Recovery of High Order Accuracy in Radial Basis Function Approximations of Discontinuous Problems

Radial basis function(RBF) methods have been actively developed in the last decades. The advantages of RBF methods are that these methods are mesh-free and yield high order accuracy if the function is smooth enough. The RBF approximation for discontinuous problems, however, deteriorates its high order accuracy due to the Gibbs phenomenon. With the Gibbs phenomenon in the RBF approximation, the L∞ error remains only O(1). The main purpose of this paper is to show that high order accuracy can be recovered from the RBF approximation contaminated with the Gibbs phenomenon if a proper reconstruction method is applied. In this work, the Gegenbauer reconstruction method is used to reconstruct the RBF approximation for the recovery of high order accuracy. Several numerical examples presented in this work indicate that the Gegenbauer polynomials are Gibbs complementary to the RBF approximations and hence high order convergence can be recovered from the RBF approximations for discontinuous problems. These results also indicate that the Gegenbauer reconstruction method which was originally developed for the polynomial approximation works for non-polynomial basis methods such as the RBF method. Numerical examples including the linear and nonlinear hyperbolic partial differential equations are presented.