Absorbing Interface Condition For Non-uniform Mesh Wave Equation Simulation

This thesis studies the absorbing interface conditions for the simulation of wave equation on the non-uniform mesh. Numerical simulation plays an important role in the study of wave equations, with applications in acoustics, elastic mechanics and elec-tromagnetics. In practice, one need coupled multiscale models to provide more realistic physical phenomena. The adaptive mesh method is a general multiscale method which plays a critical role in research. A key issue in the multiscale study is to couple together models with different scales and precisions. Research usually focus on the coupling interface both for statics and dynamics problems. This thesis mainly discusses the spurious interface reflection for dynamic problems.In the numerical simulation, usually using adaptive non-uniform mesh as space discretization to improve the efficiency. The discretization will cause some problems. First of all, the non-uniform mesh wave equation simulation will lead to the consistency of numerical scheme, the nonconsistency will generate spurious, nonphysical reflection. Secondly, the time step is limited by the CFL condition and space mesh size. The uniform time step will compromise the simulation effectiveness. So we need to do the multiscale process in the time scale which is achieved by local time stepping method. Local time stepping method will bring a numerical interface. This thesis gives a proper interface condition, which satisfies the following property:1. Satisfying the mathematical requirement, the numerical schemes have consistency, stability, and convergence properties.2. Satisfying the physical requirement, the interface can eliminate the spurious reflec-tion, permit the two-way transmission of low-frequency waves and perform one-way absorbing of high-frequency waves.3. The interface condition is easy to implement in the simulation.Then for consistency of numerical simulation of non-uniform mesh and interface condition, this thesis have following works:● For the general wave equation, by using local time stepping method, based on the superposition of second order linear wave equations, we decompose the interface condition problem into two separate problems around the interface:on one of which the conventional artificial absorbing boundary conditions can be applied; for the other, the analytic solutions can be derived. The proposed interface conditions permit the two-way transmission of low-frequency waves across mesh interfaces which can be supported by both coarse and fine meshes and perform a one-way absorbing of high-frequency waves which can only be supported by fine meshes when they travel from fine mesh regions to coarse ones. Numerical examples are presented to illustrate the efficiency of the proposed absorbing interface conditions.● For the linear elasticity equations, we also discrete the time with local time stepping method, then generalize the absorbing boundary condition to the second order elastic wave equations, first order velocity-stress elastic wave equations and the first order velocity-strain elastic wave equation. We proposed the absorbing interface condition for the elastic wave equation. The numerical examples demonstrate the validity of absorbing interface condition.● For the general wave equation, we investigate a hierarchy interface condition for wave equation simulation using classical Galerkin method. The hierarchy interface condition considers the consistency problem from the perspective of the complete-ness of numerical discretization space. The incompleteness of projection space caused the spurious reflection in the interface. To achieve the completeness of projection space, we construct an extended space. The construction is hierarchy accomplished based on the demand of accuracy.We conduct a large number of numerical experiments and theoretical analysis for the two kinds interface conditions. The numerical results show that both of absorb-ing interface condition and hierarchy interface conditions can maintain the numerical scheme consistency and eliminate the spurious reflection in numerical simulation.