Two Numerical Methods Based On Quartic Spline Functions For Solving Two-dimensional Linear Hyperbolic Equations

In this dissertation, two new numerical methods based on the relationship ofquartic spline functions for solving the two-dimensional linear hyperbolic equa-tion are proposed. The main contents of the dissertation are as follows:1. First of all, the existing numerical methods and their results for solvinglinear hyperbolic equations are summarized.2. Univariate spline functions with degree n are introduced, including basicconcepts, the relationship between quartic spline function values and the second-order derivatives at the nodal points and the interpolation errors under uniformpartition.3. By using the relationship between quartic spline function values and thesecond-order derivatives at the nodal points, a new three level implicit di?erencescheme for solving the two-dimensional linear hyperbolic equation is presented.It is shown that the scheme is unconditionally stable and the truncation error isO(k2 + h4). Some numerical examples are presented to test the e?ciency of thenew method.4. Two new iteration schemes based on quartic spline interpolation in spacedirection and Pade′approximation semi-discretization in time direction for the nu-merical solution of two-dimensional linear hyperbolic equation are constructed.It is shown that they are unconditionally stable and their truncation errors canachieve O(k~5 + h~4) and O(k~7 + h~4), respectively. Numerical results are presentedto test these two new schemes.