论文标题
使用模板组成的偏微分方程的有限差近似值的准确性顺序
On the order of accuracy for finite difference approximations of partial differential equations using stencil composition
论文作者
论文摘要
模板组成使用功能组成的概念，其中两个具有衍生词的模具组成，以获得具有衍生级的模具，等于组成模板的顺序总和。在本文中，我们展示了如何应用模板组成形成有限的差异模板，以便求解部分微分方程（PDE）。我们介绍模板组成的各种特性，并研究组成模具的准确性和组成模具的精确度之间的关系。我们还介绍了组成的高阶PDE的稳定性限制与它们的紧凑版本和数值实验之间的比较，其中我们通过收敛测试来验证准确性的顺序。为了证明对PDE的应用，使用模板组成在数值上求解了涉及二维Biharmonic方程的边界值问题，并通过执行收敛测试来验证准确性的顺序。然后将该方法应用于Cahn-Hilliard相位场模型。除了针对此基准问题的2D和3D的样本结果外，还探索了可伸缩性，光谱特性和稀疏性。
Stencil composition uses the idea of function composition, wherein two stencils with arbitrary orders of derivative are composed to obtain a stencil with a derivative order equal to sum of the orders of the composing stencils. In this paper, we show how stencil composition can be applied to form finite difference stencils in order to numerically solve partial differential equations (PDEs). We present various properties of stencil composition and investigate the relationship between the order of accuracy of the composed stencil and that of the composing stencils. We also present comparisons between the stability restrictions of composed higher-order PDEs to their compact versions and numerical experiments wherein we verify the order of accuracy by convergence tests. To demonstrate an application to PDEs, a boundary value problem involving the two-dimensional biharmonic equation is numerically solved using stencil composition and the order of accuracy is verified by performing a convergence test. The method is then applied to the Cahn-Hilliard phase-field model. In addition to sample results in 2D and 3D for this benchmark problem, the scalability, spectral properties, and sparsity is explored.
