High-precision Harmony Algorithm With Source Term Hyperbolic Conservation Law Equation

The hyperbolic conservation laws equations with source terms are important models which describe the fluid motions.One main difficulty in solving them is that whether their initial conditions are smooth or not,their solutions may appear discontinuous problems,such as shock wave,vortex and contact discontinuities.The low order schemes are easy to level off when dealing with such problems,so they can not accurately simulate the states in the discontinuities.Therefore it becomes a research focus to design high order schemes to solve such problems.Among all the high order schemes,the finite difference WENO schemes and the discontinuous Galerkin schemes are commonly used to deal with the convection-diffusion equations,which are especially suitable for numerical simulation of the hyperbolic conservation laws equations with source terms.The advantages of these two high order schemes consist that they can achieve arbitrary high order precision in smooth region,maintain numerical stability in the discontinuous area,and effectively deal with various steep problems.Another main difficulty in solving the hyperbolic conservation laws equations with source terms is the treatment of source terms,which need to be balanced by the flux gradient at the steady state.Standard numerical methods may not satisfy the discrete version of this balance exactly at(or near)the steady state,and may introduce spurious oscillations,unless the mesh size is extremely refined.But this numerical procedure of mesh refinement is not applicable for the high dimensional cases due to too high computational cost.In order to save the computational cost,well-balanced schemes are designed to preserve exactly these steady state solutions up to the machine accuracy,and have been active research areas in the past two decades.The main contents of this dissertation can be summarized as follows:1.We present a high order well-balanced finite difference WENO scheme for the blood flow equation with a geometrical source term,which maintains the still blood steady state.In order to maintain the well-balanced property,we propose to reformulate the blood flow model in order that the source term has the same structure as the flux at the still blood steady state.Moreover,we apply a novel source term approximation as well as well-balanced numerical fluxes.We prove that the proposed scheme for the blood flow equation can maintain well-balanced property.Extensive numerical experiments are preformed to verify the performances of the proposed scheme such as the maintenance of well-balanced property,the genuine high order accuracy for smooth solutions,the ability to capture the perturbations of still blood steady state,and effective simulations of wave equation,propagation of a pulse and viscous damping problems.2.We present high order DG scheme for the shallow water flows through channels with irregular geometry,which maintains the still water steady state.We reformulate the shallow water equation through channels by the still steady state,and propose to construct a novel source term approximation as well as well-balanced numerical fluxes.In addition,we design a simple positivity-preserving limiter,which can ensure the resulting methods maintain the non-negativity of the cross sectional wet area.We have carried out extensive numerical simulations,which demonstrate that the proposed methods are well-balanced,efficient for the small perturbation test near the steady state solutions,positivity-preserving near the wetting and drying front,high order accurate and also good for both continuous and discontinuous solutions.3.We present high order DG scheme for the shallow water equation under temperature fields with a geometrical source term,which maintains the still water steady state.For one-dimensional case,we reformulate the shallow water equation under temperature by the still steady state,and propose to construct a novel source term approximation as well as well-balanced numerical fluxes.In addition,we prove that the proposed scheme for the one-dimensional shallow water equation under temperature can maintain well-balanced property.Then we generalized the one-dimensional scheme to the two-dimensional case.Extensive numerical results all verify that the proposed methods are well-balanced for the still steady state,keep genuinely high order accuracy in smooth regions for smooth solutions,capture the perturbations of the still water steady state,and efficiently simulate some type of dam break problems.4.We construct high order DG scheme for the Euler equations under gravitational fields,which are well-balanced for the isentropic type hydrostatic equilibrium state.For one-dimensional case,we reformulate the Euler equation in its equivalent form by the isentropic type hydrostatic equilibrium state,and propose a novel source term approximation as well as well-balanced numerical fluxes.In addition,we prove that the proposed scheme for the one-dimensional Euler equation can maintain well-balanced property.Then we generalized the one-dimensional scheme to the multi-dimensional case.Extensive numerical results are performed to test the genuine high order accuracy in smooth regions,the maintenance of well-balanced property,and the ability to capture small perturbation of the isentropic hydrostatic solution state.