A new gradient descent divergence cleaning method with optimized high-order low-dissipation TENO schemes for ideal magnetohydrodynamic simulations

Abstract

In this study, we present a new Gradient Descent Divergence Cleaning (GDDC) method, integrated with optimized high-order low-dissipation Target Essentially Non-Oscillatory (TENO) schemes, for the simulation of ideal magnetohydrodynamic (MHD) flows while ensuring a robust divergence-free constraint. The GDDC method constructs a loss function that quantifies the total divergence error. By computing the gradient of this loss function with respect to the magnetic field, the method integrates seamlessly into the Runge-Kutta time-stepping scheme to enable divergence cleaning consistent with the spatial discretization in a discrete sense. Unlike traditional divergence cleaning methods that depend on a non-physical variable ψ and associated parameters controlling its transport velocity and decay rate, the GDDC method introduces only a single parameter, η, and preserves the original hyperbolic eigensystem of the ideal MHD equations. Numerical experiments demonstrate that the method is highly insensitive to variations of η across three orders of magnitude. Additionally, we build the GDDC method upon the optimized TENO (TENO-OPT) schemes. Leveraging the TENO schemes' inherent ability to detect discontinuities, dynamic linear weights are used to combine sub-stencils into an optimal large stencil that avoids crossing discontinuities. Extensive numerical tests demonstrate that the GDDC method coupled with the 5th- and 6th-order TENO-OPT schemes effectively captures discontinuities in ideal MHD flows with low dissipation and without spurious oscillations. We developed our code using NVIDIA's CUDA Fortran and evaluated its performance on a single NVIDIA A800 GPU. For an ideal MHD simulation that employs a 5th-order TENO-OPT scheme, each Runge-Kutta time-stepping stage took approximately 1.118 seconds at a grid resolution of 0.268 billion points. Both computational time and GPU memory usage increase linearly with grid size, with roughly 4 million grid points requiring 1 GB of memory. Based on these scaling trends, the enhanced memory capacity of the NVIDIA H200 GPU could theoretically support up to approximately 0.560 billion grid points. The code is available on https://github.com/FallenCastle/MHD_GDDC_GPU.