Abstract
Discontinuous Galerkin (DG) methods provide a means to obtain high-order accurate solutions in regions of smooth fluid flow while still resolving strong shocks. These and other properties make DG methods attractive for solving problems involving hydrodynamics; e.g., the core-collapse supernova problem. With that in mind we are developing a DG solver for the general relativistic, ideal hydrodynamics equations under a 3+1 decomposition of spacetime, assuming a conformally-flat approximation to general relativity. With the aid of limiters we verify the accuracy and robustness of our code with several difficult test-problems: a special relativistic Kelvin-Helmholtz instability problem, a two-dimensional special relativistic Riemann problem, and a one- and two-dimensional general relativistic standing accretion shock (SAS) problem. We find good agreement with published results, where available. We also establish sufficient resolution for the 1D SAS problem and find encouraging results regarding the standing accretion shock instability (SASI) in 2D.
Highlights
Core-collapse supernovae are multi-physics, multi-dimensional phenomena that require sophisticated numerical methods to accurately capture all of their features, both on a macroscopic and a microscopic scale
Speaking there are three branches of physics that every realistic core-collapse supernova (CCSN) model must treat faithfully: gravity, neutrino transport, and hydrodynamics. To model these events we are developing the toolkit for high-order neutrino-radiation hydrodynamics—thornado. thornado is a software package designed to solve the equations of neutrino transport and hydrodynamics using a discontinuous Galerkin (DG) method [4, 5]
With (10) we have a set of ordinary differential equations (ODEs) we can evolve in time with an ODE integrator [30]
Summary
Core-collapse supernovae are multi-physics, multi-dimensional phenomena that require sophisticated numerical methods to accurately capture all of their features, both on a macroscopic and a microscopic scale. Speaking there are three branches of physics that every realistic CCSN model must treat faithfully: gravity, neutrino transport, and hydrodynamics. To model these events we are developing the toolkit for high-order neutrino-radiation hydrodynamics—thornado. Overview of the DG method The DG method provides a means of solving partial differential equations and obtaining highorder accurate solutions in space, on a compact stencil. This is achieved by using a high-degree polynomial representation of the solution within each element, communicating with nearestneighbors only. We give a brief overview of the DG method here
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