Research On Finite Volume Element Algorltnms Of Some Stochastic Wave Equations

The mathematical formulation of most problems in science and engineering leads either to a partial differential equation or to a set of partial differential equations. Nu-merical methods are the preferred means of solution. Among these, thanks to it’s merits such as conservation of the approximate flux over each computational cell, ad-equate accuracy and ease of implementation, the finite volume element method has been an excellent candidate. In this dissertation, some wave equations in fluid dy-namics are investigated numerically. In each case, effective discretization scheme is designed for the considered problem. Furthermore, the problems under study are investigated by the proposed numerical algorithm.Firstly, we consider a viscous wave equation with stochastic perturbation. We employ Monte Carlo method for discretization in random space and quadratic finite volume element method for discretization in physical space. This type of scheme is verified to robust and can simulate the solutions of equation with stochastic perturba-tion in a satisfactory way.Secondly, we numerically study a stochastic nonlinear damped wave equation. we incorporate finite volume element method with Monte Carlo Sampling method to survey the influence of a impurity external term and two kinds of damping effects on the propagation of solitons profile and investigate several general quantities.At last, we study a fractional steps domain decomposition scheme for a class of viscous wave equations. The global domain is divided into multiple block-divided sub-domains, and each sub-domain is partitioned into fine meshes further. At each time interval, the fractional steps domain decomposition scheme is proposed to solve the viscous wave signal, in which we use a local multilevel explicit scheme to solve the values of viscous wave signal on the interfaces of sub-domains and use the frac-tional steps implicit scheme to solve the interior values of viscous wave signal in sub-domains. The proposed scheme is an explicit-implicit domain decomposition method over multi-block sub-domains. Numerical experiments demonstrate that this approach keeps the advantages of the non-overlapping domain decompositions and the fraction-al steps technique. The local multilevel explicit schemes at interface points relax the stability requirement of the fractional steps domain decomposition method and im-prove the accuracy near the interface boundaries as well.