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

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.

💡 Research Summary

The paper presents a generalized characteristic‑mapping method (CMM) capable of handling nonlinear advection equations with source terms, and demonstrates its application to two‑dimensional ideal magnetohydrodynamics (MHD). Traditional numerical schemes for conservation laws with source terms often suffer from stiffness and loss of accuracy, especially when the source term is strong or highly nonlinear. The authors extend the previously developed CMM—originally limited to homogeneous advection—by incorporating Duhamel’s principle, which separates the solution into a pull‑back of the initial condition by the backward flow map and an integral of the source term along characteristics.

In the new framework two quantities are evolved simultaneously: (i) the backward characteristic map (X