High Order Discontinuous Galerkin Method For Shallow Water Equations

The nonlinear shallow water equations have been successfully applied to flows in rivers and coastal areas,such as open channel flows,and tidal flows in coastal water region,among others.Therefore,various numerical methods have been developed for solving these equations,such as the finite difference method,the finite volume method and the discontinuous Galerkin method.In this thesis,we focus on the development of high order discontinuous Galerkin methods for solving the nonlinear shallow water equations with friction term over variable bottom topography in one and two dimensions.There are two issues when we numerically solve the nonlinear shallow water equations with friction term over variable bottom topography.The first one is that the equations have still water stationary solutions,for which the flux is not zero but balanced by the source.However,many numerical schemes cannot maintain the stationary solutions and thus numerical spurious oscillations may be produced during the numerical simulation.The second issue is related to the non-negativity of the water depth.When considering a problem with dry or near dry areas,the numerical may produce negative water depth,and thus the computation may blow up.Therefore,well-balancing and positivity-preserving properties are desired in the reliable numerical schemes for solving the nonlinear shallow water equations.To achieve the well-balanced property of the numerical scheme easily,the nonlinear shallow water equations are first reformulated into a new form by introducing an auxiliary variable,which maintains the same eigenvalues and eigenvectors of the original equations.By choosing the values of the auxiliary variable suitably,we can prove that the scheme can exactly preserve the still-water solution,and thus it is a truly well-balanced scheme.To ensure the non-negativity of the water depth,a positivity-preserving limiter and a continuous approximation to the bottom topography are employed and we prove the positivity-preserving property of the numerical scheme under some conditions.Finally,some numerical tests are used to certificate the validity and accuracy of the numerical method.