A parametrized maximum principle preserving flux limiter for finite difference RK-WENO schemes with applications in incompressible flows
Introduction
In this paper, we consider the following scalar hyperbolic conservation laws Popular methods for solving (1.1) include finite difference/volume schemes based on high order essentially non-oscillatory (ENO) and Weighted ENO (WENO) reconstructions [2], [6] and finite element discontinuous Galerkin (DG) methods coupled with total variation diminishing (TVD) Runge–Kutta (RK) time discretization [8]. 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 [8], [9], by introducing a sliding average function . 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, although the finite difference schemes are not as compact and flexible in domains with complicated geometry.
An important property of the solution for hyperbolic conservation laws (1.1) is the strict maximum principle [4], namely , if . 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 [16], [18]. MPP 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 maintenance of high order spatial accuracy and maximum principle is theoretically proved and numerically verified when suitable CFL numbers are chosen. The techniques have recently been applied to a number of problems including the compressible/incompressible Euler equations, shallow water equations, among many others [17], [20], [13], [12]. However, it was also pointed out in [19] that it is not trivial to apply the MPP limiters to the finite difference schemes without destroying the designed order of accuracy. Also the time step size required to preserve the MPP property is smaller than the one for the original scheme, e.g. it is about of the original CFL for a third order finite volume scheme with MPP limiters [16].
In [14], Xu developed a parametrized MPP flux limiting technique to maintain the MPP property of numerical solutions of the one-dimensional scalar hyperbolic conservation laws. Compared with limiting the cell-wise reconstructed polynomials in [16], [18], in [14] the MPP property is achieved via limiting high order numerical fluxes toward first order monotone fluxes in a conservative scheme. Compared with traditional flux limiters for the TVD property (which is a stronger stability requirement than MPP), as discussed in [10], [11] and the references therein, the MPP flux limiting approach in [14] has the potential to be designed with higher than second order accuracy. The MPP requirement of 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 [15]. The MPP property is guaranteed under the same CFL time step restriction of the first order monotone scheme. However, the scheme suffers from additional time step restriction for the preservation of high order accuracy.
In this paper, following the idea in [14], we focus on developing the MPP 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 MPP flux limiters only at the final stage of the multi-stage RK time discretization. It was commented in [14] that if the MPP flux limiter is applied at each of the intermediate stage of RK method, due to the influence of the special cancellation of RK, high order temporal accuracy could be lost. With the flux limiter applied only at the final RK stage, the implementation complexity is significantly reduced. Error analysis is performed to prove the maintenance of third order spatial and temporal accuracy when the high order flux is limited toward a first order local Lax–Friedrich (LFF) flux or Godunov flux. Secondly, we apply the MPP flux limiters to the high order FD WENO method solving the incompressible Euler equation in vorticity stream-function formulation to maintain a maximum principle for vorticity. We remark that, compared with the unified high order limiting procedure for arbitrary high order reconstructed polynomials of Zhang and Shu in [16], [18], the maintenance of high order spatial and temporal accuracy so far can only be proved for the original third order finite difference scheme solving the one-dimensional nonlinear equations. We largely rely on extensive numerical experiments to verify the maintenance of high order spatial and temporal accuracy for general high order schemes, high dimensional case, and for the incompressible flow without additional time step restriction.
The rest of the paper is organized as follows. In Section 2, we will first review the high order finite difference schemes [7] and the parametrized MPP flux limiters in Xu [14]. In Section 3, we describe on how to apply the MPP limiter on the final stage of a multi-stage RK method in one- and two-dimensional cases. Third order error analysis is provided to show that the newly proposed limiter preserves high order accuracy in both space and time without excessive time step restriction. In Section 4, MPP flux limiters are proposed for high order finite difference schemes solving incompressible flow. In Section 5, we perform numerical tests on both scalar conservation problems and incompressible Euler equations to demonstrate that maximum principle is preserved with designed order of accuracy, without additional time step restriction.
Section snippets
Finite difference WENO scheme
We first briefly review the finite difference WENO scheme [7] for a simple one-dimensional hyperbolic conservation equation with initial condition . Without loss of generality, we assume periodic boundary condition. We adopt the following spatial discretization for the spatial domain where has the mesh size . Let denote the solution at grid point at continuous time t. The finite difference
One-dimensional problem
A ‘successive’ MPP limiting procedure was proposed in [14] for limiting the upper and lower bounds of solutions at internal stages of a third order TVD RK method [1]. In this section, we propose to apply the MPP 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. Moreover, the time step restriction to ensure both MPP property and high order
The MPP flux limiter for incompressible flow
Consider 2D equations describing advection in incompressible flow, in conservative form with the divergence-free condition of the velocity field The solutions of such equation enjoy the properties of mass conservation and strict maximum principle thanks to the divergence-free condition. The challenge of computing (4.1) by a conservative scheme is to preserve a discrete divergence-free condition, especially when a high order method is used. In this
Basic tests
In this section, we present numerical examples for the proposed parametrized MPP 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, denoted as “FD3RK3”, here the 3rd order finite difference scheme is the 3rd order finite difference WENO scheme but with linear weights; the other is the 5th order finite difference WENO scheme
Conclusion
In this paper, we propose to apply a parametrized flux limiters only at the final stage of a multi-stage RK finite difference WENO schemes, to achieve the MPP property for solving scalar hyperbolic conservation laws. We use a formal local truncation error analysis to prove that, the proposed limiting approach maintains third order spatial and temporal accuracy if the high order flux is limited toward a first order local Lax–Friedrich flux or a Godunov flux under the linear stability condition
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- 1
The first and second authors are supported by Air Force Office of Scientific Computing YIP grant FA9550-12-0318, NSF grant DMS-0914852 and DMS-1217008, University of Houston.
- 2
Supported by D90755 N49019 28399 – Start-up fund.