A consistent approach to solving the radiation diffusion equation

Approximate explicit analytical solutions of the heat radiation diffusion equation by applying the double integration technique of the integral-balance method have been developed. The method allows approximate closed form solutions to be developed. A problem with a step change of the surface temperature and two problems with time dependent boundary conditions have been solved. The error minimization of the approximate solutions has been developed straightforwardly by minimization of the residual function of the governing equation.

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Solutions of the radiation diffusion equation

High Energy Density Physics ◽

10.1016/j.hedp.2009.06.009 ◽

2010 ◽

Vol 6 (1) ◽

pp. 48-56 ◽

Cited By ~ 5

Author(s):

C.C. Smith

Keyword(s):

Diffusion Equation ◽

Radiation Diffusion

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The radiation diffusion equation: Explicit analytical solutions by improved integral-balance method

HERALD of Dagestan State University ◽

10.21779/2542-0321-2017-32-4-25-37 ◽

2017 ◽

Vol 32 (4) ◽

pp. 25-37

Author(s):

Jordan Hristov ◽

Keyword(s):

Diffusion Equation ◽

Analytical Solutions ◽

Balance Method ◽

Radiation Diffusion

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An exact solution for the linearized multifrequency radiation diffusion equation

Journal of Quantitative Spectroscopy and Radiative Transfer ◽

10.1016/j.jqsrt.2004.05.052 ◽

2005 ◽

Vol 91 (2) ◽

pp. 133-153 ◽

Cited By ~ 5

Author(s):

A.I. Shestakov ◽

J.H. Bolstad

Keyword(s):

Exact Solution ◽

Diffusion Equation ◽

Radiation Diffusion

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Weighted Interior Penalty Method with Semi-Implicit Integration Factor Method for Non-Equilibrium Radiation Diffusion Equation

Communications in Computational Physics ◽

10.4208/cicp.190612.010313a ◽

2013 ◽

Vol 14 (5) ◽

pp. 1287-1303 ◽

Cited By ~ 2

Author(s):

Rongpei Zhang ◽

Xijun Yu ◽

Jiang Zhu ◽

Abimael F. D. Loula ◽

Xia Cui

Keyword(s):

Diffusion Equation ◽

Galerkin Method ◽

Discontinuous Galerkin ◽

Discontinuous Galerkin Method ◽

Implicit Integration ◽

Interior Penalty ◽

Radiation Diffusion ◽

Factor Method ◽

Equilibrium Radiation ◽

Non Equilibrium

AbstractWeighted interior penalty discontinuous Galerkin method is developed to solve the two-dimensional non-equilibrium radiation diffusion equation on unstructured mesh. There are three weights including the arithmetic, the harmonic, and the geometric weight in the weighted discontinuous Galerkin scheme. For the time discretization, we treat the nonlinear diffusion coefficients explicitly, and apply the semi-implicit integration factor method to the nonlinear ordinary differential equations arising from discontinuous Galerkin spatial discretization. The semi-implicit integration factor method can not only avoid severe timestep limits, but also takes advantage of the local property of DG methods by which small sized nonlinear algebraic systems are solved element by element with the exact Newton iteration method. Numerical results are presented to demonstrate the validity of discontinuous Galerkin method for high nonlinear and tightly coupled radiation diffusion equation.

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Analytic solutions of the nonlinear radiation diffusion equation with an instantaneous point source in non-homogeneous media

Physics of Fluids ◽

10.1063/5.0050422 ◽

2021 ◽

Vol 33 (5) ◽

pp. 057105

Author(s):

Menahem Krief

Keyword(s):

Diffusion Equation ◽

Point Source ◽

Analytic Solutions ◽

Radiation Diffusion ◽

Nonlinear Radiation ◽

Instantaneous Point ◽

Homogeneous Media

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An efficient solving method for nonlinear convection diffusion equation

International Journal of Numerical Methods for Heat &amp Fluid Flow ◽

10.1108/hff-10-2016-0403 ◽

2018 ◽

Vol 28 (1) ◽

pp. 173-187 ◽

Cited By ~ 2

Author(s):

Xia Cui ◽

GuangWei Yuan ◽

ZhiJun Shen

Keyword(s):

Theoretical Analysis ◽

Diffusion Equation ◽

Induction Hypothesis ◽

Function Analysis ◽

Second Order ◽

Nonlinear Convection ◽

Radiation Diffusion ◽

Content Type ◽

Convection Diffusion ◽

Convection Diffusion Equation

Purpose This paper aims to provide a well-behaved nonlinear scheme and accelerating iteration for the nonlinear convection diffusion equation with fundamental properties illustrated. Design/methodology/approach A nonlinear finite difference scheme is studied with fully implicit (FI) discretization used to acquire accurate simulation. A Picard–Newton (PN) iteration with a quadratic convergent ratio is designed to realize fast solution. Theoretical analysis is performed using the discrete function analysis technique. By adopting a novel induction hypothesis reasoning technique, the L∞ (H1) convergence of the scheme is proved despite the difficulty because of the combination of conservative diffusion and convection operator. Other properties are established consequently. Furthermore, the algorithm is extended from first-order temporal accuracy to second-order temporal accuracy. Findings Theoretical analysis shows that each of the two FI schemes is stable, its solution exists uniquely and has second-order spatial and first/second-order temporal accuracy. The corresponding PN iteration has the same order of accuracy and quadratic convergent speed. Numerical tests verify the conclusions and demonstrate the high accuracy and efficiency of the algorithms. Remarkable acceleration is gained. Practical implications The numerical method provides theoretical and technical support to accelerate resolving convection diffusion, non-equilibrium radiation diffusion and radiation transport problems. Originality/value The FI schemes and iterations for the convection diffusion problem are proposed with their properties rigorously analyzed. The induction hypothesis reasoning method here differs with those for linearization schemes and is applicable to other nonlinear problems.

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