The CPR Method In Treating Discontinuities

Proposing a high order method which is numerically stable and can handle discontinuities is an important researching topic in computational fluid dynamics(CFD).The newly proposed correction procedure via reconstruction(CPR)method is a high order scheme which is also stable for the smooth problem.However,CPR is merely a high order linear scheme which does not have the property to treat the discontinuity.Therefore it is a necessity to make a modification of CPR to solve the discontinuous problems,thereby making it more applicable in practice.We introduce two approaches,adding limiters and using hybrid scheme.Several numerical experiments are tested to verify the feasibility.The main works are as follows:1.We investigate the mathematical formulation of the CPR method and the relation between CPR and DG method;2.The TVB limiter designed for DG method is introduced and used in the CPR method to make the discontinuous problem trackable.Then we propose another limiter to detect the discontinuity more accurately.Numerical experiments are implemented to compare these two limiters;3.A hybrid CPR/WENO scheme is proposed.The main idea is to calculate the equa-tion using CPR method firstly,and then finding out the troubled cells and recalculate it by the WENO method.Through numerical tests this hybrid method is verified to be effective and can tackle discontinuities well.