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]

Efficient Explicit Time Stepping of High Order Discontinuous Galerkin Schemes for Waves

This work presents algorithms for the efficient implementation of discontinuous Galerkin methods with explicit time stepping for acoustic wave propagation on unstructured meshes of quadrilaterals or hexahedra. A crucial step towards efficiency is to evaluate operators in a matrix-free way with sum-factorization kernels. The method allows for general curved geometries and variable coefficients. Temporal discretization is carried out by low-storage explicit Runge-Kutta schemes and the arbitrary derivative (ADER) method. For ADER, we propose a flexible basis change approach that combines cheap face integrals with cell evaluation using collocated nodes and quadrature points. Additionally, a degree reduction for the optimized cell evaluation is presented to decrease the computational cost when evaluating higher order spatial derivatives as required in ADER time stepping. We analyze and compare the performance of state-of-the-art Runge-Kutta schemes and ADER time stepping with the proposed optimizations. ADER involves fewer operations and additionally reaches higher throughput by higher arithmetic intensities and hence decreases the required computational time significantly. Comparison of Runge-Kutta and ADER at their respective CFL stability limit renders ADER especially beneficial for higher orders when the Butcher barrier implies an overproportional amount of stages. Moreover, vector updates in explicit Runge--Kutta schemes are shown to take a substantial amount of the computational time due to their memory intensity

Discontinuous collocation methods and gravitational self-force applications

Numerical simulations of extereme mass ratio inspirals, the mostimportant sources for the LISA detector, face several computational challenges. We present a new approach to evolving partial differential equations occurring in black hole perturbation theory and calculations of the self-force acting on point particles orbiting supermassive black holes. Such equations are distributionally sourced, and standard numerical methods, such as finite-difference or spectral methods, face difficulties associated with approximating discontinuous functions. However, in the self-force problem we typically have access to full a-priori information about the local structure of the discontinuity at the particle. Using this information, we show that high-order accuracy can be recovered by adding to the Lagrange interpolation formula a linear combination of certain jump amplitudes. We construct discontinuous spatial and temporal discretizations by operating on the corrected Lagrange formula. In a method-of-lines framework, this provides a simple and efficient method of solving time-dependent partial differential equations, without loss of accuracy near moving singularities or discontinuities. This method is well-suited for the problem of time-domain reconstruction of the metric perturbation via the Teukolsky or Regge-Wheeler-Zerilli formalisms. Parallel implementations on modern CPU and GPU architectures are discussed.Comment: 29 pages, 5 figure

A realizability-preserving high-order kinetic scheme using WENO reconstruction for entropy-based moment closures of linear kinetic equations in slab geometry

We develop a high-order kinetic scheme for entropy-based moment models of a one-dimensional linear kinetic equation in slab geometry. High-order spatial reconstructions are achieved using the weighted essentially non-oscillatory (WENO) method, and for time integration we use multi-step Runge-Kutta methods which are strong stability preserving and whose stages and steps can be written as convex combinations of forward Euler steps. We show that the moment vectors stay in the realizable set using these time integrators along with a maximum principle-based kinetic-level limiter, which simultaneously dampens spurious oscillations in the numerical solutions. We present numerical results both on a manufactured solution, where we perform convergence tests showing our scheme converges of the expected order up to the numerical noise from the numerical optimization, as well as on two standard benchmark problems, where we show some of the advantages of high-order solutions and the role of the key parameter in the limiter

High-order implicit palindromic discontinuous Galerkin method for kinetic-relaxation approximation

We construct a high order discontinuous Galerkin method for solving general hyperbolic systems of conservation laws. The method is CFL-less, matrix-free, has the complexity of an explicit scheme and can be of arbitrary order in space and time. The construction is based on: (a) the representation of the system of conservation laws by a kinetic vectorial representation with a stiff relaxation term; (b) a matrix-free, CFL-less implicit discontinuous Galerkin transport solver; and (c) a stiffly accurate composition method for time integration. The method is validated on several one-dimensional test cases. It is then applied on two-dimensional and three-dimensional test cases: flow past a cylinder, magnetohydrodynamics and multifluid sedimentation

Stability of Correction Procedure via Reconstruction With Summation-by-Parts Operators for Burgers' Equation Using a Polynomial Chaos Approach

In this paper, we consider Burgers' equation with uncertain boundary and initial conditions. The polynomial chaos (PC) approach yields a hyperbolic system of deterministic equations, which can be solved by several numerical methods. Here, we apply the correction procedure via reconstruction (CPR) using summation-by-parts operators. We focus especially on stability, which is proven for CPR methods and the systems arising from the PC approach. Due to the usage of split-forms, the major challenge is to construct entropy stable numerical fluxes. For the first time, such numerical fluxes are constructed for all systems resulting from the PC approach for Burgers' equation. In numerical tests, we verify our results and show also the advantage of the given ansatz using CPR methods. Moreover, one of the simulations, i.e. Burgers' equation equipped with an initial shock, demonstrates quite fascinating observations. The behaviour of the numerical solutions from several methods (finite volume, finite difference, CPR) differ significantly from each other. Through careful investigations, we conclude that the reason for this is the high sensitivity of the system to varying dissipation. Furthermore, it should be stressed that the system is not strictly hyperbolic with genuinely nonlinear or linearly degenerate fields