Stablity Of Boundary Conditions For Numerical Wave Simulation

Numerical wave simulation has been an area of active research, which is related to fields such as mechanics, geophysics, electromagnetism and etc. Significant achievements were made both in basic and application-oriented research. However, even for numerical simulation of linear wave propagation, the local instability problems have not been well resolved, which are caused mainly due to artificial boundary conditions, physical boundary conditions or material interfaces. In this paper, the formation mechanism of local instability and ways of its elimination were studied via methods of stability analysis for numerical wave simulation.1. Based on Lax equivalent theorem, the fundamental theorem in the analysis of numerical solution for initial-boundary value problems of partial differential equations, the Lax-stability was first divided into strong Lax-stability and weak Lax-stability according to different manners of convergence, techniques of the stability analysis were then classified into the strong stability analysis and the weak stability analysis. The importance role of the former played in the local instability analysis was elucidated emphatically. While for the latter, practical value of its combination with the finite element technique in constructing stable scheme of numerical wave simulation, was stressed.2. Some results on the research of strong stability analysis of Local instability problems initiated by transmitting boundary were got. First, through a one dimensional finite discrete mode, the explanation of the mechanism of high-frequency oscillation instability was further improved by proving the condition under which the module of reflection factor will be greater than 1 in high frequency wave band. Second, for the mechanism of another kind of high-frequency local instability was clarified via a SH semi-infinite discrete model, which was caused by the improper combination of multi-transmitting boundary with interior node discrete formula. Based on that, an effective way of its elimination was provided. Third, the zero-frequency drift local instability was systematically discussed, and its instable phenomenon observed in numerical simulation was analytically explained.3. Based on interface points' analytical solution in an short time window, which was deduced by combination of the fact about the finiteness of wave velocity and solutions of Cauchy problem of wave equation, A method of constructing an high order explicit recursion formula for interface points was provided for one dimensional bar with interfaces. Thus a simulation scheme consists of the same order formula for both interior point and interface point was given. The stability of the new second order scheme for an infinite discrete model with an interface was proved via extended strong stability analysis, which was originally used in semi-infinite discrete model. Finally, the theoretical results were verified by numerical test.4. In order to solve the local instability problem arising in Orthogonal Anisotropic wave simulation when Perfect Matched Layer was used, a scheme constructed by the combination of first order multi-transmitting boundary and finite element method was provided and its stability was proved by weak stability analysis.5. A scheme for implementing Multi-Transmitting Formula (MTF) is proposed by combining MTF into the control equation of the interior nodes adjacent to the artificial boundary. Compared with the original scheme, the new one improves not only accuracy of the boundary condition but also reveals clearly a relation between the order of truncation error of the ABC and the extended mesh solution, which is commonly used as a benchmark to test ABCs. Limitation is then clarified for improving accuracy of numerical simulation of wave motion only by increasing the accuracy order of an ABC.