A review of Space-time conservation element solution element (CESE) schemes

The space-time conservation element and solution element (CESE) method is one of the recent advancements in CFD. This paper attempts to review various schemes along with their limitations starting with a non-dissipative a -scheme. The following a- e scheme has an added numerical dissipation term. This numerical dissipation is capable of damping out numerical instabilities that arise only from the smooth region of the solution but fails to suppress numerical wiggles. The a-e-α-β scheme is augmented with the ability to suppress the wiggles introduced due to discontinuities, using another added term to the solution. Hence a-e-α-β scheme is capable of capturing both small disturbances and sharp discontinuities simultaneously. Its advantage over a-e scheme as well as the suppression of numerical wiggle at the discontinuity will be demonstrated using Sod's shock tube problem. All of these CESE schemes described so far become excessively diffusive at low Courant number and Mach number. Hence the construction of a Courant number insensitive scheme which has a weighted average sense. Its advantage over a-e-α-β in terms of stability will be explained further in the paper using Sod's shock tube problem as an example and its ability to resolve contact discontinuities without any ad-hoc parameter over CFL number ranging from one to close to 0.001