Space–time streamline upwind Petrov–Galerkin methods for the Boltzmann transport equation

• Published in: Computer methods in applied mechanics and engineering Vol. 195; no. 33-36; pp. 4334 - 4357
• Main Authors: Pain, C.C., Eaton, M.D., Smedley-Stevenson, R.P., Goddard, A.J.H., Piggott, M.D., de Oliveira, C.R.E.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.07.2006; Elsevier
• Subjects: Classical statistical mechanics; Computational techniques; Exact sciences and technology; Fluid dynamics; Fundamental areas of phenomenology (including applications); General theory; Kinetic theory; Mathematical methods in physics; Physics; Space–time Petrov–Galerkin; Statistical physics, thermodynamics, and nonlinear dynamical systems; Time-dependent radiation transport; Time-dependent radiation transport; Space–time Petrov–Galerkin; Discontinuity; Transport equation; Boltzmann equation; Step method; Steady state; Upwind scheme; Galerkin-Petrov method; Space time Petrov-Galerkin: Time-dependent radiation transport; Modelling; Transport processes; Kinetic theory; Non linear effect; Finite difference method
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2005.09.005

This paper presents a space–time streamline upwind Petrov–Galerkin formulation for the time-dependent Boltzmann transport equation. Conventional linear space–time formulations typically take the distance across space–time elements as the representative distance across which dissipation is introduced. The proposed method however, calculates the representative distances based on the gradient of the solution in space–time. Therefore, the method in its most general form is non-linear. The calculation of distances based on the solution gradient means that even if small time-steps are used the method will attempt to optimise the numerical accuracy while eliminating any spurious oscillations. Unlike conventional non-linear Petrov–Galerkin formulations the proposed class of methods introduces dissipation only in the streamline direction (in space–time). Conventional non-linear discontinuity capturing schemes add dissipation in the steepest gradient direction. In order to demonstrate the accuracy and robustness (in terms of suppressing spurious oscillations), the method is applied to a series of one-group, 2-D, fixed source, steady-state and time-dependent radiation transport problems.