A High-Order Finite Volume GENO Scheme with Implicit Time Integration for Three-Temperature Radiation Diffusion Equations

Abstract

This study presents a high-order finite volume scheme capable of large time-step integration for three-temperature radiation diffusion (3TRD) equations, where conservation is naturally achieved through energy update. To handle local large gradients and discontinuities in temperature, a central generalized ENO (GENO) reconstruction is developed for diffusion systems, which achieves essentially non-oscillatory reconstruction for discontinuous solutions. Compared to conventional nonlinear reconstruction methods, its most distinctive feature is the central-type symmetric sub-stencils, which ensure consistency between the numerics and the isotropic nature of thermal diffusion. Additionally, the central GENO method provides smooth states of temperature and temperature gradient at interfaces, facilitating the evaluation of numerical fluxes. Furthermore, interface flux evaluation for cases with discontinuous physical property parameters is modeled. To address the extremely small time-step issue caused by stiff diffusion and source terms, a dual-time-stepping method based on implicit time discretization is developed for the first time in 3TRD systems, with the advantage of decoupling temporal discretization from complex nonlinear spatial discretization. A series of numerical examples validates the high accuracy, physical property preservation, strong robustness, and large time-step integration capability of the present high-order central GENO scheme.

Figures (9)

• Figure 1: Schematic of high-order reconstruction on 3D structured grids. The stencil cells used in the 1D central GENO reconstruction for the "1D + 2D" reconstruction strategy. The face with blue edges denotes the target interface to be reconstructed.
• Figure 2: Schematic of high-order reconstruction on 3D structured grids. 2D reconstruction stencil for the "1D + 2D" stage-by-stage strategy. Green dots denote Gaussian quadrature points on interfaces for flux evaluation.
• Figure 3: Accuracy test: Convergence of the $L_1$ and $L_\infty$ error norms for the electron ($T_e$), ion ($T_i$), and radiation ($T_r$) temperatures under mesh refinement.
• Figure 4: 2D model problem for verifying the bound-preserving property: Radiation temperature distributions along $x=0.15$ at $t=0.5$ computed by the 4th-order GENO and 4th-order linear schemes, with a close-up view across the material interface shown on the right.
• Figure 5: 2D model problem for evaluating high accuracy performance: Contours of $T_r$ at $t=5$ computed by the 4th-order GENO scheme (left) and the linear 2nd-order central scheme (right).