High-order schemes for solving hyperbolic conservation laws, such as the magnetohydrodynamic (MHD) equations, often suffer from uncontrolled numerical oscillations near nonlinear discontinuities. We present a novel approach to handle such discontinuities within the framework of the space-time conservation-element and solution-element (CESE) scheme. This method exploits a unique feature of the CESE scheme—the simultaneous solution of spatial derivatives along with conservative variables. We identify troubled cells (those containing discontinuities) and selectively limit high-order derivatives within them to suppress spurious oscillations. Central to this approach is a simple yet effective discontinuity indicator, which is highly sensitive to sharp gradients: its magnitude is of order unity near discontinuities, contrasting with values of order Δ3 (where Δ is the grid spacing) in smooth regions. Notably, the indicator is locally defined and computed within a single cell, requiring no information from neighboring cells, thereby making the method well-suited for parallel computing. We validate the method using a suite of benchmark tests involving strong MHD discontinuities in both two and three dimensions. Results demonstrate that the method robustly suppresses nonphysical oscillations while maintaining high-order accuracy in smooth regions. This makes it a promising tool for accurate and efficient simulation of MHD flows involving shocks and current sheets.
Zhang et al. (2025) studied this question.