Solving neutron transport problems with sharp layers on the Shishkin mesh

If there exist sharp boundary layers and/or interior layers in the solution of neutron transport problems, numerical instability and oscillations may occur when the mesh is not fine enough around the sharp layers. In this work, we propose to solve the S N neutron transport equation by resolving the sharp boundary and interior layers with the piecewise uniform Shishkin mesh. The challenge is to determine the layer thickness for the transition point of the Shishkin mesh. We derive a mathematical expression for calculating the layer thickness based on the eigenvalues of the S N transport equation. Numerical results for the diamond difference (DD) and discontinuous Galerkin (DG) methods, are presented to show the improvement in the accuracy, stability, and efficiency with the Shishkin mesh.