Some Numerical Research On Viscoelastic Equations And Sobolev Equations

Difference schemes are given,based on the idea of finite difference,to solve viscoelastic equations and Sobolev equations.In chapter 1,the background and research content of this paper are introduced.In chapter 2,firstly three-time-level implicit schemes(S1)are presented for two-dimensional viscoelastic equations.In order to improve the efficiency of computation,alternating direction implicit(ADI)schemes(S2)are obtained,based on the S1.It is proved by the discrete energy method that the S1 and S2 are uniquely solvable and unconditionally stable.The convergence of the schemes are proved in l2 norm.The order of the schemes are forth-order in space and second-order in time.Finally,numerical results also demonstrate that the scheme is efficient and accurate.In chapter 3,S1 and S2 are popularized to three-dimensional viscoelastic equations.Three-time-level implicit schemes(S3)and ADI schemes(S4)are proposed for three-dimensional viscoelastic equations.It is proved that the S3 and S4 are uniquely solvable and unconditionally stable.The convergence of the schemes is proved in l2 norm.The order of the schemes are forth-order in space and second-order in time.Numerical results also demonstrate that the scheme is efficient and accurate.In chapter 4,we present two-time-level implicit schemes(S5)for Sobolev equations.The order of the schemes are forth-order in space and second-order in time.It is proved that the S5 are uniquely solvable.Uniqueness,convergence and stability are also proved in l? norm.Numerical experiments agree with theoretical analysis.