Second-order Hyperbolic Equations With Alternating Finite Element Analysis

A large number of physical problems of significant interest, such as the propagation of waves, arc modeled by hyperbolic equations. And so the investigation of hyperbolic equations has been the object of an increasing interest of a number of specialists, engineers as well as mathematicians. As a result, the theory and the computer simulations show that the hyperbolic equations are interesting from a mathematical point of view as well as from a physical point of view.In this paper, we consider the hyperbolic equations with initial-boundary condition under the conduction of Prof. Yuan. By comprehensively employing finite element method (FEM) and alternating direction implicit method (ADI), we put forward corresponding schemes and finally give rigorous convergence analysis.Alternating direction Galerkin method is based on ADI and FEM, and inherit all advantage from both of them. Their effectiveness from ADI is due to the fact that they reduce the solution of a multidimensional problem to the solution of sets of independent one-dimensional problems. The corresponding matrices are independent of time and require only one decomposition. The storage requirements for these matrices are associated with one-dimensional problems rather than the full multidimensional problem, and so the storage requirements are quite low . Their high accuracy from FEM is also quite interesting. Besides, because of the natural parellelism, they are fit for super-computers to get the solution of large problem.This dissertation contains three parts. In first part, we introduce the development and some results about the ADI Galerkin method of hyperbolic equation. The brief outline of this dissertation is also put into this part.In the second part, we formulate an ADI Galerkin scheme for nonlinear hyperbolic system, which involves three levels. The main idea is that we disturb the system with appropriate items and get a good linear systems, the coefficient matrix of which is a tensor product of some small matrices, and finally resolve these smaller systems instead. At the same time, we put the other items on the left side, so the coefficient matrices are independent of time and require only one decomposition. Because both the coefficient matrix and the storage requirement are associated with one-dimensional problem, the scheme has lower operations and requirement. After the error equations are analyzed, the H₀¹-error estimate and stability are derived.In the last part, by variable substitution an ADI Galerkin scheme involving two levels is formulated for a class of second order linear hyperbolic equation with mixed initial-boundary condition. After φ is substituted for (?) the item is discretized...