A new numerical algorithm for the solutions of hyperbolic partial differential equations in $(2+1)$-dimensional space

TL;DR

This paper introduces a novel numerical algorithm, MTB-DQM, based on modified trigonometric cubic B-splines, for efficiently solving hyperbolic PDEs in (2+1) dimensions, demonstrating stability and high accuracy through various tests.

Contribution

The paper presents a new modified trigonometric cubic B-spline differential quadrature method for hyperbolic PDEs, including stability analysis and comparison with existing methods.

Findings

MTB-DQM is stable for telegraph equations.

The method achieves high accuracy in test problems.

Numerical solutions outperform some existing methods.

Abstract

This paper deals with a construction of new algorithm: the modified trigonometric cubic B-Spline differential quadrature (MTB-DQM) for space discretization together with a time integration algorithm" for numerical computation of the hyperbolic equations. Specially, MTB-DQM has been implemented for the initial value system of the telegraph equations together with both Dirichlet and Neumann type boundary conditions. The MTB-DQM is a DQM based on modified trigonometric cubic B-splines as new base functions. The problem has been reduced into an amenable system of ordinary differential equations adopting MTB-DQM. The resulting system of ordinary differential equations is solved using time integration algorithms. Further, the stability of MTB-DQM is studied by computing the eigenvalues of the coefficients matrices for various grid points, which confirmed the stability of MTB-DQM for the telegraphic equations. The accuracy of the method has been illustrated in terms of the various discrete error norms for six test problems of the telegraph equation. A comparison of computed numerical solutions with that obtained by the other methods has been carried out for various time levels considering various space sizes