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

Affordable, Entropy Conserving and Entropy Stable Flux Functions for the Ideal MHD Equations

In this work, we design an entropy stable, finite volume approximation for the ideal magnetohydrodynamics (MHD) equations. The method is novel as we design an affordable analytical expression of the numerical interface flux function that discretely preserves the entropy of the system. To guarantee the discrete conservation of entropy requires the addition of a particular source term to the ideal MHD system. Exact entropy conserving schemes cannot dissipate energy at shocks, thus to compute accurate solutions to problems that may develop shocks, we determine a dissipation term to guarantee entropy stability for the numerical scheme. Numerical tests are performed to demonstrate the theoretical findings of entropy conservation and robustness.Comment: arXiv admin note: substantial text overlap with arXiv:1509.06902; text overlap with arXiv:1007.2606 by other author

Entropy Stable Finite Volume Approximations for Ideal Magnetohydrodynamics

This article serves as a summary outlining the mathematical entropy analysis of the ideal magnetohydrodynamic (MHD) equations. We select the ideal MHD equations as they are particularly useful for mathematically modeling a wide variety of magnetized fluids. In order to be self-contained we first motivate the physical properties of a magnetic fluid and how it should behave under the laws of thermodynamics. Next, we introduce a mathematical model built from hyperbolic partial differential equations (PDEs) that translate physical laws into mathematical equations. After an overview of the continuous analysis, we thoroughly describe the derivation of a numerical approximation of the ideal MHD system that remains consistent to the continuous thermodynamic principles. The derivation of the method and the theorems contained within serve as the bulk of the review article. We demonstrate that the derived numerical approximation retains the correct entropic properties of the continuous model and show its applicability to a variety of standard numerical test cases for MHD schemes. We close with our conclusions and a brief discussion on future work in the area of entropy consistent numerical methods and the modeling of plasmas

On the convergence of a shock capturing discontinuous Galerkin method for nonlinear hyperbolic systems of conservation laws

In this paper, we present a shock capturing discontinuous Galerkin (SC-DG) method for nonlinear systems of conservation laws in several space dimensions and analyze its stability and convergence. The scheme is realized as a space-time formulation in terms of entropy variables using an entropy stable numerical flux. While being similar to the method proposed in [14], our approach is new in that we do not use streamline diffusion (SD) stabilization. It is proved that an artificial-viscosity-based nonlinear shock capturing mechanism is sufficient to ensure both entropy stability and entropy consistency, and consequently we establish convergence to an entropy measure-valued (emv) solution. The result is valid for general systems and arbitrary order discontinuous Galerkin method.Comment: Comments: Affiliations added Comments: Numerical results added, shortened proo

Hybrid Entropy Stable HLL-Type Riemann Solvers for Hyperbolic Conservation Laws

It is known that HLL-type schemes are more dissipative than schemes based on characteristic decompositions. However, HLL-type methods offer greater flexibility to large systems of hyperbolic conservation laws because the eigenstructure of the flux Jacobian is not needed. We demonstrate in the present work that several HLL-type Riemann solvers are provably entropy stable. Further, we provide convex combinations of standard dissipation terms to create hybrid HLL-type methods that have less dissipation while retaining entropy stability. The decrease in dissipation is demonstrated for the ideal MHD equations with a numerical example.Comment: 6 page