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

TL;DR

The paper addresses numerically solving the 1D dual-phase-lag heat equation on an unbounded domain by deriving high-order local artificial boundary conditions via the Laplace transform and Padé approximation, reducing the problem to a bounded domain. It establishes $L^2$-stability for the reduced problem, introduces an auxiliary variable to lower time-derivative order, and develops a Crank–Nicolson finite difference scheme with rigorous stability and second-order convergence in space and time. Two numerical examples validate the method: manufactured-solution tests confirm accuracy, and a Dirichlet-BC comparison demonstrates the ABC’s efficiency and effectiveness in confining computations to a small domain without sacrificing accuracy on regions of interest. The approach provides a practical framework for simulating DPL heat conduction on unbounded domains with reliable stability and precision.

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 Padé 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.

Figures (2)

• Figure 1: Numerical solutions at $t=14$ (left) and $t=16$ (right) for Scheme I and Scheme II with $h=\tau=1/64$.
• Figure 2: Numerical solutions at $t=18$ (left) and $t=20$ (right) for Scheme I and Scheme II with $h=\tau=1/64$.

Theorems & Definitions (10)

• Theorem 2.1
• proof
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• proof
• Theorem 3.1
• Theorem 3.2
• proof