An explicit computational approach for a three-dimensional system of nonlinear elastodynamic sine-Gordon problem

TL;DR

This paper introduces a high-order explicit numerical method for solving three-dimensional nonlinear elastodynamic sine-Gordon equations, emphasizing stability, convergence, and practical efficiency through theoretical analysis and numerical validation.

Contribution

The paper develops a novel explicit computational scheme combining interpolation and finite element methods for 3D sine-Gordon problems, with proven stability and second-order temporal and third-order spatial accuracy.

Findings

The method is stable under a specific time step restriction.

Numerical results confirm second-order temporal and third-order spatial convergence.

The approach is computationally efficient and practically applicable.

Abstract

This paper proposes an explicit computational method for solving a three-dimensional system of nonlinear elastodynamic sine-Gordon equations subject to appropriate initial and boundary conditions. The time derivative is approximated by interpolation technique whereas the finite element approach is used to approximate the space derivatives. The developed numerical scheme is so-called, high-order explicit computational technique. The new algorithm efficiently treats the time derivative term and provides a suitable time step restriction for stability and convergence. Under this time step limitation, both stability and error estimates of the proposed approach are deeply analyzed using a constructed strong norm. The theoretical studies indicate that the developed approach is temporal second-order convergent and spatially third-order accurate. Some numerical examples are carried out to confirm the theory, to validate the computational efficiency and to demonstrate the practical applicability of the new computational technique.