The Finite Difference Schemes For Two Kinds Of Damped Wave Equations

Damped wave equation is an important research topic in distributed parameter control theory,and is widely used in actual production and living.Due to the complexity of the system,its analytical solution is difficult to solve,which brings a lot of difficulties for the practical applications.Therefore,the study of the numerical solution has an important sig-nificance in both theory and application.In this thesis the finite difference schemes for two kinds of damped wave equation are studied.Firstly,this thesis establishes a fully discrete implicit finite difference scheme for the following one-dimensional wave equation with boundary damping.The order of local truncation error of the difference scheme is improved by adding higher or-der terms at boundary points in order to obtain whole second accuracy.A priori estimate for the proposed difference scheme is constructed by discrete multiplier method and Gronwall inequality.The accuracy of the difference scheme is proved to be of second order in infinite norm with respect to time and space.The proposed difference scheme is unconditionally stable with respect to initial conditions and right-hand term.Numerical experiments verify the theoretical results.Secondly,a fully discrete implicit finite difference scheme of the following two dimen-sional boundary damped wave equation on rectangular domain is derived in this thesis,where a five-point center scheme is employed at internal nodes,and the second order four-point scheme is constructed by introducing high-order items at the internal nodes of the two damped boundaries,and a second order three-point difference format is constructed at the intersection of the two damped boundaries.A priori estimates for the difference scheme is constructed by discrete multiplier method and Gronwall inequality,by which we prove that the solution of the difference scheme is second order convergent in L2 norm with respect to time and space,and is unconditionally stable with respect to initial conditions and right-hand term.The theoretical results are verified by numerical experiments.Finally,in this thesis a fully discrete implicit finite difference scheme is constructed for the following one-dimensional wave equation with internal delayed damping.The damped term with delay is treated by a proposed second order finite difference scheme.The solution of the difference scheme is not only second order convergent in infinite norm with respect to time and space,but also unconditionally stable with respect to initial condi-tions,which is proved by mathematical induction method using discrete multiplier method and Gronwall inequality.Numerical experiments verify the theoretical results.