An Up-wind Discontinuous Galerkin Method For Current And Silt Sediment In Two-dimensions Space

Current and slit sediment model of Yellow River estuary plays an importantrole for quantitative prediction of the river bed evolution, the sediment depositionin estuary, the formation and development of the delta and development of coastalsediments due to the movement of water and sedimentation. The intensive study ofthis problem will help us to understand the mechanism and morphology of sedimenttransport clearly. It will be of great significance for project approachment such asthe treatment of the estuary silting, the delay of the bank erosion and the controlof the seawater invasion. In recent years, people have paid much attention to itssolution, and established the mathematical model for current and slit sedimentation.Finite diference method, finite element method and finite volume element methodare used numerically simulate the mathematical model. However the numericalanalysis theory is forming.In this paper, based on conservation law and the real background of YellowRiver estuary, we are devoted to derive the mathematical model for the sedimentdeposit, in which a first-order hyperbolic equation is used to describe flow continuity,convection-dominated parabolic equations are used to describe flow movement andsilt continuity, and a first-order equation is used to describe bed deformation. Initialconditions and boundary conditions are prescribed to close the system.The standard finite diference method, finite element method and finite vol- ume element method do not reflect the conservation property and the convection-dominated characteristics, so they do not achieve ideal approximation. Hence toconstruct high performance numerical scheme is a hot-spot in engineering and com-putational mathematics.By borrowing high merits of discontinuous Galerkin method and up-wind tech-nique we attempt to construct a high performance up-wind discontinuous Galerkinmethod, which preserves local conservation property and avoids excessive amountof nonphysical dispersion and numerical difusion near the fronts.Numerical analysis done in this paper shows that the up-wind discontinuousGalerkin method realizes our expectations. In fact, by using L2projection andinduction hypothesis technique to prove that the discrete solution of water depth ispositive, we obtain the existence and uniqueness of the discrete solution and subop-timal error estimate in L2norm. Numerical experiment confirms these theoreticsfindings in this paper.