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

An implicit Riemann solver for the one- and two-dimensional time-dependent spherical harmonics approximation (P_n) 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.