Abstract:

Preserving well-balanced property is an important property for shallow water matrixs. The schemes with this property can capture small perturbations of steady state accurately in theory. For the shallow water matrixs with source terms, the first step is to construct an appropriate numerical dissipative operator and select a special discretization of source terms to accurately balance the non-zero flux and source terms, which is to obtain a class of high order well-balanced entropy stable schemes to keep balance. The new idea is to put forward the well-balanced preserving theorems for the high-order entropy conservative schemes and for the high-order entropy stable schemes. The detail proof process is also clearly given. Finally, several typical numerical examples show that new schemes can well deal with the small perturbation problems of steady-state solutions.

Key words: Well-balanced, Steady-state problems with small perturbations, Entropy stable schemes, Shallow water matrixs