Fast and accurate discrete ordinates methods for multidimensional radiative transfer. Part I, basic methods

Balsara, Dinshaw S.

doi:10.1016/s0022-4073(00)00114-x

Cited by 95 publications

(46 citation statements)

References 63 publications

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“…The work of Jameson [28] and Gropp et al [25] has already demonstrated the importance of multilevel processing for the Euler equations. The work of Balsara [6] has already demonstrated the importance of multigrid methods for the equations of radiative transfer. Thus the importance of multigrid processing for the MHD equations seems almost evident.…”

Section: ) Just As Berger and Colellamentioning

confidence: 99%

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Divergence-Free Adaptive Mesh Refinement for Magnetohydrodynamics

Balsara

2001

Journal of Computational Physics

250

329

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Section: ) Just As Berger and Colellamentioning

confidence: 99%

“…For that reason we would discourage the use of such solvers for AMR-MHD applications. 6 While eqn. (1.2) implies divergence-free evolution of magnetic fields, the equations of MHD are indeed hyperbolic equations and can be viewed on a dimension by dimension basis as having a conservative form.…”

Section: I) Introductionmentioning

confidence: 99%

Divergence-Free Adaptive Mesh Refinement for Magnetohydrodynamics

Balsara

2001

Journal of Computational Physics

250

329

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“…Their use for the solution of non-linear hyperbolic problems is currently extremely limited. Among the very few astrophysical studies that have adopted multigrid for hyperbolic problems, are those of Fabiani Bendicho et al (1997) and Balsara (2001), who have both focused on the stationary radiation-transport equation (which is a single scalar equation). The purpose of our paper is to present the construction of a family of FAS multigrid cycling schemes for a hyperbolic system, namely the Euler equations, and to evaluate their convergence and stability characteristics.…”

Section: A47 Page 2 Of 30mentioning

confidence: 99%

On multigrid solution of the implicit equations of hydrodynamics

Kifonidis

Müller

2012

A&A

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Aims. We describe and study a family of new multigrid iterative solvers for the multidimensional, implicitly discretized equations of hydrodynamics. Schemes of this class are free of the Courant-Friedrichs-Lewy condition. They are intended for simulations in which widely differing wave propagation timescales are present. A preferred solver in this class is identified. Applications to some simple stiff test problems that are governed by the compressible Euler equations, are presented to evaluate the convergence behavior, and the stability properties of this solver. Algorithmic areas are determined where further work is required to make the method sufficiently efficient and robust for future application to difficult astrophysical flow problems. Methods. The basic equations are formulated and discretized on non-orthogonal, structured curvilinear meshes. Roe's approximate Riemann solver and a second-order accurate reconstruction scheme are used for spatial discretization. Implicit Runge-Kutta (ESDIRK) schemes are employed for temporal discretization. The resulting discrete equations are solved with a full-coarsening, nonlinear multigrid method. Smoothing is performed with multistage-implicit smoothers. These are applied here to the time-dependent equations by means of dual time stepping. Results. For steady-state problems, our results show that the efficiency of the present approach is comparable to the best implicit solvers for conservative discretizations of the compressible Euler equations that can be found in the literature. The use of red-black as opposed to symmetric Gauss-Seidel iteration in the multistage-smoother is found to have only a minor impact on multigrid convergence. This should enable scalable parallelization without having to seriously compromise the method's algorithmic efficiency. For time-dependent test problems, our results reveal that the multigrid convergence rate degrades with increasing Courant numbers (i.e. time step sizes). Beyond a Courant number of nine thousand, even complete multigrid breakdown is observed. Local Fourier analysis indicates that the degradation of the convergence rate is associated with the coarse-grid correction algorithm. An implicit scheme for the Euler equations that makes use of the present method was, nevertheless, able to outperform a standard explicit scheme on a time-dependent problem with a Courant number of order 1000. Conclusions. For steady-state problems, the described approach enables the construction of parallelizable, efficient, and robust implicit hydrodynamics solvers. The applicability of the method to time-dependent problems is presently restricted to cases with moderately high Courant numbers. This is due to an insufficient coarse-grid correction of the employed multigrid algorithm for large time steps. Further research will be required to help us to understand and overcome the observed multigrid convergence difficulties for time-dependent problems.

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“…Although various quadrature sets for the DOM have been proposed [3,6,1], the most common choice of the sets is the level symmetric even (LSE) set [6], which is determined to satisfy the symmetry to all directions over the unit sphere. It has been shown that numerical calculation of the RTE using the LSE set is accurate especially for isotropic scattering as in the case of neutron transport.…”

Section: Introductionmentioning

confidence: 99%

Renormalization of the highly forward-peaked phase function using the double exponential formula for radiative transfer

et al. 2016

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Numerical calculation of photon migration in biological tissue using the radiative transfer equation (RTE) has attracted great interests in biomedical optics and imaging. Because biological tissue is a highly forward-peaked scattering medium, renormalization of the phase function in numerical calculation of the RTE is crucial. This paper proposes a simple approach of renormalizing the phase function by the double exponential formula, which was heuristically modified from the original one. Firstly, the validity of the proposed approach was tested by comparing numerical results for an average cosine of the polar scattering angle calculated by the proposed approach with those by the conventional approach in highly forward-peaked scattering. The results show that calculation of the average cosine converged faster using the proposed approach than using the conventional one as a total number of discrete angular directions increases. Next, the accuracy of the numerical solutions of the RTE using the proposed approach was examined by comparing the numerical solutions with the analytical solutions of the RTE in a homogeneous medium with highly forward-peaked scattering. It was found that the proposed approach reduced the errors of the numerical solutions from those using the conventional one especially at a small value of the total number of the directions. This result suggests that the proposed approach has a possibility to

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Fast and accurate discrete ordinates methods for multidimensional radiative transfer. Part I, basic methods

Cited by 95 publications

References 63 publications

Divergence-Free Adaptive Mesh Refinement for Magnetohydrodynamics

Divergence-Free Adaptive Mesh Refinement for Magnetohydrodynamics

On multigrid solution of the implicit equations of hydrodynamics

Renormalization of the highly forward-peaked phase function using the double exponential formula for radiative transfer

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