Multimesh anisotropic adaptivity for the Boltzmann transport equation

This article presents a new adaptive finite element based method for the solution of the spatial dimensions of the Boltzmann transport equation. The method applies a curvature based error metric to locate the under and over resolved regions of a solution and this, in turn, is used to guide the refinement and coarsening of the spatial mesh. The error metrics and re-meshing procedures are designed such that they enable anisotropic resolution to form in the mesh should it be appropriate to do so. The adaptive mesh enables the appropriate resolution to be applied throughout the whole domain of a problem and so increase the efficiency of the solution procedure. Another new approach is also described that allows independent adaptive meshes to form for each of the energy group fluxes. The use of independent meshes can significantly improve computational efficiency when solving problems where the different group fluxes require high resolution over different regions. The mesh to mesh interpolation is made possible through the use of a ‘supermeshing’ procedure that ensures the conservation of particles when calculating the group to group scattering sources. Finally it is shown how these methods can be incorporated within a solver to resolve both fixed source and eigenvalue problems. A selection of both fixed source and eigenvalue problems are solved in order to demonstrate the capabilities of these methods.

► We solve the Boltzmann transport equation using anisotropically adaptive finite element meshes. ► The finite element mesh is resolved with minimal user input. ► Anisotropic adaptivity uses less elements than adaptive mesh refinement for the same finite element error. ► This paper also demonstrates the use of separate meshes for each energy group within the multigroup discretisation. ► The methods are applied to a range of fixed source and eigenvalue problems.