A strong two-stage explicit/implicit approach combined with mixed finite element methods for a three-dimensional nonlinear radiation-conduction model in anisotropic media

TL;DR

This paper introduces a novel two-stage explicit/implicit mixed finite element method for simulating 3D nonlinear radiation-conduction in anisotropic media, achieving stability, second-order temporal accuracy, and fourth-order spatial convergence.

Contribution

It develops a new predictor-corrector scheme combining explicit and implicit methods with mixed finite elements for complex nonlinear heat transfer models.

Findings

The method is stable under specific time step conditions.

Numerical results confirm second-order temporal and fourth-order spatial accuracy.

The approach is computationally efficient and applicable to practical problems.

Abstract

This paper develops a strong computational approach to simulate a three-dimensional nonlinear radiation-conduction model in optically thick media, subject to suitable initial and boundary conditions. The space derivatives are approximated by the mixed finite element method ($\mathcal{P}_{p}/\mathcal{P}_{p-1}$), while the interpolation technique is employed in two stages to approximate the time derivative. The proposed strategy is so-called, a strong two-stage explicit/implicit computational technique combined with mixed finite element method. Specifically, the new algorithm should be observed as a predictor-corrector numerical scheme. Additionally, it efficiently treats the time derivative term and provides a necessary requirement on time step for stability. Under this time step limitation, the stability is deeply analyzed whereas the convergence order is numerically computed in the $L^{2}$-norm. The theoretical results suggest that the developed approach is stable and temporal second-order accurate. Some numerical experiments are performed to confirm the theory, to establish that the constructed method is spatial fourth-order convergent and to demonstrate the practical applicability and computational efficiency of the numerical scheme.