Time-discretization of a plasma-neutral MHD model with a semi-implicit leapfrog algorithm

TL;DR

This paper explores advanced time-discretization methods for plasma-neutral MHD models, comparing semi-implicit leapfrog, Crank-Nicolson, and operator-splitting techniques to improve accuracy and computational efficiency.

Contribution

It introduces and evaluates operator-splitting methods for integrating atomic physics into semi-implicit leapfrog MHD models, enhancing accuracy and reducing computational costs.

Findings

Douglas-Rachford inspired coupling reduces time-discretization error.

Operator-splitting methods enable parallelization and lower computational cost.

Second-order-in-time methods achieve accuracy comparable to Crank-Nicolson with nonlinear iteration.

Abstract

The semi-implicit leapfrog time-discretization is a workhorse algorithm for initial-value MHD codes to bridge between vastly separated time scales. Inclusion of atomic interactions with neutrals breaks the functional structure of the MHD equations that exploited by the leapfrog. We address how to best integrate atomic physics into the semi-implicit leapfrog. Following the Crank-Nicolson method, one approach is to time-center the atomic interactions in the linear solver and use a Newton method to include the nonlinear contributions. Alternatively, another family of methods are based on operator-splitting the terms associated with the atomic interactions using a Strang-splitting technique. These methods naturally break equations into constituent ODE and PDE parts and preserve the structure exploited by the semi-implicit leapfrog. We study the accuracy and efficiency of these methods through a battery of 0D and 1D cases and show that a second-order-in-time Douglas-Rachford inspired coupling between the ODE and PDE advances is effective in reducing the time-discretization error to be comparable to that of Crank-Nicolson with Newton iteration of the nonlinear terms. Splitting ODE and PDE parts results in independent matrix solves for each field which reduces the computational cost considerably and provides parallelization over species relative to Crank-Nicolson.