Hyperbolic Equations With Discontinuous Coefficients:Numerical Schemes And Numerical Error Estimates

Hyperbolic equations with interface problems are important for physical and engineering applications,yet the mathematical research and numerical methods are still underdeveloped.Most of the papers in this area focus on capturing the interface,developing high-resolution and stable numerical methods and estimating the numerical errors of these schemes.Scientists pay close attention to convection equations,the Liouville equation,transport equations and the Schr?dinger equation.This dissertation is divided into four chapters.We introduce the background and recent work in chapter 1,and then present our own research work in the following three chapters.In chapter 1,we introduce the background knowledge about interface problems.we present several examples such as quantum tunnelling,high frequency elastic waves in heterogeneous media,models of glaciation,wave interactions on the ocean's free surface.We also introduce the surface hopping method,Hamiltonian-preserving schemes,immersed interface methods and so on.And we summarize the importance and innovation of this paper.In chapter 2,we present a modified surface hopping method for the Schr?dinger equation with conical crossings in the Eulerian formulation.The time-dependent Schr?dinger equation mathematically describes the fundamental physical behavior while it is too expensive to solve.Jin,Qi and Zhang proposed an Eulerian surface hopping method based on the semiclassical limit of the Born–Oppenheimer approximation of the Schr?dinger equation,namely,the system of linear Liouville equations.In order to preserve energy at the crossing manifold,we modify the surface hopping method with jump operators and obtain more accurate numerical results.In chapter 3,we estimate the numerical error of the new method proposed in chapter 2which is similar to a Hamiltonian-preserving scheme for the Liouville equation with piecewise constant potentials introduced by Jin and Wen.They also give?~1-error estimates on that scheme established in[X.Wen and S.Jin,SIAM J.Num.Anal.46,2688–2714,2008].Motivated by the work of[S.Jin and P.Qi,Sci.China Math.56,2773–2782,2013],we provide a much simpler analysis and obtain the same half-order convergence rate.We formulate the Liouville equation with discretized velocities into a series of linear convection equations with piecewise constant coefficients,and rewrite the numerical scheme into some immersed interface upwind schemes.?~1-error estimates are then evaluated by comparing the derived equations and schemes.In chapter 4,we study the Liouville euqation with moving intreface.This equation satisfies the conservation of mass in phase space.We develop a numerical method coupled with a high-resolution scheme for a class of moving interface conservation law problems.This approach is derived on a uniform Cartesian grid and allows a standard hyperbolic stability condition.We present some numerical experiments to study convergence and accuracy.In conclusion,we present a modified surface hopping method for the Schr?dinger equation with conical crossings using jump operators and obtain better numerical results.And we also provide a much simple analysis and obtain an half-order convergence rate for the Scheme es-tablished in[X.Wen and S.Jin,SIAM J.Num.Anal.46,2688–2714,2008].And we develop an immersed interface finite volume method for a class of moving interface conservation law problems.