Numerical Methods To Solve Partial Differential Equations Based On Spline Functions

In this dissertation, three methods based on univariate and bivariate splines are prensented to solve differential equations.In Chapter 1, univariate spline functions and some related basic relationships as well as multivariate spline spaces are briefly introduced.In Chapter 2, combined with spline functions, an unconditional stable sub-domain precise integration scheme containing parameterα> 0(α<< h) for the initial-boundary value and periodic initial value problem of four order parabolic equation is presented, the local truncation error is O(τ~ 2 +ατ~2+ h~4). The numerical example shows that the accuracy are much better than some previous methods.In Chapter 3, based on subdomain precise integration method and combined the cubic spline function approximation, the spline subdomain precise integration (SSPI) scheme is presented to solve convection-diffusion equations, and then analysis of the stability of the SSPI scheme .It is showed that the accuracy of the method is much better than previous methods.In Chapter 4, based on the method presented in Chapter 3, the spline sub-domain precise integration alternating segment method containing parameterα>0 for the first initial-boundary value problem of convection-diffusion equation is presented. The method is unconditionally stable. The numerical example shows that the accuracy of the method is satisfactory and can be conveniently used to solve the second and third initial-boundary value problems.