Positivity-Preserving Finite Difference WENO 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 non-oscillatory (WENO) methods for scalar hyperbolic conservation laws, to develop a class of high order positivity-preserving finite difference WENO methods for the ideal magnetohydrodynamic (MHD) equations. Our schemes, 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.Comment: 21 pages, 28 figure

Notes on the Discontinuous Galerkin methods for the numerical simulation of hyperbolic equations 1 General Context 1.1 Bibliography

The roots of Discontinuous Galerkin (DG) methods is usually attributed to Reed and Hills in a paper published in 1973 on the numerical approximation of the neutron transport equation [18]. In fact, the adventure really started with a rather thoroughfull series of five papers by Cockburn and Shu in the late 80's [7, 5, 9, 6, 8]. Then, the fame of the method, which could be seen as a compromise between Finite Elements (the center of the method being a weak formulation) and Finite Volumes (the basis functions are defined cell-wise, the cells being the elements of the primal mesh) increased and slowly investigated successfully all the domains of Partial Differential Equations numerical integration. In particular, one can cite the ground papers for the common treatment of convection-diffusion equations [4, 3] or the treatment of pure elliptic equations [2, 17]. For more information on the history of Discontinuous Galerkin method, please refer to section 1.1 of [15]. Today, DG methods are widely used in all kind of manners and have applications in almost all fields of applied mathematics. (TODO: cite applications and structured/unstructured meshes, steady/unsteady, etc...). The methods is now mature enough to deserve entire text books, among which I cite a reference book on Nodal DG Methods by Henthaven and Warburton [15] with the ground basis of DG integration, numerical analysis of its linear behavior and generalization to multiple dimensions. Lately, since 2010, thanks to a ground work of Zhang and Shu [26, 27, 25, 28, 29], Discontinuous Galerkin methods are eventually able to combine high order accuracy and certain preservation of convex constraints, such as the positivity of a given quantity, for example. These new steps forward are very promising since it brings us very close to the "Ultimate Conservative Scheme", [23, 1]

High-order accurate physical-constraints-preserving finite difference WENO schemes for special relativistic hydrodynamics

The paper develops high-order accurate physical-constraints-preserving finite difference WENO schemes for special relativistic hydrodynamical (RHD) equations, built on the local Lax-Friedrich splitting, the WENO reconstruction, the physical-constraints-preserving flux limiter, and the high-order strong stability preserving time discretization. They are extensions of the positivity-preserving finite difference WENO schemes for the non-relativistic Euler equations. However, developing physical-constraints-preserving methods for the RHD system becomes much more difficult than the non-relativistic case because of the strongly coupling between the RHD equations, no explicit expressions of the primitive variables and the flux vectors, in terms of the conservative vector, and one more physical constraint for the fluid velocity in addition to the positivity of the rest-mass density and the pressure. The key is to prove the convexity and other properties of the admissible state set and discover a concave function with respect to the conservative vector replacing the pressure which is an important ingredient to enforce the positivity-preserving property for the non-relativistic case. Several one- and two-dimensional numerical examples are used to demonstrate accuracy, robustness, and effectiveness of the proposed physical-constraints-preserving schemes in solving RHD problems with large Lorentz factor, or strong discontinuities, or low rest-mass density or pressure etc.Comment: 39 pages, 13 figure