Invariant Numerical Schemes for Solution of Hyperbolic Partial Differential Equations

In this study, we extend our earlier work on Lie symmetry preservation in numerical schemes (Ozbenli and Vedula, JCP 349, 2017) and demonstrate the implementation of the proposed methods to more general problems including (but not limited to) hyperbolic PDEs. Considering non-invariant base numerical schemes that are obtained from compact schemes or defect correction procedures, we present a method to develop high order accurate invariant schemes that inherit Lie symmetry groups (such as translation, scaling, Galilean and projection groups) of underlying PDEs. Performance of the proposed method is evaluated through applications to numerical solution of linear/nonlinear advection equations and Euler equations for description of inviscid compressible flows (in 1D and 2D). For the case of Euler equations, we demonstrate the construction and performance of invariant forms of Lax-Friedrichs scheme and van Leer flux vector splitting scheme. Our preliminary results indicate that the invariant schemes often perform considerably better than their non-invariant counterparts. In particular it can be shown that invariant schemes often have significantly better performance with reference to error measures based on symmetries than standard (non-invariant) schemes.