Numerical Simulation for the Two-Dimensional Partial Differential Equations

Abstract

The present paper is devoted to the development of a new schemes for solving the two-dimensional heat equation and two-dimensional wave equation by the finite difference method (FDM) and non-standard finite difference method (NSFDM). The stability analysis for the two schemes is discussed and provides conditional stability conditions. The local truncation errors of both schemes are calculated, so confirm their consistency and convergence of the schemes. Numerical results are obtained for two different test cases for each equation and are compared with exact solutions and other numerical results revealing absolute errors and generating convergence tables. The graphical results show the spatial-temporal behavior of solutions, emphasizing the effectiveness of the presented schemes. When we compare the FDM with the NSFDM, the NSFDM technique outperforms the latter in terms of numerical artifact mitigation. This paper presents a comprehensive methodology for solving multidimensional PDEs while maintaining theoretical rigor (stability, error analysis) and practical validation (numerical benchmarks), with applications in computational physics and engineering.