An Alternative Finite Difference WENO-Like Scheme with Physical Constraint Preservation for Divergence-Preserving Hyperbolic Systems

Abstract Alternative finite difference Weighted Essentially Non-Oscillatory (AFD-WENO) schemes allow us to very efficiently update hyperbolic systems even in complex geometries. Recent innovations in AFD-WENO methods allow us to treat hyperbolic systems with non-conservative products almost as efficiently as conservation laws. However, some PDE systems, like computational electrodynamics (CED), magnetohydrodynamics (MHD), and relativistic magnetohydrodynamics (RMHD), have involution constraints that require divergence-free or divergence-preserving evolution of vector fields. In such situations, a Yee-style collocation of variables proves indispensable, and that collocation is retained in this work. In previous works, only higher order finite volume (FV) discretization of such involution-constrained systems was possible. In this work, we show that substantially more efficient AFD-WENO methods have been extended to encompass divergence-preserving hyperbolic PDEs. Our method retains the Yee-style collocation of normal components of the divergence-free/preserving vector field. However, the variables that require zone-centered evolution are evolved with AFD-WENO methods. Since those variables make up the bulk of the primal variables for the PDE of interest, this results in substantial savings in computational complexity. Even the volumetric reconstruction of the divergence-free/preserving vector field is bypassed. Instead, we realize that any divergence-preserving update of a vector field must have a general form. This general form looks closely like the familiar induction equation that is well known in CED or MHD. We exploit the generality of that form to extract the edge-centered variables that are needed in the update of the facially averaged vector field components. The two-dimensional (2D) Riemann solver is used to provide us with multidimensionally stabilized versions of these update terms. The generality of our approach is demonstrated by the fact that problems in CED, MHD, and RMHD can all be solved by the same general AFD-WENO algorithm that is presented here. Spatial accuracies up to the ninth order of accuracy are demonstrated. Several stringent test problems from CED, MHD, and RMHD are shown. We also show that the algorithm takes well to the physical constraint-preserving (PCP) formulation of AFD-WENO schemes that was presented by the authors. The efficient and time-explicit PCP strategy for divergence-preserving PDEs that we have presented here extends the applicability of our method to very stringent MHD and RMHD problems.