On the performance of exponential integrators for problems in magnetohydrodynamics

Abstract

Exponential integrators have been introduced as an efficient alternative to explicit and implicit methods for integrating large stiff systems of differential equations. Over the past decades these methods have been studied theoretically and their performance was evaluated using a range of test problems. While the results of these investigations showed that exponential integrators can provide significant computational savings, the research on validating this hypothesis for large scale systems and understanding what classes of problems can particularly benefit from the use of the new techniques is in its initial stages. Resistive magnetohydrodynamic (MHD) modeling is widely used in studying large scale behavior of laboratory and astrophysical plasmas. In many problems numerical solution of MHD equations is a challenging task due to the temporal stiffness of this system in the parameter regimes of interest. In this paper we evaluate the performance of exponential integrators on large MHD problems and compare them to a state-of-the-art implicit time integrator. Both the variable and constant time step exponential methods of EpiRK-type are used to simulate magnetic reconnection and the Kelvin--Helmholtz instability in plasma. Performance of these methods, which are part of the EPIC software package, is compared to the variable time step variable order BDF scheme included in the CVODE (part of SUNDIALS) library. We study performance of the methods on parallel architectures and with respect to magnitudes of important parameters such as Reynolds, Lundquist, and Prandtl numbers. We find that the exponential integrators provide superior or equal performance in most circumstances and conclude that further development of exponential methods for MHD problems is warranted and can lead to significant computational advantages for large scale stiff systems of differential equations such as MHD.

References in corpus (9)

• A Krylov subspace algorithm for evaluating the phi-functions appearing in exponential integrators
• MHD simulations of jet acceleration from Keplerian accretion disks: the effects of disk resistivity
• The Leja method revisited: backward error analysis for the matrix exponential
• Plasmoids formation during simulations of coaxial helicity injection in the National Spherical Torus Experiment
• The Acceleration Mechanism of Resistive MHD Jets Launched from Accretion Disks
• A splitting approach for the Kadomtsev--Petviashvili equation
• A new approach to constructing efficient stiffly accurate exponential propagation iterative methods of Runge-Kutta type (EPIRK)
• Exponential Integrators on Graphic Processing Units
• A modern resistive magnetohydrodynamics solver using C++ and the Boost library

Cited by in corpus (11)

• KIOPS: A fast adaptive Krylov subspace solver for exponential integrators
• Exponential methods for solving hyperbolic problems with application to kinetic equations
• High-order numerical solutions to the shallow-water equations on the rotated cubed-sphere grid
• An adaptive step size controller for iterative implicit methods
• Efficient adaptive step size control for exponential integrators
• LeXInt: Package for Exponential Integrators employing Leja interpolation
• Exponential Integrators for Resistive Magnetohydrodynamics: Matrix-free Leja Interpolation and Efficient Adaptive Time Stepping
• Leveraging Mixed Precision in Exponential Time Integration Methods
• Exploring exponential time integration for strongly magnetized charged particle motion
• Post-Moore Technologies for Plasma Simulation: A Community Roadmap
• Nyström Type Exponential Integrators for Strongly Magnetized Charged Particle Dynamics