Abstract
As is well-known, the advantage of the high-order compact difference scheme (H-OCD) is that it is unconditionally stable and convergent on the order O ( τ 2 + h 4 ) (where τ is the time step size and h is the mesh size), under the maximum norm for a class of nonlinear delay partial differential equations with initial and Dirichlet boundary conditions. In this article, a new numerical gradient scheme based on the collocation polynomial and Hermite interpolation is presented. The convergence order of this kind of method is also O ( τ 2 + h 4 ) under the discrete maximum norm when the spatial step size is twice the one of H-OCD, which accelerates the computational process. In addition, some corresponding analyses are made and the Richardson extrapolation technique is also considered in the time direction. The results of numerical experiments are consistent with the theoretical analysis.
Highlights
A great deal of effort has been devoted to the development of numerical approximations to heat equation problems
These results show that the convergence order in time direction can reach O(τ 4 ), which is consistent with the theoretical analysis
Many people have devoted themselves to the development of numerical approximations of heat equation problems
Summary
A great deal of effort has been devoted to the development of numerical approximations to heat equation problems (see [1,2,3,4]). Mathematics 2019, 7, 93 mesh grid by cubic and bi-cubic Hermite interpolation According to these intermediate points, a new explicit scheme on the gradient of the discrete solutions of the heat equation is deduced based on the collocation polynomial. This greatly reduces the amount of calculation for the same accuracy as the high-order compact difference schemes. 2. One-Dimensional Numerical Gradient Schemes Based on the Local Hermite Interpolation and Collocation Polynomial. For the convenience of description, let us firstly consider the one-dimensional case and generalize to the two-dimensional case
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