Toward Asymptotic Diffusion Limit Preserving High-Order, Low-Order Method

Abstract

Recent development of the high-order, low-order (HOLO) method has shown promising results for solving thermal radiative transfer problems. The HOLO algorithm is a moment-based acceleration, similar to the well-known nonlinear diffusion acceleration and coarse-mesh finite difference methods. In this work, we introduce a new spatial-differencing scheme for the low-order (LO) system based on the corner-balance method and analyze an asymptotic diffusion property for a one-dimensional gray equation. An asymptotic analysis indicates that the new spatial-differencing scheme possesses the equilibrium diffusion limit. Numerical examples demonstrate significant improvements in the solution accuracy compared to the LO finite-volume discretization with a discontinuous source reconstruction.

Keywords:

• High-order
• low-order method
• asymptotic analysis
• corner balance
• radiation hydrodynamics

Notes

^a Equation (12) becomes linear when the right side is fixed. We also assume the HO and LO systems have the same initial conditions.

^b A discretely consistent system gives the same reaction rate. Thus, integral quantities can be directly computed using the LO system. Hence, the LO system can be employed for multiphysics coupling.

^c Note that because the consistency terms γ^± are defined in terms of the HO variables, we keep the right side of Eqs. (46)(46) $\frac{{ϵ}^2}{c}\frac{\mathrm{\partial }{F}_{LO,i-\frac{1}{2}}}{\mathrm{\partial }t}+\frac{cϵ}{3}\frac{E_{LO,i}^L-E_{LO,i-1}^R}{\left(\mathrm{\Delta }{x}_{i-\frac{1}{2}}/2\right)}+{\sigma }_{i-\frac{1}{2}}{F}_{LO,i-\frac{1}{2}}={\gamma }_{HO,i-\frac{1}{2}}^{+}c{E}_{LO,i-1}^R-{\gamma }_{HO,i-\frac{1}{2}}^{-}c{E}_{LO,i}^L,$(46) and (47)(47) $\frac{{ϵ}^2}{c}\frac{\mathrm{\partial }{F}_{LO,i}}{\mathrm{\partial }t}+\frac{cϵ}{3}\frac{E_{LO,i}^R-E_{LO,i}^L}{\left(\mathrm{\Delta }{x}_i/2\right)}+{\sigma }_i{F}_{LO,i}={\gamma }_{HO,i}^{+}c{E}_{LO,i}^L-{\gamma }_{HO,i}^{-}c{E}_{LO,i}^R,$(47) as $O({ϵ}^0)$. Using $O({ϵ}^1)$-scaling in the right side of Eqs. (46)(46) $\frac{{ϵ}^2}{c}\frac{\mathrm{\partial }{F}_{LO,i-\frac{1}{2}}}{\mathrm{\partial }t}+\frac{cϵ}{3}\frac{E_{LO,i}^L-E_{LO,i-1}^R}{\left(\mathrm{\Delta }{x}_{i-\frac{1}{2}}/2\right)}+{\sigma }_{i-\frac{1}{2}}{F}_{LO,i-\frac{1}{2}}={\gamma }_{HO,i-\frac{1}{2}}^{+}c{E}_{LO,i-1}^R-{\gamma }_{HO,i-\frac{1}{2}}^{-}c{E}_{LO,i}^L,$(46) and (47)(47) $\frac{{ϵ}^2}{c}\frac{\mathrm{\partial }{F}_{LO,i}}{\mathrm{\partial }t}+\frac{cϵ}{3}\frac{E_{LO,i}^R-E_{LO,i}^L}{\left(\mathrm{\Delta }{x}_i/2\right)}+{\sigma }_i{F}_{LO,i}={\gamma }_{HO,i}^{+}c{E}_{LO,i}^L-{\gamma }_{HO,i}^{-}c{E}_{LO,i}^R,$(47) yields the identical result.

^d We assume that the full- and half-range discrete angular integrations closely approximate the analytic integrals.