A Second-Order Unsplit Godunov Scheme for Cell-Centered MHD: the CTU-GLM scheme

TL;DR

This paper introduces a second-order, dimensionally unsplit Godunov scheme for multi-dimensional magneto-hydrodynamics that conserves key physical quantities and effectively manages divergence errors, offering a simpler alternative to existing methods.

Contribution

It presents a novel cell-centered, second-order Godunov scheme for MHD that is robust, accurate, and easier to implement than constrained transport methods.

Findings

The scheme accurately solves 2D and 3D MHD problems.

It effectively dampens divergence errors using Dedner's cleaning technique.

The method is competitive with existing constrained transport approaches.

Abstract

We assess the validity of a single step Godunov scheme for the solution of the magneto-hydrodynamics equations in more than one dimension. The scheme is second-order accurate and the temporal discretization is based on the dimensionally unsplit Corner Transport Upwind (CTU) method of Colella. The proposed scheme employs a cell-centered representation of the primary fluid variables (including magnetic field) and conserves mass, momentum, magnetic induction and energy. A variant of the scheme, which breaks momentum and energy conservation, is also considered. Divergence errors are transported out of the domain and damped using the mixed hyperbolic/parabolic divergence cleaning technique by Dedner et al. (J. Comput. Phys., 175, 2002). The strength and accuracy of the scheme are verified by a direct comparison with the eight-wave formulation (also employing a cell-centered representation) and with the popular constrained transport method, where magnetic field components retain a staggered collocation inside the computational cell. Results obtained from two- and three-dimensional test problems indicate that the newly proposed scheme is robust, accurate and competitive with recent implementations of the constrained transport method while being considerably easier to implement in existing hydro codes.