Hybrid radial basis function methods of lines for the numerical solution of viscous Burgers’ equation

Abstract

An efficient and accurate hybrid radial basis function (HRBF) method is proposed to numerically solve the quasi-linear viscous Burgers’ equation. Two different RBFs are combined: an infinite smooth RBF defined by a shape parameter and a piecewise smooth RBF independent of the shape parameter. This combination formulates a family of HRBF methods, which is then used to approximate the spatial operator of the viscous Burgers equation. An efficient high-order solver is then used to integrate in time the resulting initial value problem. The developed numerical scheme is tested on some benchmark problems varying the Reynolds number. Accuracy and efficiency is validated using $$L_{\infty },~ L_{2}$$ and $$L_{\text {rms}}$$ error norms, as well as the number of nodes N over the domain of influence. A rigorous comparison is conducted against well-known numerical schemes to manifest improved accuracy of the proposed scheme. Eigenvalue stability analysis of the proposed technique is thoroughly discussed and confirmed by numerical examples. The proposed scheme is found a well-behaved alternative to RBF (Kansa) and RBF-PS methods with improved accuracy and good conditioning properties.