Ozbenli E, Vedula P
School of Aerospace and Mechanical Engineering, University of Oklahoma, 865 Asp Ave., Norman, Oklahoma 73019, USA.
Phys Rev E. 2020 Feb;101(2-1):023303. doi: 10.1103/PhysRevE.101.023303.
In this paper we propose a method, which is based on equivariant moving frames, for development of high-order accurate invariant compact finite-difference schemes that preserve Lie symmetries of underlying partial differential equations. In this method, variable transformations that are obtained from the extended symmetry groups of partial differential equations (PDEs) are used to transform independent and dependent variables and derivative terms of compact finite-difference schemes (constructed for these PDEs) such that the resulting schemes are invariant under the chosen symmetry groups. The unknown symmetry parameters that arise from the application of these transformations are determined through selection of convenient moving frames. In some cases, owing to selection of convenient moving frames, numerical representation of invariant (or symmetry-preserving) compact numerical schemes is found to be notably simpler than that of standard, noninvariant compact numerical schemes. Further, the accuracy of these invariant compact schemes can be arbitrarily set to a desired order by considering suitable compact finite-difference algorithms. Application of the proposed method is demonstrated through construction of invariant compact finite-difference schemes for some common linear and nonlinear PDEs (including the linear advection-diffusion equation in one or two dimensions, the inviscid Burgers' equation in one dimension, viscous Burgers' equation in one or two dimensions, spherical Burgers' equation in one dimension, and shallow water equations in two dimensions). Results obtained from our numerical simulations indicate that invariant compact finite-difference schemes not only inherit selected symmetry properties of underlying PDEs, but are also comparably more accurate than the standard, noninvariant base numerical schemes considered here.
在本文中，我们提出了一种基于等变移动标架的方法，用于开发高阶精确不变紧致有限差分格式，该格式能保持基础偏微分方程的李对称性。在这种方法中，从偏微分方程（PDEs）的扩展对称群得到的变量变换被用于变换紧致有限差分格式（为这些PDEs构建）的自变量、因变量和导数项，使得所得格式在所选择的对称群下是不变的。通过选择合适的移动标架来确定这些变换应用中出现的未知对称参数。在某些情况下，由于选择了合适的移动标架，发现不变（或保对称）紧致数值格式的数值表示比标准的、非不变紧致数值格式的数值表示显著更简单。此外，通过考虑合适的紧致有限差分算法，这些不变紧致格式的精度可以任意设定为所需的阶数。通过为一些常见的线性和非线性PDEs（包括一维或二维的线性平流扩散方程、一维的无粘伯格斯方程、一维或二维的粘性伯格斯方程、一维的球面伯格斯方程以及二维的浅水方程）构建不变紧致有限差分格式，展示了所提出方法的应用。我们数值模拟得到的结果表明，不变紧致有限差分格式不仅继承了基础PDEs的选定对称性质，而且比这里考虑的标准非不变基础数值格式具有更高的精度。
