MESHLESS METHOD OF LINES FOR NUMERICAL SOLUTIONS OF NONLINEAR TIME DEPENDENT PARTIAL DIFFERENTIAL EQUATIONS

Abstract

Much of useful work being done today for numerical solution of partial differential equations involves nonlinear equations arising in different fields of science and engineering. Most widely used numerical techniques are finite differences, finite elements, spectral methods and collocation methods. However these methods face some limitations like construction of regular grid for irregular and complex geometries, slow convergence rate, stability and low accuracy. Another class of methods known as mesh free methods is expected to be superior than conventional mesh based methods in providing more accurate and stable numerical solution without any connective mesh. Meshless methods using radial basis functions are more flexible with high convergence rate. These methods provide very accurate numerical solution with low spatial resolution. The research presented in this dissertation is based on meshless method of lines using radial basis functions for numerical solutions of nonlinear time dependent partial differential equations (PDEs) namely, Generalized Kuramoto-Sivashinsky (GKS) equation, Kawahara type equations, Equal Width (EW) equation, Modified Equal Width (MEW) equation and Nonlinear Schrodinger (NLS) equation. First the spatial derivatives are discretize converting given PDE in to a system of first order ordinary differential equations (ODEs), which is then solved by classical fourth order Runge-Kutta (RK-4) scheme. Eigenvalue stability and convergence properties of the method are also discussed. Accuracy of the method is measured in terms of L2 and L error norms and conservative properties of mass, momentum and energy. The present method is used to solve various numerical examples available in the literature and results are compared with existing numerical techniques. The method shows superior accuracy, ease of implementation and efficiency of meshless method.