High-order finite volume methods for magnetohydrodynamics with applications in computational astrophysics

Abstract

Finite volume (FV) methods with high-order accuracy have attracted the interest of computational astrophysicists who are interested in modeling the most challenging physical regimes. With their large numerical diffusivity, popular second-order accurate FV codes are incapable of resolving many such problems using current high-performance computing (HPC) hardware. High-order FV methods may perform more efficiently than their lower-order counterparts while continuing to offer the robust shock-capturing properties that are essential for simulating the highly compressible flows that often occur in astrophysical phenomena.

We present a novel fourth-order accurate finite volume method for the solution of ideal magnetohydrodynamics (MHD). The numerical method combines high-order quadrature rules in the solution to semi-discrete formulations of hyperbolic conservation laws with the upwind constrained transport (UCT) framework to ensure that the divergence-free constraint of the magnetic field is satisfied. A novel implementation of UCT that uses the piecewise parabolic method (PPM) for the reconstruction of magnetic fields at cell corners in 2D is introduced. The resulting scheme can be expressed as the extension of the second-order accurate constrained transport (CT) Godunov-type scheme that is currently used in the Athena++ astrophysics code. After validating the base algorithm on a series of hydrodynamics test problems, we present the results of multidimensional MHD test problems which demonstrate formal fourth- order convergence for smooth problems, robustness for discontinuous problems, and improved accuracy relative to the second-order scheme.

The fourth-order FV method is implemented within the open-source Athena++ framework. A comprehensive set of validation and performance tests are added and directly integrated in the collaborative development environment using continuous integration services. Other automated tools and practices are established to ensure the manageability of the codebase as it matures.

We apply the solver to a set of computationally demanding 2D benchmarks based on the Kelvin- Helmholtz instability. The fourth-order and several related high- order FV methods produce significantly more accurate solutions than the second-order method. By comparing the computational performance of these schemes on modern multi-core, distributed-memory architectures, we show that the high-order solvers are capable of much greater efficiency (time- to-solution for a given level of accuracy) than the second-order alternatives.