We propose and analyze a class of robust, uniformly high-order accurate discontinuous Galerkin (DG) schemes for multidimensional relativistic magnetohydrodynamics (RMHD) on general meshes. A distinct feature of the schemes is their physical-constraint-preserving (PCP) property, i.e., they are proven to preserve the subluminal constraint on the fluid velocity and the positivity of density, pressure, and specific internal energy. Developing PCP high-order schemes for RMHD is highly desirable but remains a challenging task, especially in the multidimensional cases, due to the inherent strong nonlinearity in the constraints and the effect of the magnetic divergence-free condition. Inspired by some crucial observations at the PDE level, we construct the provably PCP schemes by using the locally divergence-free DG schemes of the recently proposed symmetrizable RMHD equations as the base schemes, a limiting technique to enforce the PCP property of the DG solutions, and the strong-stability-preserving methods for time discretization. We rigorously prove the PCP property by using a novel "quasi-linearization" approach to handle the highly nonlinear physical constraints, technical splitting to offset the influence of divergence error, and sophisticated estimates to analyze the beneficial effect of the additional source term in the symmetrizable RMHD system. Several two-dimensional numerical examples are provided to confirm the PCP property and to demonstrate the accuracy, effectiveness and robustness of the proposed PCP schemes.
We propose a constraint preserving discontinuous Galerkin method for ideal compressible MHD in two dimensions and using Cartesian grids, which automatically maintains the global divergence-free property. The approximation of the magnetic field is achieved using Raviart-Thomas polynomials and the DG scheme is based on evolving certain moments of these polynomials which automatically guarantees divergence-free property. We also develop HLL-type multi-dimensional Riemann solvers to estimate the electric field at vertices which are consistent with the 1-D Riemann solvers. When limiters are used, t…
In the numerical simulation of ideal MHD, keeping the pressure and density positive is essential for both physical considerations and numerical stability. This is a challenge, due to the underlying relation between such positivity-preserving (PP) property and the magnetic divergence-free (DF) constraint as well as the strong nonlinearity of the MHD equations. This paper presents the first rigorous PP analysis of the central discontinuous Galerkin (CDG) methods and constructs arbitrarily high-order PP CDG schemes for ideal MHD. By the recently developed geometric quasilinearization (GQL) approa…
The second paper of this series presents two robust entropy stable shock-capturing methods for discontinuous Galerkin spectral element (DGSEM) discretizations of the compressible magneto-hydrodynamics (MHD) equations. Specifically, we use the resistive GLM-MHD equations, which include a divergence cleaning mechanism that is based on a generalized Lagrange multiplier (GLM). For the continuous entropy analysis to hold, and due to the divergence-free constraint on the magnetic field, the GLM-MHD system requires the use of non-conservative terms, which need special treatment. Hennemann et al. [DOI…
A discontinuous Galerkin method for solving the fluid and MHD equations in astrophysical simulations
Harvard University（哈佛大学）；Kavli Institute for Cosmology（卡弗里宇宙学研究所）；Institute of Astronomy（天文学研究所）；Heidelberger Institut für Theoretische Studien（海德堡理论研究所）
We present a provably stable discontinuous Galerkin spectral element method for the incompressible Navier-Stokes equations with artificial compressibility and variable density. Stability proofs, which include boundary conditions, that follow a continuous entropy analysis are provided. We define a mathematical entropy function that combines the traditional kinetic energy and an additional energy term for the artificial compressiblity, and derive its associated entropy conservation law. The latter allows us to construct a provably stable split-form nodal Discontinuous Galerkin (DG) approximation…
This paper presents high order accurate discontinuous Galerkin (DG) methods for wave problems on moving curved meshes with general choices of basis and quadrature. The proposed method adopts an arbitrary Lagrangian-Eulerian (ALE) formulation to map the acoustic wave equation from the time-dependent moving physical domain onto a fixed reference domain. For moving curved meshes, weighted mass matrices must be assembled and inverted at each time step when using explicit time stepping methods. We avoid this step by utilizing an easily invertible weight-adjusted approximation. The resulting semi-di…
In this work we present a framework for enforcing discrete maximum principles in discontinuous Galerkin (DG) discretizations. The developed schemes are applicable to scalar conservation laws as well as hyperbolic systems. Our methodology for limiting volume terms is similar to recently proposed methods for continuous Galerkin approximations, while DG flux terms require novel stabilization techniques. Piecewise Bernstein polynomials are employed as shape functions for the DG spaces, thus facilitating the use of very high order spatial approximations. We discuss the design of a new, provably inv…
The Galerkin difference (GD) basis is a set of continuous, piecewise polynomials defined using a finite difference like grid of degrees of freedom. The one dimensional GD basis functions are naturally extended to multiple dimensions using the tensor product constructions to quadrilateral elements for discretizing partial differential equations. Here we propose two approaches to handling complex geometries using the GD basis within a discontinuous Galerkin finite element setting: (1) using non-conforming, curvilinear GD elements and (2) coupling affine GD elements with curvilinear simplicial el…
Solving the relativistic magnetohydrodynamics equations with ADER discontinuous Galerkin methods, a posteriori subcell limiting and adaptive mesh refinement
This paper presents a class of novel high-order accurate discontinuous Galerkin (DG) schemes for the compressible Euler equations under gravitational fields. A notable feature of these schemes is that they are well-balanced for a general hydrostatic equilibrium state, and at the same time, provably preserve the positivity of density and pressure. In order to achieve the well-balanced and positivity-preserving properties simultaneously, a novel DG spatial discretization is carefully designed with suitable source term reformulation and a properly modified Harten-Lax-van Leer contact (HLLC) flux.…
A free-energy stable nodal discontinuous Galerkin approximation with summation-by-parts property for the Cahn-Hilliard equation
ETSIAE-UPM - School of Aeronautics, Universidad Politécnica de Madrid（马德里理工大学航空学院）；Center for Computational Simulation, Universidad Politécnica de Madrid（马德里理工大学计算模拟中心）；Department of Mathematics, Florida State University（佛罗里达州立大学数学系）；Computational Science Research Center, San Diego State University（圣地亚哥州立大学计算科学研究中心）
We build a multi-element variant of the smoothness increasing accuracy conserving (SIAC) shock capturing technique proposed for single element spectral methods by Wissink et al. (B.W. Wissink, G.B. Jacobs, J.K. Ryan, W.S. Don, and E.T.A. van der Weide. Shock regularization with smoothness-increasing accuracy-conserving Dirac-delta polynomial kernels. Journal of Scientific Computing, 77:579--596, 2018). In particular, the baseline scheme of our method is the nodal discontinuous Galerkin spectral element method (DGSEM) for approximating the solution of systems of conservation laws. It is well kn…
We present a scalable block preconditioning strategy for the trace system coming from the high-order hybridized discontinuous Galerkin (HDG) discretization of incompressible resistive magnetohydrodynamics (MHD). We construct the block preconditioner with a least squares commutator (BFBT) approximation for the inverse of the Schur complement that segregates out the pressure unknowns of the trace system. The remaining velocity, magnetic field, and Lagrange multiplier unknowns form a coupled nodal unknown block (the upper block), for which a system algebraic multigrid (AMG) is used for the approx…
Provably stable flux reconstruction (FR) schemes are derived for partial differential equations cast in curvilinear coordinates. Specifically, energy stable flux reconstruction (ESFR) schemes are considered as they allow for design flexibility as well as stability proofs for the linear advection problem on affine elements. Additionally, split forms are examined as they enable the development of energy stability proofs. The first critical step proves, that in curvilinear coordinates, the discontinuous Galerkin (DG) conservative and non-conservative forms are inherently different--even under exa…
An Entropy Stable Nodal Discontinuous Galerkin Method for the Two Dimensional Shallow Water Equations on Unstructured Curvilinear Meshes with Discontinuous Bathymetry
Universität zu Köln（科隆大学）；The Florida State University（佛罗里达州立大学）
We present a general family of subcell limiting strategies to construct robust high-order accurate nodal discontinuous Galerkin (DG) schemes. The main strategy is to construct compatible low order finite volume (FV) type discretizations that allow for convex blending with the high-order variant with the goal of guaranteeing additional properties, such as bounds on physical quantities and/or guaranteed entropy dissipation. For an implementation of this main strategy, four main ingredients are identified that may be combined in a flexible manner: (i) a nodal high-order DG method on Legendre-Gaus…
Fully Discrete Multi-Entropy Stability of High-Order Schemes for Compressible MHD: A Weak-to-Strong Framework
Southern University of Science and Technology（南方科技大学）；Shenzhen International Center for Mathematics, Southern University of Science and Technology（南方科技大学深圳数学中心）
High order accurate and explicit time-stable solvers are well suited for hyperbolic wave propagation problems. As a result of the complexities of real geometries, internal interfaces and nonlinear boundary and interface conditions, discontinuities and sharp wave fronts may become fundamental features of the solution. Thus, geometrically flexible and adaptive numerical algorithms are critical for high fidelity and efficient simulations of wave phenomena in many applications. Adaptive curvilinear meshes hold promise to minimise the effort to represent complicated geometries or heterogeneous mate…
Hypocoercivity-preserving space-time Galerkin methods for kinetic Fokker-Planck equations
Inria（法国国家信息与自动化研究所）；CERMICS, ENPC, Institut Polytechnique de Paris, CNRS（巴黎高等师范学院计算数学与计算机科学中心，巴黎理工学院，法国国家科学研究中心）；The Maxwell Institute for Mathematical Sciences and Department of Mathematics, Heriot-Watt University（赫瑞-瓦特大学麦克斯韦数学科学研究所及数学系）；Department of Mathematics, School of Applied Mathematical and Physical Sciences, National Technical University of Athens（雅典国立技术大学应用数学与物理科学学院数学系）；Institute of Applied and Computational Mathematics, FORTH（希腊基础与研究基金会应用与计算数学研究所）
In this paper, we develop a provably energy stable and conservative discontinuous spectral element method for the shifted wave equation in second order form. The proposed method combines the advantages and central ideas of very successful numerical techniques, the summation-by-parts finite difference method, the spectral method and the discontinuous Galerkin method. We prove energy-stability, discrete conservation principle, and derive error estimates in the energy norm for the (1+1)-dimensions shifted wave equation in second order form. The energy-stability results, discrete conservation prin…
