Numerical simulation of the dual-phase-lag heat conduction equation on a one-dimensional unbounded domain using artificial boundary condition

TL;DR

This paper develops a high-order artificial boundary condition and a stable, second-order convergent finite difference scheme for simulating dual-phase-lag heat conduction on unbounded domains, validated by numerical experiments.

Contribution

It introduces a novel high-order artificial boundary condition combined with a stable finite difference method for dual-phase-lag heat conduction on unbounded domains.

Findings

The boundary condition effectively truncates the unbounded domain.

The numerical scheme is unconditionally stable with second-order accuracy.

Numerical results confirm the theoretical stability and convergence.

Abstract

This paper focuses on the numerical solution of a dual-phase-lag heat conduction equation on a space unbounded domain. First, based on the Laplace transform and the Pad\'e approximation, a high-order local artificial boundary condition is constructed for the considered problem, which effectively transforms the original problem into an initial-boundary value problem on a bounded computational domain. Subsequently, for the resulting reduced problem on the bounded domain equipped with high-order local artificial boundary, a stability result based on the $L^2$-norm is derived. Next, we develop finite difference method for the reduced problem by introducing auxiliary variable to reduce the order of time derivative. The numerical analysis demonstrates that the developed numerical scheme is unconditionally stable and possesses a second-order convergence rate in both space and time. Finally, numerical results are presented to validate the effectiveness of the proposed numerical method and the correctness of the theoretical analysis.