No abstract

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.

In this paper, we utilize the maximum-principle-preserving flux limiting technique, originally designed for high order weighted essentially non-oscillatory (WENO) methods for scalar hyperbolic conservation laws, to develop a class of high order positivity-preserving finite difference WENO method for the ideal magnetohydrodynamic (MHD) equations. Our scheme, under the constrained transport (CT) framework, can achieve high order accuracy, a discrete divergence-free condition and positivity of the numerical solution simultaneously. Numerical examples in 1D, 2D and 3D are provided to demonstrate the performance of the proposed method. literature [6,11,35]. To design divergence-free methods for solving the ideal MHD equations, the CT methodology arises as one important approach, see [1,5,9,10,11,12,15,25,26, 30, 32,33,35,34, 31] for references. Following [9,15,16, 30, 31], we propose to conduct our investigation within the CT framework in this paper.Another major focus of this paper is the design of high-order schemes that preserve the positivity of the density and pressure of the MHD system. Even with divergencefree methods, negative density or/and pressure can still be observed in numerical simulations, such as those for the low-β plasma. This can lead to a complex wave speed that breaks the hyperbolicity of the system and causes the numerical simulations to break down. A lot of efforts have been dedicated addressing this issue in the literature. For instance, Balsara and Spicer [4] proposed a strategy to maintain the positivity of pressure by switching the Riemann solvers based on different wave situations. Janhunen [18] designed a new Riemann solver for the modified ideal MHD equations and demonstrated its positivity-preserving property numerically. In [36], a conservative second-order MUSCL-Hancock scheme was shown to be positivity-preserving for the 1D ideal MHD equations and the extension to multi-dimensional (multi-D) cases was constructed based on similar ideas as Powell's 8-wave formulation [28,29]. Balsara [2] developed a high-order positivity-preserving scheme for ideal MHD through limiting high-order numerical solutions by a conservative bounded solution. Another class of important methods for the ideal MHD equations is discontinuous Galerkin (DG) methods [21,22,23, 31,41]. Recently, Cheng et al. proposed positivity-preserving DG and central DG methods for the ideal MHD equations [7], in which they generalized Zhang and Shu's positivity-preserving limiters for the compressible Euler equations [42]. In [7], it was proven that the first-order Lax-Fridrichs scheme is positivity-preserving for the 1D MHD under the restriction CFL≤ 0.5. This first-order scheme also serves as the building block for the positivity-preserving scheme in this paper.Besides the aforementioned work for MHD equations, several high-order positivitypreserving schemes have been developed recently for compressible Euler equations. Zhang and Shu developed arbitrary-order positivity-preserving finite volume WENO and DG methods by limiting ...