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.

翻译：模板组合借鉴了函数组合的思想，即组合两个具有任意导数阶数的模板，以获得导数阶数等于组合模板阶数之和的模板。本文展示了如何利用模板组合构造有限差分模板，以数值求解偏微分方程（PDE）。我们介绍了模板组合的各种性质，并研究了组合后模板的精度阶与其组成模板精度阶之间的关系。我们还比较了组合高阶偏微分方程与其紧致版本之间的稳定性限制，并通过收敛性测试验证了精度阶。为了演示在偏微分方程中的应用，我们使用模板组合数值求解了涉及二维双调和方程的边值问题，并通过收敛性检验验证了精度阶。随后将该方法应用于Cahn-Hilliard相场模型。除了针对该基准问题的二维和三维示例结果外，我们还探讨了可扩展性、谱特性和稀疏性。