Algorithms based on DQM with new sets of base functions for solving parabolic partial differential equations in $(2+1)$ dimension

TL;DR

This paper introduces three modified differential quadrature methods using new sets of base functions for solving two-dimensional parabolic PDEs, demonstrating their stability and accuracy through numerical experiments.

Contribution

The paper develops and compares three novel DQM-based methods with new base functions for efficiently solving 2D parabolic PDEs, including stability analysis and validation.

Findings

All three methods are stable for convection-diffusion equations.

The methods produce accurate solutions in agreement with exact solutions.

The proposed approaches are effective for 2D parabolic PDEs.

Abstract

This paper deals with the numerical computations of two space dimensional time dependent parabolic partial differential equations by adopting adopting an optimal five stage fourth-order strong stability preserving Runge Kutta (SSP-RK54) scheme for time discretization, and three methods of differential quadrature with different sets of modified B-splines as base functions, for space discretization: namely i) mECDQM: (DQM with modified extended cubic B-splines); ii) mExp-DQM: DQM with modified exponential cubic B-splines, and iii) MTB-DQM: DQM with modified trigonometric cubic B-splines. Specially, we implement these methods on convection-diffusion equation to convert them into a system of first order ordinary differential equations,in time which can be solved using any time integration method, while we prefer SSP-RK54 scheme. All the three methods are found stable for two space convection-diffusion equation by employing matrix stability analysis method. The accuracy and validity of the methods are confirmed by three test problems of two dimensional convection-diffusion equation, which shows that the proposed approximate solutions by any of the method are in good agreement with the exact solutions.