A High-Order Meshless Method for Solving Multiphysics Problems in Complex Domains

We present a high-order meshless method based on scattered data interpolation for solving Multiphysics problems in complex domains. A complex domain is first represented by scattered points generated by any method, including as vertices of a finite element grid. The flow variables are located at the scattered points and interpolated over a local cloud of points using a radial basis function (RBF). We consider Polyharmonic splines (PHS) which do not need a shape parameter. We further append a polynomial of desired degree to achieve high order discretization accuracy. To solve the Navier-Stokes equations, the partial-derivatives are evaluated by differentiating the radial basis functions and the polynomials. We have developed an explicit fractional step method and a semi-implicit method to integrate the Navier-Stokes and energy equations in time with high spatial and second-order temporal accuracies. The explicit fractional step method is limited by the combined convective and diffusive Courant numbers while the semi-implicit method can use time steps 10-20 times the explicit time step. A three-dimensional extension for periodic problems has been developed with Fourier expansions in the periodic direction. We have demonstrated the method has high spatial discretization accuracy.