)

13 where s = (3 8λ + λ 4λ 3 ) (3.0) s = ( 3λ + λ ) + (3 8λ + λ 4λ 3 ) (3.) it is easy to check that s is real and s 0 for 0 λ and we obtain u j λ(ĥrk j+ u j ) = u(x s x) + s u x + O( x 3 + t r ) (3.) We can similarly prove (3.) if x = x j+ with u we have proved (3.). Case (c): Similar to case (b). 3. Two-dimensional problem We consider the two-dimensional scalar problem u + O( x 3 + t r ) (3.3) 0. Combining the above discussion u t + f(u) x + g(u) y = 0 u(x y 0) = u 0 (x y). (3.4) The multi-dimensional parametrized PP flux limiters for high order schemes solving (3.4) are developed in []. We refer to [] for the algorithm description and implementation details of the successive PP flux limiters for multi-dimensional problems. In this section we would like to apply the PP flux limiters at the final stage of the multi-stage RK-WENO schemes solving two-dimensional problem (3.4). As in the one-dimensional case the high order finite difference RK-WENO scheme can be written as u n+ ij = u n ij λ x (Ĥrk i+/j Ĥrk i /j) λ y (Ĝrk ij+/ Ĝrk ij /) (3.5) where Ĥrk and Ĝrk are linear combination of fluxes from RK multiple stages. Let ĥi+/j ĝ ij+/ be any first order monotone flux satisfying maximum principle u m u n ij λ x (ĥi+/j ĥi /j) λ y (ĝ ij+/ ĝ ij / ) u. (3.6) In order to ensure maximum principle we are looking for the type of limiters H i+/j = θ i+/j (Ĥrk i+/j ĥi+/j) + ĥi+/j G ij+/ = θ ij+/ (Ĝrk ij+/ ĝ ij+/ ) + ĝ ij+/ (3.7) 3

14 such that u m u n ij λ x ( H i+/j H i /j ) λ y ( G ij+/ G ij / ) u. (3.8) (3.7) and (3.8) form coupled inequalities for the limiting parameters θ i+/j θ ij+/. As for the -D case for each node (i j) the maximum principle preserving limiters can be parametrized in the sense that we can find a group of numbers Λ Lij Λ Rij Λ Dij Λ Uij such that the numerical solutions of (3.5) satisfy the PP property (3.8) with (θ i /j θ i+/j θ ij / θ ij+/ ) [0 Λ Lij ] [0 Λ Rij ] [0 Λ Dij ] [0 Λ Uij ]. For the maximum value case let Γ ij = u (u ij λ x (ĥi+/j ĥi /j) λ y (ĝ ij+/ ĝ ij / )) 0 (3.9) t when a monotone numerical flux is used under suitable CFL constraint α x + α x y t y here α x = max u f (u) and α y = max u g (u). Denote F i /j = λ x (Ĥrk i /j ĥi /j) F i+/j = λ x (Ĥrk i+/j ĥi+/j) F ij / = λ y (Ĝrk ij / ĝ (3.30) ij /) F ij+/ = λ y (Ĝrk ij+/ ĝ ij+/). The coupled inequalities (3.7) and (3.8) can be rewritten as θ i+/j F i+/j + θ i /j F i /j + θ ij+/ F ij+/ + θ ij / F ij / Γ ij (3.3) We shall now focus on decoupling the inequalities (3.3). For the single node (i j). Identify positive values out of the four locally defined numbers F i /j F i+/j F ij / F ij+/ ;. Corresponding to those positive values collectively the limiting parameters can be defined. For example if F i+/j F i /j > 0 and F ij / F ij+/ 0 then For the minimum value part let { Λ i+/j Λ i /j = min( Γ ij F i+/j +F i /j ) Λ ij / Λ ij+/ =. (3.3) Γ ij = u m (u ij λ x (ĥi+/j ĥi /j) λ y (ĝ ij+/ ĝ ij / )) 0. (3.33) 4

15 The coupled inequalities (3.7) and (3.8) can be rewritten as Γ ij θ i+/j F i+/j + θ i /j F i /j + θ ij+/ F ij+/ + θ ij / F ij /. (3.34) A similar procedure would be applied. Identify negative values out of the four locally defined numbers F i /j F i+/j F ij / F ij+/ ;. Corresponding to the negative values collectively the limiting parameters can be defined. For example if F ij / F ij+/ 0 and F i /j F i+/j < 0 then { Λ m i /j Λ m i+/j = min( Γ ij F i /j +F i+/j ) Λ m ij / Λm ij+/ =. (3.35) Namely all high order fluxes which possibly contribute (beyond that of the first order fluxes) to the overshooting or undershooting of the updated value shall be limited by the same scaling. Similarly we can find Λ ij±/ and Λm ij±/ The range of the limiting parameters satisfying PP for a single node (i j) therefore can be defined by Λ Lij = min(λ i /j Λm i /j ) Λ Rij = min(λ i+/j Λm i+/j ) Λ Uij = min(λ ij+/ Λm ij+/ ) (3.36) Λ Dij = min(λ ij / Λm ij / ). Considering the limiters from neighboring nodes finally we let { θ i+/j = min(λ Rij Λ Li+j ) θ ij+/ = min(λ Uij Λ Dij+ ). (3.37) 4 The PP flux limiter for incompressible flow Consider D equations describing advection in incompressible flow u t + (v (x y t)u) x + (v (x y t)u) y = 0 (4.) in conservative form with the divergence free condition of the velocity field x v = 0. 5

16 With such condition there exists a potential function Φ s.t. v = x Φ. The solutions of such equation enjoy the properties of mass conservation and strict maximum principle which we wish to preserve for the numerical scheme. We start from designing a first order monotone scheme in flux difference form. D ± x w ij = ±(w i±j w ij ) and D ± y w ij = ±(w ij± w ij ). Assuming α i j = t x y (Φ i j Φ i j ) 0 α ij = t x y (Φ ij Φ i j ) 0 The first order monotone scheme is designed by using the potential function Φ: u n+ ij = u n ij t ( D x ((D y Φ)u n ) + D y ((D x Φ)u n ) ) = α ij u n ij + α i j u n i j + α ij u n ij (4.) Let with α ij = (α i j + α ij ). These coefficients α are positive under the CFL condition α x t x + α y t y here α x = max D y Φ / y and α y = max D x Φ / x. With the updated solution being the convex combination of the solution from previous time steps it is easy to check that the first order scheme (4.) satisfies the PP property. For the general case we perform a flux splitting of Φ. Friedrichs splitting we let For example with the Lax- Φ + = (Φ + α xx + α y y) Φ = (Φ (α xx + α y y)) (4.3) with α x and α y large enough so that D x (Φ + ) 0 D y (Φ + ) 0 D + x (Φ ) 0 and D + y (Φ ) 0. A first order monotone scheme satisfying the PP property can be designed as u n+ ij = u n ij t ( D + x (( D y Φ + )u n ) + D x (( D + y Φ )u n ) +D y ((D + x Φ + )u n ) + D + y ((D x Φ )u n ) ). (4.4) Similar to equation (4.) it is easy to check that the scheme (4.4) satisfies the PP property. Similar to the PP flux limiters presented in Section 3 the high order finite difference RK WENO scheme for the conservative form of incompressible flow (4.) can be written in the form of (3.5) with the PP flux limiter (3.7) where ĥ i+ j = Φ i+j Φ i+j u i+j Φ ij+ Φ ij u ij α x y y (u i+j u ij ) 6

17 ĝ ij+ = Φ i+j Φ ij u ij + Φ ij+ Φ i j+ u ij+ α y x x (un ij+ u ij ) is the numerical flux from the first order scheme (4.4) with PP property. 5 Numerical simulations 5. Basic tests. In this section we present numerical examples for the proposed parametrized PP flux limiters for high order finite difference schemes with high order RK time discretization. There are two schemes we tested. One is the 3rd order finite difference scheme with 3rd order Runge- Kutta time discretization denote as FD3RK3 ; the other is the 5th order finite difference WENO scheme with 4th order RK time discretization [7] denoted as WENO5RK4. We use the global Lax-Friedrichs scheme as the first order monotone scheme unless otherwise stated. And the CFL condition in our numerical experiments is defined to be max f (u) t CF L x for one dimensional case and max f (u) t + max x g (u) t CF L for two dimensional y case. Example 5.. Consider the D linear equation u t + u x = 0 u(x 0) = u 0 (x) (5.) on [0 π] with periodic boundary conditions. We take the initial condition to be u 0 (x) = sin 4 (x). We list the L and L errors at T = 0.5 in Tables with CF L = 0.6. The 3rd order for FD3RK3 and 5th order for WENO5RK4 schemes with the PP flux limiters are numerically observed. The minimum values are observed to be strictly non-negative from schemes with limiters. We also test CF L =.0 for FD3RK3 and report results in Tables Similar behaviors are observed. Example 5.. Consider the D nonlinear Burgers equation with periodic boundary conditions on [0 π] ( ) u u t + x = 0 u(x 0) = u 0 (x) (5.) and with the initial condition u 0 (x) = sin 4 (x). The exact solution is smooth up to t = The errors at T = 0.5 are reported in Tables The numerical solutions are observed to enjoy the PP property. The order of accuracy is maintained for both FD3RK3 and WENO5RK4. After t = 4 3 the solution develops a still shock. We 9 show the solutions at T =. in Figure 5.. For FD3RK3 at mesh N = 60 without 7

18 Table 5.: L and L error and order for FD3RK3 T = 0.5 CF L = 0.6 D linear equation with initial condition u 0 (x) = sin 4 (x) without limiters. N L error order L error order (u h ) min 0.e e-0 -.6E e e E e e E e e E e e E-06 Table 5.: L and L error and order for FD3RK3 T = 0.5 CF L = 0.6 D linear equation with initial condition u 0 (x) = sin 4 (x) with limiters. N L error order L error order (u h ) min 0.83e e E e e E e e E e e E e e E- limiters there are several large overshoots and undershoots. However with the PP flux limiters the overshoots and undershoots are completely under control with the minimum value (u h ) min = E and the maximum value (u h ) max = Without limiters the minimum value for WENO5RK4 is also negative which is (u h ) min =.753E 05 at mesh N = 60 but with limiters the minimum value is (u h ) min =.0706E. Figure 5.: Numerical solutions for D Burgers equation at T =.. esh: N = 60. CFL=0.6. Left: without limiter; Right: with limiter. Top: FD3RK3; Bottom: WENO5RK4. Example 5.3. Consider the D linear equation on [ ] [ ] with discontinuous initial condition u t + u x + u y = 0 u(x y 0) = u 0 (x y) (5.3) u 0 (x y) = { y x; y < x. We show the cuts of numerical solutions for FD3RK3 along y + x = 0 in Figure 5.. With the PP flux limiters numerical solutions are observed to be non-oscillatory without overshoots and undershoots. For WENO5RK4 without limiters the numerical solution 8

19 Table 5.3: L and L error and order for WENO5RK4 T = 0.5 CF L = 0.6 D linear equation with initial condition u 0 (x) = sin 4 (x) without limiters. N L error order L error order (u h ) min 0.0e-0.90e E e e E e e E e e E e e E-08 Table 5.4: L and L error and order for WENO5RK4 T = 0.5 CF L = 0.6 D linear equation with initial condition u 0 (x) = sin 4 (x) with limiters. N L error order L error order (u h ) min e-03.90e E e e E e e E e e E e e E-5 is observed to be non-oscillatory. However numerical values show negative values. For example with mesh (u h ) min =.0033 and (u h ) max =.0009 from the scheme without limiters but (u h ) min = and (u h ) max = from the scheme with limiters. Figure 5.: Numerical solutions for D linear problem with discontinuous initial condition for FD3RK3 at T = 0.5. esh: CFL=. Left: without limiter; Right: with limiter. Cuts along x + y = 0. Example 5.4. Consider the D Burgers equation ( ) ( ) u u u t + + = 0 u(x y 0) = u 0 (x y) (5.4) x y on [0 π] [0 π] with periodic boundary conditions. The initial condition is u 0 (x y) = sin 4 (x + y). The results are similar to the D case see Tables The cuts of the discontinuous solution along x + y = 0 are showed in Figure 5.3 at T = 0.8. With limiters the numerical solution performs much better than the solution without limiters for FD3RK3. For the WENO5RK4 scheme the numerical solution is observed to be nonoscillatory. However PP property is violated from the scheme without limiters (u h ) min = E 05. When the PP limiter is applied (u h ) min = E 5. 9

20 Table 5.5: L and L error and order for FD3RK3 T = 0.5 CF L =.0 D linear equation with initial condition u 0 (x) = sin 4 (x) without limiters. N L error order L error order (u h ) min 0.54e e E e e E e e E e e E e e E-05 Table 5.6: L and L error and order for FD3RK3 T = 0.5 CF L =.0 D linear equation with initial condition u 0 (x) = sin 4 (x) with limiters. N L error order L error order (u h ) min 0.0e e-0.66E e e E e e E e e E e e E-58 Figure 5.3: Numerical solutions for D Burgers equation at T = 0.8. esh: CFL=. Left: without limiter; Right: with limiter. Cuts along y + x = 0. Top: FD3RK3; Bottom: WENO5RK4. 5. Advection in incompressible flow Example 5.5. Consider the rigid body rotation u t (yu) x + (xu) y = 0 x [ π π] y [ π π]. (5.5) The initial condition includes a slotted disk a cone as well as a smooth hump see Figure 5.4. The cuts of the numerical solution for FD3RK3 are displayed in the Figure 5.5. With limiters the numerical solution is clearly within the range [0 ]. For WENO5RK4 without limiter the minimum value for the numerical solution is (u h ) min =.853E 04. With the limiter the minimum value is strictly within the range [0 ] with (u h ) min = 6.799E 5. To save space we omit the figure here. Figure 5.4: Plots of the initial profile for equation (5.5). esh:

21 Table 5.7: L and L error and order for FD3RK3 T = 0.5 CF L = 0.6 D Burgers equation with initial condition u 0 (x) = sin 4 (x) without limiters. N L error order L error order (u h ) min (u h ) max e-0.47e E e e E e e E e e E e e E e e E Table 5.8: L and L error and order for FD3RK3 T = 0.5 CF L = 0.6 D Burgers equation with initial condition u 0 (x) = sin 4 (x) with limiters. N L error order L error order (u h ) min (u h ) max 0 3.3e-0.47e-0.05E e e E e e E e e E e e E e e E-.000 Figure 5.5: Plots of slides of numerical solutions for equation (5.5) with FD3RK3 at T = π. esh: CFL=. Left: without limiter; Right: with limiter. Cuts along x = 0 y = 0.8 and y = from top to bottom respectively. Example 5.6. (Swirling deformation flow) Consider solving u t (cos ( x ) sin(y)g(t)u) x + (sin(x) cos ( y )g(t)u) y = 0 x [ π π] y [ π π] (5.6) with g(t) = cos(πt/t )π. The initial condition is the same as Example 5.5. We also use FD3RK3 and WENO5RK4 to compute this example for which similar results are observed. Solutions from FD3RK3 with and without PP limiters are displayed in Figure 5.6. For WENO5RK4 without the limiter the minimum and maximum values for the numerical solution are (u h ) min =.74885E 03 (u h ) max =.04. With the limiter (u h ) min =.579E 3 (u h ) max = within the range [0 ].

22 Table 5.9: L and L error and order for WENO5RK4 T = 0.5 CF L = 0.6 D Burgers equation with initial condition u 0 (x) = sin 4 (x) without limiters. N L error order L error order (u h ) min (u h ) max 0.74e-0.0e-0-6.3E e e E e e E e e E e e E e e E Table 5.0: L and L error and order for WENO5RK4 T = 0.5 CF L = 0.6 D Burgers equation with initial condition u 0 (x) = sin 4 (x) with limiters. N L error order L error order (u h ) min (u h ) max 0.7e-0.0e E e e E e e E e e E e e E e e E Figure 5.6: Plots of slides of numerical solutions for equation (5.6) with FD3RK3 at T =.5. esh: CFL=. Left: without limiter; Right: with limiter. Cuts along x = 0 y = 0.8 and y = from top to bottom respectively. Example 5.7. Consider the incompressible Euler equations ω t + (uω) x + (vω) y = 0 (5.7) ψ = ω u v = ψ y ψ x (5.8) ω(x y 0) = ω 0 (x y) u v n=given on Ω (5.9) on the domain [0 π] [0 π] with periodic boundary conditions. The initial condition ω 0 (x y) = sin(x) sin(y). The exact solution stays stationary with ω(x y t) = sin(x) sin(y). We tested the order of accuracy for WENO5RK4 in Tables We can see that the numerical solution with limiters can be within the range [ ] without affecting the order of accuracy. The numerical solution for FD3RK3 is already within the range [ ] without limiters the results with limiters are the same as those without limiters so we do not list the results here.

23 Table 5.: L and L error and order for FD3RK3 T = 0. CF L =.0 D Burgers equation with initial condition u 0 (x y) = sin 4 (x + y) without limiters. N L error order L error order (u h ) min 0.66e-0.06e-0 -.8E e e E e e E e e E e e E e e E-07 Table 5.: L and L error and order for FD3RK3 T = 0. CF L =.0 D Burgers equation with initial condition u 0 (x y) = sin 4 (x + y) with limiters. N L error order L error order (u h ) min 0.49e-0.05e-0.77E e e-0.68.E e e E e e E e e E e e E-4 Example 5.8. (The vortex patch problem) Consider the same problem as in Example 5.7 but with the initial condition given by π x 3π π y 3π; 4 4 π ω 0 (x y) = x 3π 5π y 7π; otherwise. We show the contour plots of vorticity ω and the cut along the diagonal at T = 5 in Figure 5.7. The mesh size is 8 8. For FD3RK3 without the limiter overshoots and undershoots are observed. However the numerical solutions from the scheme with limiter are observed to be in the range of [ ]. For WENO5RK4 we cannot observe any visible difference between the results without and with the limiter. The minimum and maximum values of the numerical solution without limiters are (ω h ) min =.0005 and (ω h ) max =.0005 and with limiters are (ω h ) min = and (ω h ) max = Figure 5.7: Plots of numerical solutions for Example 5.8 with FD3RK3 at T = 5. esh: 8 8. CFL=. Left: without limiter; Right: with limiter. Top: 30 equally spaced contours from -. to.; bottom: cut along the diagonal. 3

24 Table 5.3: L and L error and order for WENO5RK4 T = 0. CF L =.0 D Burgers equation with initial condition u 0 (x y) = sin 4 (x + y) without limiters. N L error order L error order (u h ) min 0.e e-0-4.6E e e E e e E e e E e e E e e E-0 Table 5.4: L and L error and order for WENO5RK4 T = 0. CF L =.0 D Burgers equation with initial condition u 0 (x y) = sin 4 (x + y) with limiters. 6 Conclusion N L error order L error order (u h ) min 0.e e E e e E e e E e e E e e E e e E-5 In this paper we propose to apply a parametrized flux limiters only on the final stage of a multi-stage RK finite difference WENO schemes to achieve the PP property for solving scalar hyperbolic conservation laws. We use a formal local truncation error analysis to prove that the proposed approach maintains both high order spatial and temporal accuracy under the linear stability condition of the original RK WENO scheme. We also propose a first order monotone scheme for solving incompressible flow problems based on which we design a high order conservative PP scheme. Numerical experiments have demonstrated the efficiency and effectiveness of our new approach. Extension to hyperbolic systems will be explored in the future. A Appendix In this appendix we prove the Theorem 3. for the finite difference scheme with a fourth order reconstruction for linear problem. Extension to higher order could be done with similar ideas but could be very technically involved. 4

25 Table 5.5: L and L error and order for WENO5RK4 T = 0. CF L =.0 incompressible Euler equations with initial condition ω 0 (x y) = sin(x)sin(y) without limiters. N L error order L error order (u h ) min u min u max (u h ) max E E E E E E E E E E E E E E E- -.60E E E E-3.68E-3 Table 5.6: L and L error and order for WENO5RK4 T = 0. CF L =.0 incompressible Euler equations with initial condition ω 0 (x y) = sin(x)sin(y) with limiters. N L error order L error order (u h ) min u min u max (u h ) max E E E E E E E E E E E E E E E E E E E-3.77E-3 A. Fourth order scheme for linear equation We are going to prove u (u j λ(ĥrk j+ u j )) O( x 4 + t r ). (A.) We only consider the case that the maximum or local maximum u interval of I j. Other cases can be considered similar to those in Theorem 3.. have is reached inside the For the high order RK flux similar to the argument in the proof of 3rd order case we Ĥ rk j+ = t t n+ h(x j+ τ)dτ + O( t r ) = t n t t and the 4th order approximate reconstructed polynomial P h (x) is 0 h(x j+ τ t n )dτ + O( t r ) (A.) P h (x) = u j + u j u j (x x j ) + u j+ u j + u j ((x x x 6 x j )(x x j+ ) +(x x j )(x x j+ 3 ) + (x x j+ )(x x j+ 3 )) + u j+ 3u j + 3u j u j ((x x 4 x 3 j 3 +(x x j 3 )(x x j )(x x j+ 3 +(x x j )(x x j+ )(x x j ) + (x x j 3 )(x x j+ ) )(x x j+ )(x x j+ 3 ) )(x x j+ 3 )). (A.3) 5

26 Let u j λ(ĥrk j+ u j ) = u j λ( t t 0 h(x j+ τ t n )dτ u j ) + O( t r ) = I + O( t r ) (A.4) with = t λ x t 0 xj+ h(x j+ τ t n )dτ x j+ λ x h(ξ t n )dξ = 4 (6u j 0u j + u j + 6u j+ ) + λ 4 (9u j + 3u j u j u j+ ) + λ 4 ( 4u j + 0u j u j + 6u j+ ) + λ3 4 (3u j 3u j + u j u j+ ). (A.5) Now performing the Taylor expansions u j = u + u (x j x ) + u (x j x ) + u (x j x ) 3 + O( x 4 ) 6 u j = u + u (x j x x) + u (x j x x) + u (x j x x) 3 + O( x 4 ) 6 u j = u + u (x j x x) + u (x j x x) + u (x j x x) 3 + O( x 4 ) 6 u j+ = u + u (x j x + x) + u (x j x + x) + u (x j x + x) 3 + O( x 4 ) 6 since u = 0 we can rewrite I as with I = u + u x R + u x 3 6 R 3 + O( x 4 ) (A.6) R = 5 6 λ + λ 3 λ3 + (λ 3λ)z + z R 3 = 4 ( 4λ + λ λ 3 + λ 4 ) + 4 (0λ + 6λ 4λ 3 )z + 4 ( 8λ + 6λ )z + z 3 and z = (x j x )/ x. Also let I = βu(x ) + ( β)u(x + c x) = u + u x ( β)c + u x 3 6 ( β)c3 + O( x 4 ). (A.7) 6

27 If we can find β [0 ] and c such that { ( β)c R ( β)c 3 = R 3 (A.8) then from (A.6) and (A.7) since u 0 we would have u I + O( x 4 ). (A.9) Combined with (A.4) we can prove (A.). In the following we are going to determine the parameters β and c satisfying (A.8). To get (A.8) we first need ( β)r3 R 3 that is ( ) R 3 β min with 0 λ. (A.0) z / R 3 From athematica and note that z we have min z ( ) R 3 = R 3 { 54 08λ+7λ 6λ 3 0 λ z 54 8λ+7λ 3 3 7λ+9λ 5λ 3 +λ 4 z 3 λ 7 3λ+9λ 3 (A.) and λ + 7λ 6λ 3 if 0 λ (A.) 54 8λ + 7λ λ + 9λ 5λ 3 + λ 4 if 0 λ (A.3) 7 3λ + 9λ ( so that we can choose β to be the minimum value of (A.0) and we have c = ) R /3. 3 β We still need to indicate that c is bounded which makes the Taylor remainder high order terms reasonable. If 0 λ z 3 from (A.) we have c < ( max0 λ z ( 8 64 ( min 0 λ R 3 ( β) max R 3 0 λ z ) /3 ) /3 ) /3 =

28 If z 3 λ we have and R 3 β = Λ(λ)( 4 ( 4 + λ λ + λ 3 ) + 4 (0 + 6λ 4λ )z + 4 ( 8 + 6λ)z + z3 λ ) (A.4) 6 5 Λ(λ) = λ 3 3 (A.5) 7λ+9λ 5λ 3 +λ ( 4 + λ λ + λ 3 ) + 4 (0 + 6λ 4λ )z + 4 ( 8 + 6λ)z 6 (A.6) z 3 λ (A.7) so that in this case c ((6 + )) /3 = 4 /3. Therefore we have proven (A.) when the maximum or local maximum u is reached inside the interval of I j. Other cases can be discussed in a very similar way. We will not repeat the algebraically tedious procedure here. References [] S. Gottlieb D. Ketcheson and C.-W. Shu High order strong stability preserving time discretizations Journal of Scientific Computing 38 (009) pp [] G.-S. Jiang and C.-W. Shu Efficient implementation of weighted ENO schemes Journal of Computational Physics 6 (996) pp [3] S. Osher Riemann solvers the entropy condition and difference approximations SIA Journal on Numerical Analysis (984) pp [4] C. Shen J.-. Qiu and A. Christlieb Adaptive esh Refinement Based on High order Finite Difference WENO Scheme for ulti-scale Simulations Journal of Computational Physics 30 (0) pp [5] C.-W. Shu Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws Advanced numerical approximation of nonlinear hyperbolic equations (998) pp [6] High order weighted essentially non-oscillatory schemes for convection dominated problems SIA Review 5 (009) pp

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