A Quasilinear Implicit Riemann Solver for the Time-Dependent Pn Equations

Abstract An implicit Riemann solver for the one- and two-dimensional time-dependent spherical harmonics approximation (Pn) to the linear transport equation is presented. This spatial discretization scheme is based on cell-averaged quantities and uses a monotonicity-preserving high resolution method to achieve second-order accuracy (away from extreme points in the solution). Such a spatial scheme requires a nonlinear method of reconstructing the slope within a spatial cell. We have devised a means of creating an implicit (in time) method without the necessity of a nonlinear solver. This is done by computing a time step using a first-order scheme and then, based on that solution, reconstructing the slope in each cell, an implementation that we justify by analyzing the model equation for the method. This quasilinear approach produces smaller errors in less time than both a first-order scheme and a method that solves the full nonlinear system using a Newton-Krylov method.

[1]  T. Brunner,et al.  Riemann solvers for time-dependent transport based on the maximum entropy and spherical harmonics closures , 2000 .

[2]  P. Holloway,et al.  An implicit riemann solver for the time-dependent Pn equations. , 2005 .

[3]  C. Kelley Iterative Methods for Linear and Nonlinear Equations , 1987 .

[4]  Ryan G. McClarren,et al.  Implicit Riemann solvers for the Pn equations , 2006 .

[5]  Bingjing Su,et al.  Variable Eddington Factors and Flux Limiters in Radiative Transfer , 2001 .

[6]  A. Goddard,et al.  A HIGH-ORDER RIEMANN METHOD FOR THE BOLTZMANN TRANSPORT EQUATION , 2003 .

[7]  B. Su,et al.  A nonlinear diffusion theory for particle transport in strong absorbers , 2002 .

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

[9]  Thomas M. Evans,et al.  Comparison of four parallel algorithms for domain decomposed implicit Monte Carlo , 2006, J. Comput. Phys..

[10]  R. F. Warming,et al.  The modified equation approach to the stability and accuracy analysis of finite-difference methods , 1974 .

[11]  James Paul Holloway,et al.  One-dimensional Riemann solvers and the maximum entropy closure , 2001 .

[12]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme , 1974 .

[13]  N. A. Gentile,et al.  Obtaining identical results on varying numbers of processors in domain decomposed particle Monte Carlo simulations. , 2006 .

[14]  R. Chaplin NUCLEAR REACTOR THEORY , 2022 .

[15]  Finite Element Based Riemann Solvers for Time-Dependent and Steady-State Radiation Transport , 2003 .

[16]  R. Courant,et al.  Über die partiellen Differenzengleichungen der mathematischen Physik , 1928 .