Towards Robust Solvers for Nuclear Fusion Simulations Using JOREK: A Numerical Analysis Perspective

TL;DR

This paper analyzes the mathematical properties of linear systems in JOREK, a code for nuclear fusion simulations, to improve solver robustness and convergence in large-scale MHD modeling.

Contribution

It provides the first thorough mathematical analysis of JOREK's linear systems, diagnosing convergence issues and guiding the development of more effective solver strategies.

Findings

Spectral analysis reveals causes of convergence problems.

Diagnosis enables targeted improvements in solver methods.

Analysis paves the way for advanced projection techniques.

Abstract

One of the most well-established codes for modeling non-linear Magnetohydrodynamics (MHD) for tokamak reactors is JOREK, which solves these equations with a B\'ezier surface based finite element method. This code produces a highly sparse but also very large linear system. The main solver behind the code uses the Generalized Minimum Residual Method (GMRES) with a physics-based preconditioner, but even with the preconditioner there are issues with memory and computation costs and the solver does not always converge well. This work contains the first thorough study of the mathematical properties of the underlying linear system. It enables us to diagnose and pinpoint the cause of hampered convergence. In particular, analyzing the spectral properties of the matrix and the preconditioned system with numerical linear algebra techniques, will open the door to research and investigate more performant solver strategies, such as projection methods.