An efficient numerical scheme for a 3D spherical dynamo equation

TL;DR

This paper introduces a fast and stable numerical scheme for solving the 3D spherical dynamo equations, combining semi-implicit time discretization with spectral methods to efficiently handle the divergence-free condition.

Contribution

The paper presents a novel semi-implicit spectral scheme that reduces the computational complexity of 3D spherical dynamo simulations by transforming the problem into a sequence of 1D radial equations.

Findings

The scheme remains stable and bounded regardless of the number of unknowns.

Numerical results validate the efficiency and accuracy of the proposed method.

Abstract

We develop an efficient numerical scheme for the 3D mean-field spherical dynamo equation. The scheme is based on a semi-implicit discretization in time and a spectral method in space based on the divergence-free spherical harmonic functions. A special semi-implicit approach is proposed such that at each time step one only needs to solve a linear system with constant coefficients. Then, using expansion in divergence-free spherical harmonic functions in the transverse directions allows us to reduce the linear system at each time step to a sequence of one-dimensional equations in the radial direction, which can then be efficiently solved by using a spectral-element method. We show that the solution of fully discretized scheme remains bounded independent of the number of unknowns, and present numerical results to validate our scheme.