Meshless Euler solver using radial basis functions for solving inviscid compressible flows

The goal of this research work is to develop a meshless Euler solver using radial basis functions (RBFs). Meshless methods attempt to address the problems in computational methods arising due to their mesh dependence. The present meshless method uses the differential quadrature (DQ) technique, which states that the derivatives of a function at a point can be approximated by a linear combination of the function values at a set of scattered points or nodes in its neighborhood. The derivative evaluation is dependent only on the nodal distribution and independent of the function. RBFs are used as basis functions for the DQ technique.; The local radial basis function-differential quadrature (RBF-DQ) method is used to develop a meshless Euler solver for inviscid compressible flows. An Euler solver should take into account the direction of wave propagation associated with the hyperbolic PDEs. Hence second order a Rusanov solver is employed to evaluate fluxes at the mid-points, analogous to flux evaluation at cell interface in the finite volume method. The DQ technique is then applied to these upwind fluxes to approximate the flux gradients. Thus the conservative form of the Euler equation in differential form is discretized using RBF-DQ technique. The solver is applied to and validated by various steady state compressible flows. The present meshless Euler solver using RBFs captures the flow physics both qualitatively and quantitatively.