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A two-stage explicit/implicit approach combined with mixed finite element methods for a radiation-conduction model in optically thick anisotropic media

Eric Ngondiep

TL;DR

This paper introduces a three-dimensional nonlinear SP3 radiation–conduction model for optically thick, anisotropic media and develops a two-stage explicit/implicit predictor–corrector algorithm built on a mixed finite element discretization ($\mathcal{P}_{p}/\mathcal{P}_{p-1}/\mathcal{P}_{p-1}$). The method decouples time derivatives via a two-stage interpolation to yield a stable, high-accuracy scheme: a predictor step with explicit treatment and a corrector step with implicit treatment, analyzed under a time-step restriction. Theoretical results indicate spatial fourth-order and temporal second-order convergence in $L^2$-norm, complemented by numerical experiments across 3D examples that confirm stability and efficiency relative to existing approaches. These findings suggest the approach is well-suited for fast, accurate simulations of radiative-conduction processes in anisotropic media.

Abstract

This paper develops a two-stage explicit/impicit computational technique combined with a mixed finite element method for solving a nonlinear radiation-conduction problem in anisotropic media, subject to suitable initial and boundary conditions. The space derivatives are approximated by the mixed finite element method ($\mathcal{P}_{p}/\mathcal{P}_{p-1}/\mathcal{P}_{p-1}$), while the interpolation technique is employed in two stages to approximate the time derivative. The proposed strategy is so-called, a 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 obtained in the $L^{2}$-norm. The theoretical results suggest that the developed approach is spatial fourth-order convergent and temporal second-order accurate. Some numerical experiments are carried out to confirm the theoretical results and to demonstrate the practical applicability of the new algorithm.

A two-stage explicit/implicit approach combined with mixed finite element methods for a radiation-conduction model in optically thick anisotropic media

TL;DR

This paper introduces a three-dimensional nonlinear SP3 radiation–conduction model for optically thick, anisotropic media and develops a two-stage explicit/implicit predictor–corrector algorithm built on a mixed finite element discretization (). The method decouples time derivatives via a two-stage interpolation to yield a stable, high-accuracy scheme: a predictor step with explicit treatment and a corrector step with implicit treatment, analyzed under a time-step restriction. Theoretical results indicate spatial fourth-order and temporal second-order convergence in -norm, complemented by numerical experiments across 3D examples that confirm stability and efficiency relative to existing approaches. These findings suggest the approach is well-suited for fast, accurate simulations of radiative-conduction processes in anisotropic media.

Abstract

This paper develops a two-stage explicit/impicit computational technique combined with a mixed finite element method for solving a nonlinear radiation-conduction problem in anisotropic media, subject to suitable initial and boundary conditions. The space derivatives are approximated by the mixed finite element method (), while the interpolation technique is employed in two stages to approximate the time derivative. The proposed strategy is so-called, a 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 obtained in the -norm. The theoretical results suggest that the developed approach is spatial fourth-order convergent and temporal second-order accurate. Some numerical experiments are carried out to confirm the theoretical results and to demonstrate the practical applicability of the new algorithm.

Paper Structure

This paper contains 6 sections, 2 theorems, 97 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Mathematical formulation of $SP_{3}$ approximations of nonlinear radiation-conduction equations
  3. Detailed description of the new computational technique
  4. Stability analysis of the developed computational approach
  5. Numerical experiments
  6. General conclusions and future works

Key Result

Lemma 4.1

For any $w,\rho\in W_{2}^{1}(\Omega)$, it holds for every positive constants $\epsilon$ and $\epsilon_{1}$.

Figures (3)

  • Figure 1: Graphs of exact and computed solutions for Temperature $(T\text{\,and\,}T_{h})$, related approximate radiation intensity $(\phi_{1h}\text{\,and\,}\phi_{2h})$ and errors $(E_{T}\text{\,and\,}E_{\phi_{j}})$, associated with Example 1.
  • Figure 2: Graphs of analytical and approximate solutions for Temperature $(T\text{\,and\,}T_{h})$, related approximate radiation intensity $(\phi_{1h}\text{\,and\,}\phi_{2h})$ and errors $(E_{T}\text{\,and\,}E_{\phi_{j}})$, corresponding to Example 2.
  • Figure 3: Graphs of analytical and numerical solutions for Temperature $(T\text{\,and\,}T_{h})$, related approximate radiation intensity $(\phi_{1h}\text{\,and\,}\phi_{2h})$ and errors $(E_{T}\text{\,and\,}E_{\phi_{j}})$, corresponding to Example 3.

Theorems & Definitions (4)

  • Lemma 4.1
  • proof
  • Theorem 4.1
  • proof