论文标题
用于初始边界值热方程的误差抑制方案
Error Inhibiting Schemes for Initial Boundary Value Heat Equation
论文作者
论文摘要
有限差异（FD）方案广泛用于科学和工程中，用于近似偏微分方程（PDE）的解决方案。 FD方案的错误分析依赖于估计每个时间步骤的截断误差。这种方法通常会导致一个全局误差，该错误的顺序是截断误差的相同顺序。对于经典的FD方案，全局误差确实与截断误差相同。特定类别的FD方案是块有限差（BFD）方案，其中网格分为块。此类方案的结构类似于不连续的Galerkin（DG）方法的结构，并允许居住截断误差。最近，大量精力用于设计BFD方案的全局误差比截断误差更快。在本文中，我们详细阐述了Arxiv：1711.07926中介绍的使用周期性边界条件的热方程。我们将这种方法推广到使用Dirichlet或Neumann边界条件的热方程设计BFD方案，其全局误差比截断误差更快。此后，此类方案称为误差抑制方案。我们提供明确的误差分析，包括稳定性和提议方案的收敛证明。我们使用几个数值示例说明了我们的方法，这些示例与标准FD方案相比证明了我们方法的效率。
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.
