Well-Balanced High Order Finite Difference WENO Schemes for a First-Order Z4 Formulation of the Einstein Field Equations

In this work we aim at developing a new class of high order accurate well-balanced finite difference (FD) Weighted Essentially Non-Oscillatory (WENO) methods for numerical general relativity, which can be applied to any first-order reduction of the Einstein field equations, even if non-conservative terms are present. We choose the first-order non-conservative Z4 formulation of the Einstein equations, which has a built-in cleaning procedure that accounts for the Einstein constraints and that has already shown its ability in keeping stationary solutions stable over long timescales. Upon the introduction of auxiliary variables, the vacuum Einstein equations in first order form constitute a PDE system of 54 equations that is naturally nonconservative. We show how to design FD–WENO schemes that can handle non-conservative products. Different variants of FD–WENO are discussed, with an eye to their suitability for higher order accurate formulations for numerical general relativity. We successfully solve a set of fundamental tests of numerical general relativity with up to ninth order spatial accuracy. Due to their intrinsic robustness, flexibility and ease of implementation, finite difference WENO schemes can effectively replace traditional central finite differencing in any first–order formulation of the Einstein field equations, without any artificial viscosity. When used in combination with well–balancing, the new numerical schemes preserve stationary equilibrium solutions of the Einstein equations exactly. This is particularly relevant in view of the numerical study of the quasi-normal modes of oscillations of relevant astrophysical sources. In conclusion, general relativistic high–energy astrophysics could benefit from this new class of numerical schemes and the ecosystem of desirable capabilities built around them.