Mixed Crank-Nicolson and Galerkin Methods for Solving Nonlinear Hyperbolic Partial Differential Equation

Marwa Ahmed

Department of Mathematics, College of Basic Education, Diyala University

W. A. Ibrahim

S. J. M. Al-Qaisi

DOI: https://doi.org/10.47831/mjpas.v2i4.107

Keywords: NonlinearHyperbolicPartial DifferentialEquation, method of Galerkin finite elements, Crank-Nicolson Scheme, Convergent


Abstract

In this work,the approximation solution for nonlinear hyperbolic partial differential equation(NLHPDE) is obtained by using the mixed Crank-Nicolson (CN) schemeand the GalerkinMethod(GM) and it is symbolized by (MCNGM).At first theCNis utilized for the variable of time to obtain the discrete weak form for the NLHPDE, and then the GM is utilized which reduces the DWF into theGalerkin nonlinear algebraic system (GNLAS) at each step of time. Through utilizing the predictor-corrector techniques which are symbolized by the obtained GNLAS is transformed into Galerkin linear algebraic system (GLAS) which is solved by applying the Cholesky method. The convergence of the method is studied. Some examples are given to illustrate the efficiency and the accuracy for the proposed method.