Numerical solution of the heat conduction problem with memory

TL;DR

This paper develops a stable numerical method for solving heat conduction problems with memory effects, transforming complex nonlocal models into simpler local systems, enabling efficient simulations of thermal processes with memory.

Contribution

It introduces a stable two-level scheme for first-order evolution equations modeling heat conduction with memory, simplifying nonlocal problems into local ones under certain conditions.

Findings

Proves unconditional stability of the proposed numerical scheme.

Transforms nonlocal integrodifferential equations into local first-order systems.

Demonstrates numerical solution for a 1D heat conduction problem with memory.

Abstract

It is necessary to use more general models than the classical Fourier heat conduction law to describe small-scale thermal conductivity processes. The effects of heat flow memory and heat capacity memory (internal energy) in solids are considered in first-order integrodifferential evolutionary equations with difference-type kernels. The main difficulties in applying such nonlocal in-time mathematical models are associated with the need to work with a solution throughout the entire history of the process. The paper develops an approach to transforming a nonlocal problem into a computationally simpler local problem for a system of first-order evolution equations. Such a transition is applicable for heat conduction problems with memory if the relaxation functions of the heat flux and heat capacity are represented as a sum of exponentials. The correctness of the auxiliary linear problem is ensured by the obtained estimates of the stability of the solution concerning the initial data and the right-hand side in the corresponding Hilbert spaces. The study's main result is to prove the unconditional stability of the proposed two-level scheme with weights for the evolutionary system of equations for modeling heat conduction in solid media with memory. In this case, finding an approximate solution on a new level in time is not more complicated than the classical heat equation. The numerical solution of a model one-dimensional in space heat conduction problem with memory effects is presented.