For Solving Differential Equations Of High-precision Spline Difference Method

In this thesis, the applications of spline functions in the numerical solutions of differential equations are introduced in the first chapter. Then in the second chapter, a brief discussion to cubic, quartic, quintic, sextic polynomial spline functions and the corresponding relationship between splines are given. Local interpolation error by splines also discussed.In the third chapter, based on the classical quartic spline functions, a sixth-order difference scheme is presented for solving the singularly-perturbed boundary value problem with constant coefficients. At the same time, another new difference scheme is proposed by using sextic spline functions, the truncation error can reach to O(h~8) at the interior nodal points and O(h~5) at the end nodal points.In the fourth chapter, based on the classical quartic spline functions, a new method is given for solving the two-point boundary value problem with Neumann boundary conditions. The truncation error of this new method to deal with the derivatives can reach O(h~4) at least, while the accuracy of the classical difference scheme is second-order only.In the fifth chapter, by using quartic spline functions, we present an unconditionally stable difference scheme to solve the second-order linear hyperbolic equations, and the truncation error of this scheme is O(k~2 + h~4).At the same time, in the sixth chapter, two high semi-discrete methods with high accuracy to solve the hyperbolic equations are given, and the truncation errors of the two methods are O(k~5+ h~4) and O(k~7 + h~4), respectively.At last, in the seventh and eighth chapters, we consider the numerical solution of a class of parabolic equations based on quartic spline functions. For different boundary conditions, we will give some different numerical methods.