RBF-FD Based Method of Lines with an Optimal Constant Shape Parameter for Unsteady PDEs

Abstract

It is known that, the RBF-FD scheme combined with standard Runge-Kutta methods in Method of Lines (MOL) provides robust and efficient numerical solutions for unsteady Partial Differential Equations (PDE). If the infinitely smooth RBF like multiquadric is used as basic function in RBF-FD scheme, then the scaling (shape) parameter present in them plays a substantial role in obtaining the accurate numerical solutions. In this work, an optimization process is proposed to determine a constant optimal scaling parameter of basis function in RBF-FD scheme, through method of lines. In this process, the error function is written in terms of local truncation error and then an (near) optimal shape parameter is found by minimizing the error function. The proposed optimization process is validated for one dimensional heat and wave equations.