Radial-basis-function-based numerical methods for solving compressible flow equations at different Mach numbers

We present the results of a meshless numerical method, based on radial basis functions (RBFs), for the compressible Navier–Stokes equations at different Mach numbers. The method is based on an RBF– Finite Difference (RBF–FD) discretization of the domain using scattered node sets. Local RBF–FD stencils weights are computed at each node location. The RBF–FD stencil weights encode linear operators like differentiation and interpolation and ultimately comprise global differentiation matrices. We use polyharmonic spline RBFs with appended polynomials for the weight generation process. Boundary conditions are enforced via ghost nodes whose values are computed via the local RBF–FD system at each time step. A hyperviscosity operator stabilizes the resulting system of equations. The governing equations are transformed into a system of ODEs using the global differential operators, boundary conditions, and hyperviscosities. The system is solved via an explicit time integrator. We investigate several benchmark problems to determine the efficacy of this approach for simulating high-Mach flows.