A New Well-Balanced Finite Volume CWENO Scheme for Shallow Water Equations over Bottom Topography (Advances in Applied Mathematics and Mechanics, 2023)
In this article, we develop a new well-balanced finite volume centralweighted essentially non-oscillatory (CWENO) scheme for one- and two-dimensionalshallow water equations over uneven bottom. The well-balanced property is ofparamount importance in practical applications, where many studied phenomena canbe regarded as small perturbations to the steady state. To achieve the well-balancedproperty, we construct numerical fluxes by means of a decomposition algorithm basedon a novel equilibrium preserving reconstruction procedure and we avoid applyingthe traditional hydrostatic reconstruction technique accordingly. This decompositionalgorithm also helps us realize a simple source term discretization. Both rigorous theoretical analysis and extensive numerical examples all verify that the proposed schememaintains the well-balanced property exactly. Furthermore, extensive numerical results strongly suggest that the resulting scheme can accurately capture small perturbations to the steady state and keep the genuine high-order accuracy for smooth solutionsat the same time.