Exact explicit time integration of hyperbolic partial differential equations with mesh free radial basis functions

This study is a progress report that examines the numerical solution of inviscid hyperbolic partial differential equations (PDEs) without the need for upwind differencing and other numerical artifacts. The fixed frame PDEs are locally transformed by rotating and translating the coordinate system at each local discretization point. These transformations yield a simpler PDE system that is effectively linearized. It is assumed that in this transformed local frame within a time interval, ΔtΔt, the dependent variables are products of the spatial dependent radial basis functions (RBFs), and the time dependent expansion coefficients, χ(t)χ(t). This linearization is exploited by transforming the PDEs into systems of linear ordinary differential equations (ODEs) in terms of the expansion coefficients. The affine space decomposition is used to obtain an ODE system of NiNi ODEs in NiNi unknowns that can be integrated exactly in time. Then the entire set of N expansion coefficients is found. Numerical results show that hyperbolic PDEs can be integrated in time without upwinding and the root mean square errors between the exact and numerical solutions are indeed very small.