A steady-state, single phase subchannel solver was created for the purpose of integration into a multi-physics nuclear fuel performance code. Since applications of such a code include full nuclear reactor core flow simulation, a thorough investigation of efficient solution techniques is a requirement. Execution time profiling found that formation of the Jacobian matrix required by the nonlinear Newton solve was found to be the most time-consuming step in solution of the subchannel equations, so several techniques were tested to minimize the time spent on this task, such as finite difference and the formation of an approximate Jacobian. Simple Jacobian lagging was shown to be very effective at reducing the total time computing the Jacobian throughout the Newton iteration process. Various linear solution techniques were investigated with the subchannel equations, such as the generalized minimal residual method (GMRES) and the aggregation- based algebraic multigrid method (AGMG). A number of physics-based preconditioners were created, based on a simplified formulation with no crossflow between subchannels, and it was found that of the preconditioners developed for this research, the most promising was a preconditioner that fully decoupled the subchannels by ignoring crossflow, conduction, and turbulent momentum exchange between subchannels. This independence between subchannels makes the task of parallelization in the preconditioner to be very feasible. However, AGMG clearly proved to be the most efficient linear solution technique for the subchannel equations, solving the linear systems in less than 5 percent of the time required for preconditioned GMRES.

This thesis discusses properties arising when finite differences are implemented forsolving the two dimensional wave equation on media with various properties. Both homogeneous and heterogeneous surfaces are considered. The time derivative of the wave equation is discretised using a weighted central difference scheme, dependenton a variable parameter gamma. Stability and convergence properties are studied forsome different values of gamma. The report furthermore features an introduction to solving large sparse linear systems of equations, using so-called multigrid methods.The linear systems emerge from the finite difference discretisation scheme. Aconclusion is drawn stating that values of gamma in the unconditionally stable region provides the best computational efficiency. This holds true as the multigrid based numerical solver exhibits optimal or near optimal scaling properties.