Zhengfu Xu

Parametrized maximum principle preserving flux limiters for high order schemes solving hyperbolic conservation laws: one-dimensional scalar problem

In this paper, we present a class of parametrized limiters used to achieve strict maximum principle for high order numerical schemes applied to hyperbolic conservation laws computation. By decoupling a sequence of parameters embedded in a group of explicit inequalities, the numerical fluxes are locally redefined in consistent and conservative formulation. We will show that the global maximum principle can be preserved while the high order accuracy of the underlying scheme is maintained. The parametrized limiters are less restrictive on the CFL number when applied to high order finite volume scheme. The less restrictive limiters allow for the development of the high order finite difference scheme which preserves the maximum principle. Within the proposed parametrized limiters framework, a successive sequence of limiters are designed to allow for significantly large CFL number by relaxing the limits on the intermediate values of the multistage Runge-Kutta method. Numerical results and preliminary analysis for linear and nonlinear scalar problems are presented to support the claim. The parametrized limiters are applied to the numerical fluxes directly. There is no increased complexity to apply the parametrized limiters to different kinds of monotone numerical fluxes.

Parametrized Maximum Principle Preserving Flux Limiters for High Order Schemes Solving Multi-Dimensional Scalar Hyperbolic Conservation Laws

In this paper, we will extend the strict maximum principle preserving flux limiting technique developed for one dimensional scalar hyperbolic conservation laws to the two-dimensional scalar problems. The parametrized flux limiters and their determination from decoupling maximum principle preserving constraint is presented in a compact way for two-dimensional problems. With the compact fashion that the decoupling is carried out, the technique can be easily applied to high order finite difference and finite volume schemes for multi-dimensional scalar hyperbolic problems. For the two-dimensional problem, the successively defined flux limiters are developed for the multi-stage total-variation-diminishing Runge–Kutta time-discretization to improve the efficiency of computation. The high order schemes with successive flux limiters provide high order approximation and maintain strict maximum principle with mild Courant-Friedrichs-Lewy constraint. Two dimensional numerical evidence is given to demonstrate the capability of the proposed approach.

Parametrized Positivity Preserving Flux Limiters for the High Order Finite Difference WENO Scheme Solving Compressible Euler Equations

In this paper, we develop parametrized positivity satisfying flux limiters for the high order finite difference Runge–Kutta weighted essentially non-oscillatory scheme solving compressible Euler equations to maintain positive density and pressure. Negative density and pressure, which often leads to simulation blow-ups or nonphysical solutions, emerges from many high resolution computations in some extreme cases. The methodology we propose in this paper is a nontrivial generalization of the parametrized maximum principle preserving flux limiters for high order finite difference schemes solving scalar hyperbolic conservation laws (Liang and Xu in J Sci Comput 58:41–60, 2014; Xiong et al. in J Comput Phys 252:310–331, 2013; Xu in Math Comput 83:2213–2238, 2014). To preserve the maximum principle, the high order flux is limited towards a first order monotone flux, where the limiting procedures are designed by decoupling linear maximum principle constraints. High order schemes with such flux limiters are shown to preserve the high order accuracy via local truncation error analysis and by extensive numerical experiments with mild CFL constraints. The parametrized flux limiting approach is generalized to the Euler system to preserve the positivity of density and pressure of numerical solutions via decoupling some nonlinear constraints. Compared with existing high order positivity preserving approaches (Zhang and Shu in Proc R Soc A Math Phys Eng Sci 467:2752–2776, 2011; J Comput Phys 230:1238–1248, 2011; J Comput Phys 231:2245–2258, 2012), our proposed algorithm is positivity preserving by the design; it is computationally efficient and maintains high order spatial and temporal accuracy in our extensive numerical tests. Numerical tests are performed to demonstrate the efficiency and effectiveness of the proposed new algorithm.

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

In Xu (2013) [14], 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 (MPP). In this paper, we continue along this line of research, but propose to apply the parametrized MPP flux limiter only to the final stage of any explicit RK method. Compared with the original work (Xu, 2013) [14], the proposed new approach has several advantages: First, the MPP property is preserved with high order accuracy without as much time step restriction; Second, the implementation of the parametrized flux limiters is significantly simplified. Analysis is performed to justify the maintenance of third order spatial/temporal accuracy when the MPP flux limiters are applied to third order finite difference schemes solving general nonlinear problems. We further apply the limiting procedure to the simulation of the incompressible flow: the numerical fluxes of a high order scheme are limited toward that of a first order MPP scheme which was discussed in Levy (2005) [3]. The MPP property is guaranteed, while designed high order of spatial and temporal accuracy for the incompressible flow computation is not affected via extensive numerical experiments. The efficiency and effectiveness of the proposed scheme are demonstrated via several test examples.

High order maximum principle preserving semi-Lagrangian finite difference WENO schemes for the Vlasov equation

In this paper, we propose the parametrized maximum principle preserving (MPP) flux limiter, originally developed in [37], to the semi-Lagrangian finite difference weighted essentially non-oscillatory scheme for solving the Vlasov equation. The MPP flux limiter is proved to maintain up to fourth order accuracy for the semi-Lagrangian finite difference scheme without any time step restriction. Numerical studies on the Vlasov-Poisson system demonstrate the performance of the proposed method and its ability in preserving the positivity of the probability distribution function while maintaining the high order accuracy.

Positivity-preserving finite difference weighted ENO schemes with constrained transport for ideal magnetohydrodynamic equations

In this paper, we utilize the maximum-principle-preserving flux limiting technique, originally designed for high-order weighted essentially nonoscillatory (WENO) methods for scalar hyperbolic conservation laws, to develop a class of high-order positivity-preserving finite difference WENO methods for the ideal magnetohydrodynamic equations. Our scheme, under the constrained transport framework, can achieve high-order accuracy, a discrete divergence-free condition, and positivity of the numerical solution simultaneously. Numerical examples in one, two, and three dimensions are provided to demonstrate the performance of the proposed method.

High order operator splitting methods based on an integral deferred correction framework

Integral deferred correction (IDC) methods have been shown to be an efficient way to achieve arbitrary high order accuracy and possess good stability properties. In this paper, we construct high order operator splitting schemes using the IDC procedure to solve initial value problems (IVPs). We present analysis to show that the IDC methods can correct for both the splitting and numerical errors, lifting the order of accuracy by r with each correction, where r is the order of accuracy of the method used to solve the correction equation. We further apply this framework to solve partial differential equations (PDEs). Numerical examples in two dimensions of linear and nonlinear initial-boundary value problems are presented to demonstrate the performance of the proposed IDC approach.

An Explicit High-Order Single-Stage Single-Step Positivity-Preserving Finite Difference WENO Method for the Compressible Euler Equations

In this work we construct a high-order, single-stage, single-step positivity-preserving method for the compressible Euler equations. Space is discretized with the finite difference weighted essentially non-oscillatory method. Time is discretized through a Lax–Wendroff procedure that is constructed from the Picard integral formulation of the partial differential equation. The method can be viewed as a modified flux approach, where a linear combination of a low- and high-order flux defines the numerical flux used for a single-step update. The coefficients of the linear combination are constructed by solving a simple optimization problem at each time step. The high-order flux itself is constructed through the use of Taylor series and the Cauchy–Kowalewski procedure that incorporates higher-order terms. Numerical results in one- and two-dimensions are presented.

High Order Maximum Principle Preserving Finite Volume Method for Convection Dominated Problems

In this paper, we investigate the application of the maximum principle preserving (MPP) parameterized flux limiters to the high order finite volume scheme with Runge–Kutta time discretization for solving convection dominated problems. Such flux limiter was originally proposed in Xu (Math Comput 83:2213–2238, 2014) and further developed in Xiong et al. (J Comput Phys 252:310–331, 2013) for finite difference WENO schemes with Runge–Kutta time discretization for convection equations. The main idea is to limit the temporal integrated high order numerical flux toward a first order MPP monotone flux. In this paper, we generalize such flux limiter to high order finite volume methods solving convection-dominated problems, which is easy to implement and introduces little computational overhead. More importantly, for the first time in the finite volume setting, we provide a general proof that the proposed flux limiter maintains high order accuracy of the original WENO scheme for linear advection problems without any additional time step restriction. For general nonlinear convection-dominated problems, we prove that the proposed flux limiter introduces up to