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.
A Narrow-stencil finite difference method for approximating viscosity solutions of fully nonlinear elliptic partial differential equations with applications to Hamilton-Jacobi-Bellman equations
Oscillatory behaviour of the RBF-FD approximation accuracy under increasing stencil size
Jožef Stefan Institute（约瑟夫·斯蒂芬研究所）；Parallel and Distributed Systems Laboratory（并行与分布式系统实验室）；Faculty of Mathematics and Physics, University of Ljubljana（卢布尔雅那大学数学物理学院）；Jožef Stefan International Postgraduate School（约瑟夫·斯蒂芬国际研究生院）
Some observations regarding the RBF-FD approximation accuracy dependence on stencil size
Jožef Stefan Institute（约瑟夫·斯特凡研究所）；University of Ljubljana, Faculty of Mathematics and Physics（卢布尔雅那大学数学与物理学院）；International Postgraduate School Jožef Stefan（约瑟夫·斯特凡国际研究生院）
A stencil scaling approach for accelerating matrix-free finite element implementations
Ludwig-Maximilians-Universität München（路德维希-马克西米利安慕尼黑大学）；Technische Universität München（慕尼黑工业大学）；Friedrich-Alexander-Universität Erlangen-Nürnberg（埃尔朗根-纽伦堡弗里德里希·亚历山大大学）
On discontinuous Galerkin discretizations of second-order derivatives
Princeton Plasma Physics Laboratory（普林斯顿等离子体物理实验室）；Max-Planck/Princeton Center for Plasma Physics, Princeton University（马克斯·普朗克/普林斯顿等离子体物理中心，普林斯顿大学）；Department of Astrophysical Sciences, Princeton University（普林斯顿大学天体物理科学系）
