StaRMAP---A Second Order Staggered Grid Method for Spherical Harmonics Moment Equations of Radiative Transfer

We present a simple method to solve spherical harmonics moment systems, such as the the time-dependent PN and SPN equations, of radiative transfer. The method, which works for arbitrary moment order N, makes use of the specific coupling between the moments in the PN equations. This coupling naturally induces staggered grids in space and time, which in turn give rise to a canonical, second-order accurate finite difference scheme. While the scheme does not possess TVD or realizability limiters, its simplicity allows for a very efficient implementation in Matlab. We present several test cases, some of which demonstrate that the code solves problems with ten million degrees of freedom in space, angle, and time within a few seconds. The code for the numerical scheme, called StaRMAP (Staggered grid Radiation Moment Approximation), along with files for all presented test cases, can be downloaded so that all results can be reproduced by the reader.

[1]  John K. Reid,et al.  The Multifrontal Solution of Indefinite Sparse Symmetric Linear , 1983, TOMS.

[2]  Shi Jin.  ASYMPTOTIC PRESERVING (AP) SCHEMES FOR MULTISCALE KINETIC AND HYPERBOLIC EQUATIONS: A REVIEW , 2010 .

[3]  R. E. Marshak,et al.  Note on the Spherical Harmonic Method As Applied to the Milne Problem for a Sphere , 1947 .

[4]  Martin Frank,et al.  Diffusive Corrections to PN Approximations , 2009, Multiscale Model. Simul..

[5]  James Paul Holloway,et al.  On solutions to the Pn equations for thermal radiative transfer , 2008, J. Comput. Phys..

[6]  B. Galapol,et al.  HOMOGENEOUS INFINITE MEDIA TIME-DEPENDENT ANALYTICAL BENCHMARKS , 2001 .

[7]  Ryan G. McClarren,et al.  Theoretical Aspects of the Simplified Pn Equations , 2010 .

[8]  Ryan G. McClarren,et al.  Positive P N Closures. , 2010 .

[9]  Roger Grimes,et al.  The influence of relaxed supernode partitions on the multifrontal method , 1989, TOMS.

[10]  Julien Langou,et al.  A Class of Parallel Tiled Linear Algebra Algorithms for Multicore Architectures , 2007, Parallel Comput..

[11]  SeiboldBenjamin,et al.  StaRMAP---A Second Order Staggered Grid Method for Spherical Harmonics Moment Equations of Radiative Transfer , 2014 .

[12]  Ryan G. McClarren,et al.  Semi-implicit time integration for PN thermal radiative transfer , 2008, J. Comput. Phys..

[13]  G. C. Pomraning,et al.  Linear Transport Theory , 1967 .

[14]  L. C. Henyey,et al.  Diffuse radiation in the Galaxy , 1940 .

[15]  J. Reid,et al.  Monitoring the stability of the triangular factorization of a sparse matrix , 1974 .

[16]  I. Bohachevsky,et al.  Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics , 1959 .

[17]  Randall J. LeVeque,et al.  Python Tools for Reproducible Research on Hyperbolic Problems , 2009, Computing in Science & Engineering.

[18]  G. Strang.  On the Construction and Comparison of Difference Schemes , 1968 .

[19]  Ryan G. McClarren,et al.  Robust and accurate filtered spherical harmonics expansions for radiative transfer , 2010, J. Comput. Phys..

[20]  James Paul Holloway,et al.  Two-dimensional time dependent Riemann solvers for neutron transport , 2005 .

[21]  Ryan G. McClarren,et al.  Positive PN Closures , 2010, SIAM J. Sci. Comput..

[22]  Gordon L. Olson,et al.  Second-order time evolution of PN equations for radiation transport , 2009, J. Comput. Phys..

[23]  Benjamin Seibold,et al.  Optimal prediction for radiative transfer: A new perspective on moment closure , 2008, 0806.4707.

[24]  Jim E. Morel,et al.  Analysis of Ray-Effect Mitigation Techniques , 2003 .

[25]  Benjamin Seibold,et al.  Asymptotic derivation and numerical investigation of time-dependent simplified PN equations , 2013, J. Comput. Phys..

[26]  Patrick Knupp,et al.  Code Verification by the Method of Manufactured Solutions , 2000 .

[27]  Edward W. Larsen,et al.  The P N Theory as an Asymptotic Limit of Transport Theory in Planar Geometry—I: Analysis , 1991 .

[28]  Shi Jin,et al.  Efficient Asymptotic-Preserving (AP) Schemes For Some Multiscale Kinetic Equations , 1999, SIAM J. Sci. Comput..