Numerical Gradient Schemes for Heat Equations Based on the Collocation Polynomial and Hermite Interpolation

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.