High-order Compact Difference Schemes For Solving Parabolic Equations

A lot of phenomena can be described by parabolic equations such as diffusion, transfusion and heat condution etc. It is very difficult, even sometimes impossible to get their exact solutions in the theory because of the complexity of the problems. So it has important theoretical significance and practial value to look for various numerical methods with good stability, higher order accuracy comparably, less amount of calculation and storage.Firstly, univariate spline functions with degree n and the relationships between second-order derivatives of quartic and sextic spline functions at the nodal points under uniform partition are introduced. Let quartic and sextic spline functions interpolate smooth enough function and it obtains the truncation errors of second-order derivatives of the interpolation function and the original function at the nodal points being of O(h4) and O(h6). Secondly, the two relationships of second-order derivatives based on quartic and sextic spline interpolation are used for the spatial discretization. The one-dimensional parabolic equation is transformed into two kinds of ordinary differential equations with respect to time t. And two exact solutions containing exponential matrixs are given. Then, the exponential matrixs are approximated by the (2,2) Padd and (3,3) Pade approximations after the temporal discretization, therefore two high-order compact difference schemes are proposed for solving the one-dimensional parabolic equation. One scheme is fourth order accuracy in both time and space and the other is sixth order accuracy in both time and space. Thirdly, the relationship of second-order derivative based on quartic spline interpolation of two variables and (2,2) Pade approximation, a high-order compact difference scheme is proposed for solving the two-dimensional parabolic equation. The scheme is four order accuracy in both time and space. Fourthly, the relationship of second-order derivative based on quartic spline interpolation of three variables and (2,2) Pade approximation, a high-order compact difference scheme is proposed for solving the three-dimensional parabolic equation. The scheme is four order accuracy in both time and space. And it proves that four schemes are unconditionally stable. At last, numerical experiments with exact solution are carried out to demonstrate theirs accuracy and stability.