An overset grid method for global geomagnetic induction

Abstract

A new finite difference solution to the global geomagnetic induction problem is developed and tested, based on a modified Lorenz gauge of the magnetic vector and electric scalar potentials and implementing a novel, overset 'Yin-Yang' grid that avoids unnecessary mesh refinement at the geographic poles. Previously used in whole-earth mantle convection models, the overset grid is built from a pair of partially overlapping mid-latitude latitude-longitude (lat/lon) grids, one of which is rotated with respect to the other for complete coverage of the sphere. Because of this symmetry, only one set of finite difference templates is required for global discretization of the governing Maxwell equations, a redundancy that is exploited for computational efficiency and multithreaded parallelization. Comparisons between solutions obtained by the proposed method show excellent agreement with those obtained by independent integral equation methods for 1-D, 2-D and 3-D problem geometries. The computational footprint of the method is minimized through a (non-symmetric) matrix-free BiCG-STAB iterative solver which computes finite difference matrix coefficients 'on the fly' as needed, rather than pulling stored values from memory. Scaling of the matrix-free BiCG-STAB algorithm with problem size shows behaviour similar to that seen with the (symmetric) QMR algorithm used in the Cartesian case from which the present algorithm is based. The proposed method may therefore provide a competitive addition to the existing body of global-scale geomagnetic induction modelling algorithms, allowing for resource-efficient forward modelling as the kernel for large-scale computing such as inversion of geomagnetic response functions, computational hypothesis testing and parametric studies of mantle geodynamics and physiochemical state. © The Author 2014. Published by Oxford University Press on behalf of The Royal Astronomical Society.