High Order Well-Balanced Compact Finite Difference Schemes For Shallow Water Equations

Shallow water equations(SWEs)with bottom source term have important applications in hydraulic engineering.It’s necessary to perform special treatment to the source term when conducting numerical simulation,which is due to the fact that the hyperbolic conservation laws with source terms will maintain some steady state solutions when the source term is exactly balanced by the flux gradient at steady state.Numerical methods may not satisfy the discrete version of balance exactly at the steady state,then may introduce spurous oscillations.It’s well-known that developing well-balanced schemes for balance law is useful for reducing numerical errors.Like general hyperbolic conservation law equations,no matter whether the initial conditions are smooth or not,the solution may cause discontinuous situation,such as shock and contact discontinuities.Therefore,a weighted compact nonlinear scheme(WCNS)is applied to solve pre-balanced SWEs.First,the shallow water equation is transformed into the pre-balanced form,then a highorder WCNS scheme is constructed to solve it.The advantage of this method is that no spliting of the bottom topography or the source term is needed to satisfy the well-balanced property,which greatly simplifies the difficulty of construction.This dissertation is devoted to construct well-balanced WCNS for SWEs on Cartesian grids and curvilinear grids,respectively.To validate the method,a series of numerical tests have been presented in aspects of the well-balanced property,accuracy and shock-capturing capability.The main contributions of this dissertation are divided into two parts:1.We have developed a well-balanced scheme based on WCNS for shallow water equations.To the best our knowledge,it is the first time that WCNS is applied to solve prebalanced SWEs directly.This algorithm has the property of well-balancing,of which the core thought is the surface gradient method,replacing the water depth into the water surface level,so that the general shallow water equation becomes the equivalent pre-balanced shallow water equation.For the semi-discrete system,the third order strong stability preserving(SSP)Runge-Kutta scheme is employed.At the same time,a series of numerical experiments are carried out to illustrate the well-balanced property,high-order accuracy and good shock-capturing ability of the scheme.2.A well-balanced weighted compact nonlinear scheme on curvilinear grids is also proposed for two-dimensional shallow water equations in a pre-balanced form provided that geometric conservation laws are satisfied.To minimize the numerical error,a socalled symmetrical conservative metric method is adopted to compute the transformation metrics and Jacobian,which maintains the accuracy and stability of the original scheme on curvilinear grids.And the accuracy of the algorithm,the well-balancing property and the problems on the curvilinear grid are analyzed and verified by the theoretical derivation and numerical experiments.