A fully implicit dynamo model for long-term evolution of the geomagnetic field

Abstract:

In this work, we present a Newton-Krylov-Schwarz (NKS) based parallel implicit solver for the governing equations of Earth’s dynamo. NKS is a general purpose parallel solver for nonlinear systems and has been widely applied to solve different kinds of nonlinear problems. All previously published dynamo models treat nonlinear terms of dynamo governing equations in an explicit or semi-explicit manner, consequently, the numerical schemes are constrained by the Courant–Friedrichs–Lewy (CFL) condition. To ensure numerical stability, time step sizes should be rather small, especially for high-resolution dynamo simulations, which makes it impractical to use high-resolution dynamo models to study long term evolution of the geomagnetic field, such as reversals and superchrons. To avoid the time step size constraint imposed by CFL numbers associated with fine spacial mesh sizes, we try a fully implicit method and focus on efficiently solving the large nonlinear system at each time step on large scale parallel computers. Our algorithm begins with a discretization of the governing equations on an unstructured tetrahedron mesh with a stable finite element method in space and a fully implicit backward difference scheme in time. At each time step, an inexact Newton method is employed to solve the discretized large sparse nonlinear system while in the Newton steps, a domain decomposition preconditioned Krylov method is used to solve the Jacobian system which is constructed analytically in order to obtain the desired performance. Our numerical model is tested against known standard dynamo solutions at a moderate Ekman number. Additionally, numerical experiments show that our model has super-linear scalability with over eight thousand processors for dynamo problems with tens of millions of unknowns.