A Characteristic Mapping Method with Source Terms: Applications to Ideal Magnetohydrodynamics

TL;DR

This paper presents a generalized characteristic mapping method with source terms for ideal magnetohydrodynamics, achieving high accuracy and resolution in simulating complex flow features.

Contribution

It introduces a recursive semi-Lagrangian characteristic mapping approach incorporating source terms, enhancing computational efficiency and accuracy in MHD simulations.

Findings

Achieves third-order accuracy in space and time.

Demonstrates high-resolution simulation of fine-scale current sheets.

Provides error estimates indicating third-order convergence.

Abstract

This work introduces a generalized characteristic mapping method designed to handle non-linear advection with source terms. The semi-Lagrangian approach advances the flow map, incorporating the source term via the Duhamel integral. We derive a recursive formula for the time decomposition of the map and the source term integral, enhancing computational efficiency. Benchmark computations are presented for a test case with an exact solution and for two-dimensional ideal incompressible magnetohydrodynamics (MHD). Results demonstrate third-order accuracy in both space and time. The submap decomposition method achieves exceptionally high resolution, as illustrated by zooming into fine-scale current sheets. An error estimate is performed and suggests third order convergence in space and time.