Error Inhibiting Schemes for Initial Boundary Value Heat Equation

TL;DR

This paper introduces Error Inhibiting Schemes for the heat equation with various boundary conditions, achieving faster convergence of global error than traditional finite difference methods through a novel block scheme approach.

Contribution

It generalizes previous methods to Dirichlet and Neumann boundary conditions, providing explicit error analysis and demonstrating improved efficiency over standard schemes.

Findings

Error Inhibiting Schemes outperform standard FD schemes in convergence speed.

The schemes are stable and convergent for different boundary conditions.

Numerical examples confirm the efficiency of the proposed method.

Abstract

Finite Difference (FD) schemes are widely used in science and engineering for approximating solutions of partial differential equations (PDEs). Error analysis of FD schemes relies on estimating the truncation error at each time step. This approach usually leads to a global error whose order is of the same order of the truncation error. For classical FD schemes the global error is indeed of the same order as the truncation error. A particular class of FD schemes is the Block Finite Difference (BFD) schemes, in which the grid is divided into blocks. The structure of such schemes is similar to the structure of the Discontinuous Galerkin (DG) method, and allows inhabitation of the truncation errors. Recently, much effort was devoted to design BFD schemes whose global error converges faster than the truncation error. In this paper, we elaborate the approach presented in arXiv:1711.07926 for the heat equation with periodic boundary conditions. We generalize this methodology to design BFD schemes for the heat equation with Dirichlet or Neumann boundary conditions, whose global error converges faster than the truncation error. Such schemes are henceforth called Error Inhibiting Schemes. We provide an explicit error analysis, including proofs of stability and convergence of the proposed schemes. We illustrate our approach using several numerical examples, which demonstrate the efficiency of our method in comparison to standard FD schemes.