Short note on the mass matrix for Gauss-Lobatto grid points

The mass matrix for Gauss-Lobatto grid points is usually approximated by Gauss-Lobatto quadrature because this leads to a diagonal matrix that is easy to invert. The exact mass matrix and its inverse are full. We show that the exact mass matrix \emph{and} its inverse differ from the approximate diagonal ones by a simple rank-1 update (outer product). They can thus be applied to an arbitrary vector in $O(N)$ operations instead of $O(N^2)$

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

Unified discontinuous Galerkin scheme for a large class of elliptic equations

We present a discontinuous Galerkin internal-penalty scheme that is applicable to a large class of linear and nonlinear elliptic partial differential equations. The unified scheme can accommodate all second-order elliptic equations that can be formulated in first-order flux form, encompassing problems in linear elasticity, general relativity, and hydrodynamics, including problems formulated on a curved manifold. It allows for a wide range of linear and nonlinear boundary conditions, and accommodates curved and nonconforming meshes. Our generalized internal-penalty numerical flux and our Schur-complement strategy of eliminating auxiliary degrees of freedom make the scheme compact without requiring equation-specific modifications. We demonstrate the accuracy of the scheme for a suite of numerical test problems. The scheme is implemented in the open-source SpECTRE numerical relativity code

Simulating magnetized neutron stars with discontinuous Galerkin methods

Discontinuous Galerkin methods are popular because they can achieve high order where the solution is smooth, because they can capture shocks while needing only nearest-neighbor communication, and because they are relatively easy to formulate on complex meshes. We perform a detailed comparison of various limiting strategies presented in the literature applied to the equations of general relativistic magnetohydrodynamics. We compare the standard minmod/$\Lambda\Pi^N$ limiter, the hierarchical limiter of Krivodonova, the simple WENO limiter, the HWENO limiter, and a discontinuous Galerkin-finite-difference hybrid method. The ultimate goal is to understand what limiting strategies are able to robustly simulate magnetized TOV stars without any fine-tuning of parameters. Among the limiters explored here, the only limiting strategy we can endorse is a discontinuous Galerkin-finite-difference hybrid method

SpECTRE: A Task-based Discontinuous Galerkin Code for Relativistic Astrophysics

We introduce a new relativistic astrophysics code, SpECTRE, that combines a discontinuous Galerkin method with a task-based parallelism model. SpECTRE's goal is to achieve more accurate solutions for challenging relativistic astrophysics problems such as core-collapse supernovae and binary neutron star mergers. The robustness of the discontinuous Galerkin method allows for the use of high-resolution shock capturing methods in regions where (relativistic) shocks are found, while exploiting high-order accuracy in smooth regions. A task-based parallelism model allows efficient use of the largest supercomputers for problems with a heterogeneous workload over disparate spatial and temporal scales. We argue that the locality and algorithmic structure of discontinuous Galerkin methods will exhibit good scalability within a task-based parallelism framework. We demonstrate the code on a wide variety of challenging benchmark problems in (non)-relativistic (magneto)-hydrodynamics. We demonstrate the code's scalability including its strong scaling on the NCSA Blue Waters supercomputer up to the machine's full capacity of 22,380 nodes using 671,400 threads.Comment: 41 pages, 13 figures, and 7 tables. Ancillary data contains simulation input file

Discontinuous galerkin methods for general relativistic hydrodynamics

Diese Arbeit dokumentiert die Implementierung und Überprüfung einer neuen numerischen Methodik zur allgemeinrelativistischen Simulation von Neutronensternen. Hierbei wurden die GHG-Gleichungen mit Hilfe einer pseudospektralen Penalty-Methode diskretisiert, um die Metrik-Variablen zu evolvieren. Zur Behandlung der hydrodynamischen Variablen wurde sowohl eine diskontinuierliche Galerkin-Methode, als auch ein Finite-Volumen-Verfahren auf krummlinigen Gittern implementiert. Zusätzlich wurden dem Code Techniken zur Schockerkennung und -behandlung hinzugefügt. Diese Bestandteile wurden in umfangreichen Tests validiert. Hierbei wurden numerische Fehler in der Massenerhaltung, in der Einhaltung von physikalischen Zwangsbedingungen und im Vergleich mit analytischen Lösungen betrachtet. Die zugehörigen Konvergenzordnungen aller Methoden wurden untersucht. Dabei wurde eine Konvergenzordnung ~2 bei aktiver Schockbehandlung und mögliche Konvergenzordnungen >2 für die reine diskontinuierliche Galerkin-Methode festgestellt. Weiterhin wurde bestätigt, dass die numerische Einhaltung der metrischen Identitäten essentiell für die Massenerhaltung in Fluiddynamik-Simulationen ist. Eine aktive Schockbehandlung erlaubte die stabile und numerisch konvergente Simulation von statischen, rotierenden oder oszillierenden Neutronensternen. Dank der vorgestellten Modifikationen des bamps Codes ist dieser nun in der Lage, die frontale Kollision zweier Neutronensterne zu simulieren. Neben der Extraktion des zugehörigen Gravitationswellensignals wurde die Simulation genutzt, um das nahezu ideale Skalierungsverhalten der Rechenzeit von bamps aufzuzeigen. Abschließend wurde der neuartige Duale-Blätterung-Formalismus angewandt, um die Wellengleichung auf einem numerischen Gitter zu lösen, welches die lichtartige Zukunftsunendlichkeit beinhaltet. Wir zeigen erste erfolgreiche Tests dieses Verfahrens