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

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.

💡 Research Summary

The paper presents a novel, second‑order accurate, single‑step Godunov scheme for solving the magnetohydrodynamics (MHD) equations in multiple dimensions. The core of the method is the dimensionally unsplit Corner Transport Upwind (CTU) algorithm originally introduced by Colella, which allows simultaneous computation of fluxes in all coordinate directions without resorting to directional splitting. By embedding CTU within a cell‑centered framework, the authors retain a compact data layout where all primary variables—including the magnetic field components—are stored at cell centers, simplifying integration with existing hydrodynamics codes.

A major challenge in MHD simulations is the preservation of the divergence‑free condition (∇·B = 0). To address this, the scheme incorporates the Generalized Lagrange Multiplier (GLM) divergence‑cleaning technique of Dedner et al. (2002). An auxiliary scalar field ψ is introduced; its evolution equation contains both hyperbolic transport terms, which propagate divergence errors away from their source at the speed of artificial “cleaning waves,” and parabolic damping terms, which attenuate the errors exponentially. This mixed hyperbolic/parabolic approach ensures that any spurious magnetic monopole generated by discretization is quickly expelled from the computational domain, eliminating the need for staggered magnetic field representations used in constrained‑transport (CT) methods.

Two variants of the algorithm are examined. The first, fully conservative version, preserves mass, momentum, total energy, and magnetic induction exactly (up to truncation error) by using the standard Godunov fluxes together with the GLM source terms. The second variant relaxes momentum and energy conservation in order to reduce computational overhead; despite this relaxation, the scheme remains robust for a wide range of test problems because the dominant source of instability—divergence error—is still actively cleaned.

The authors validate the CTU‑GLM scheme against a suite of standard 2‑D and 3‑D MHD benchmarks. In the 2‑D Orszag‑Tang vortex, the method reproduces the intricate vortex cascade and magnetic reconnection patterns with L1 errors 10–20 % lower than those obtained with a traditional cell‑centered eight‑wave formulation. In the 3‑D blast‑wave test, the scheme captures the spherical shock front, the anisotropic expansion caused by the background magnetic field, and the subsequent formation of a high‑pressure cavity, again matching or surpassing the accuracy of constrained‑transport implementations at comparable grid resolutions. Additional tests involving rotating helically magnetized columns and open‑boundary outflows demonstrate that the GLM cleaning efficiently transports divergence errors out of the domain, even when complex boundary conditions are present.

Performance-wise, the CTU‑GLM algorithm requires fewer lines of code and less auxiliary data than CT methods because it avoids staggered magnetic field storage and the associated face‑centered flux reconstructions. The unsplit nature of CTU also eliminates the directional bias inherent in split‑step schemes, leading to more isotropic error distribution and better preservation of symmetry in multidimensional flows.

In summary, the paper introduces a practical, high‑accuracy MHD solver that combines the strengths of unsplit Godunov‑type CTU integration with GLM divergence cleaning. It delivers second‑order convergence, maintains the key conservation properties of the MHD system (or offers a controlled relaxation for efficiency), and demonstrates robustness across a broad spectrum of challenging test cases. Its relative implementation simplicity, together with competitive accuracy relative to both eight‑wave and constrained‑transport approaches, makes it an attractive candidate for incorporation into existing astrophysical and plasma‑physics simulation frameworks.