AM Seminar: Local, Matrix-Free Poisson Solve and its Projection Scheme for a High-Order Finite Volume MHD Method

Presenter: Dongwook Lee, Professor, Applied Mathematics, UC Santa Cruz

Description: We present a fifth-order accurate multidimensional finite volume MHD scheme using a divergence-cleaning projection method to preserve the magnetic field’s solenoidal constraint. The projection step employs a new hyperbolic-variant successive over-relaxation (SORh) iteration to solve the Poisson equation. Unlike conventional global FFT approaches, SORh is a local method with performance comparable to a highly optimized FFT solver. Integrated into the MHD framework via a fifth-order Gaussian-process (GP) approximation, SORh features a matrix-free, single-stencil-pass-per-sweep structure. This allows straightforward extensions to mixed/non-periodic boundary conditions, variable-coefficient elliptic operators, and adaptive meshes. Benchmark tests, including smooth circularly polarized Alfvén waves and shock-dominated problems, validate that the new method matches highly optimized spectral FFT projection in accuracy, stability, and efficiency.