Fifth-Order Accurate Multidimensional Finite Volume MHD Scheme Based on Divergence Cleaning

Speaker: Dongwook Lee, Department of Applied Mathematics, University of California
October 29, 2026

Presentation

Abstract

We present a fifth-order accurate multidimensional finite volume MHD scheme based on a divergence-cleaning projection method. The new cleaning method numerically preserves the solenoidal constraint on the magnetic field. Our projection method utilizes a new hyperbolic-variant successive over-relaxation (SORh) iteration designed specifically to solve the Poisson equation arising in the projection step. Unlike the conventional FFT-based global approach, SORh is a local method whose overall performance is nonetheless comparable to a highly optimized FFT solver. We integrate the new SORh scheme into an MHD scheme within the projection framework, introducing a fifth-order accurate algorithm based on the Gaussian-process (GP) approximation. In comparison to global FFT-based Poisson solvers, SORh features a matrix-free, single-stencil-pass-per-sweep structure for its local operations, admitting straightforward extension to mixed and non-periodic boundary conditions, variable-coefficient elliptic operators, and adaptive meshes. The new SORh MHD algorithm is validated across a suite of benchmark test problems, including the smooth circularly polarized Alfvén wave for convergence and several shock-dominated problems, on which the new method is on par with the highly optimized spectral FFT projection in terms of accuracy, stability, and efficiency.

Biography

Dongwook Lee is a professor in the Department of Applied Mathematics at UC Santa Cruz. Dongwook’s research interests emphasize developing numerical schemes of high-order shock-capturing methods for computational magnetohydrodynamics (MHD) and gas dynamics on large-scale computing architectures. Dongwook applies these numerical methods to investigate nonlinear flow problems in astrophysics and high-energy-density physics. Before joining UC Santa Cruz, he worked as a senior applied mathematician at the FLASH Center for Computational Science at the University of Chicago. Dongwook was the primary code architect of the unsplit hydrodynamics and magnetohydrodynamics solvers in the FLASH code from 2006 to 2014. Dongwook received his Ph.D. in Applied Mathematics and Scientific Computation from the University of Maryland at College Park in 2006.

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