Solving neutron transport problems with sharp boundary and interior layers using 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 the piecewise uniform Shishkin mesh for solving the S_N transport equation by resolving the sharp boundary and interior layers. The challenge is determining the layer thickness (or mesh transition point) for 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), step characteristic (SC) and modified step characteristic (mSC) methods are presented to show the improvement in the accuracy, stability, and efficiency when using the Shishkin mesh. (authors)