Reduced Order Models for the Angular Discretisation of the Boltzmann Transport Equation

Abstract

This chapter will present two methods of solver acceleration which aim to reduce the computational cost of converging solutions to the BTE. Both methods utilise the DPOD and RDPOD methods presented in chapters 3 and 4, respectively. The first method presented is a simple multigrid algorithm which aims to accelerate the solution of the ROMs. Since POD basis functions are hierarchical, low order basis functions generally contribute more to the overall solution than higher order basis functions. The multigrid method solves DPOD and RDPOD models on a coarse grid, then increases the number of basis functions in the model while retaining the solution coefficients of existing basis functions. This allows the low order basis functions to be resolved at a reduced computational cost, before introducing higher order basis functions which contribute relatively small corrections to the overall solution and therefore require fewer iterations to converge to a given level of accuracy. The second method presented aims to accelerate the solution of full order models. The ROMs developed in previous chapters are used to produce an initial solution to a full order model, which is then iterated further to improve the accuracy of the solution. Since the reduced order models have significantly fewer degrees of freedom, they are able to converge to a given level of accuracy more quickly than the full order model. However, the process of dimensionality reduction inevitably introduces error due to the approximations made, which reduces the maximum accuracy which the models can attain. By iterating ROM solutions in the full order model, this error can be eliminated while retaining some of the computational cost reductions provided by the ROMs. The chapter is split into two sections, the first describing the multigrid method, and the second describing the use of ROMs to accelerate full order solutions. Each section is organised as follows: First, the motivation and theory behind the method is described. Numerical results for a range of problems are then presented to demonstrate its effectiveness. Finally, conclusions are drawn on the utility of the method in question, based on the numerical results.