School of Aerospace and Mechanical Engineering, University of Oklahoma, Norman, Oklahoma 73019, USA.
Phys Rev E. 2017 Dec;96(6-1):063304. doi: 10.1103/PhysRevE.96.063304. Epub 2017 Dec 4.
In this paper, we propose a method to construct invariant finite-difference schemes for solution of partial differential equations (PDEs) via consideration of modified forms of the underlying PDEs. The invariant schemes, which preserve Lie symmetries, are obtained based on the method of equivariant moving frames. While it is often difficult to construct invariant numerical schemes for PDEs due to complicated symmetry groups associated with cumbersome discrete variable transformations, we note that symmetries associated with more convenient transformations can often be obtained by appropriately modifying the original PDEs. In some cases, modifications to the original PDEs are also found to be useful in order to avoid trivial solutions that might arise from particular selections of moving frames. In our proposed method, modified forms of PDEs can be obtained either by addition of perturbation terms to the original PDEs or through defect correction procedures. These additional terms, whose primary purpose is to enable symmetries with more convenient transformations, are then removed from the system by considering moving frames for which these specific terms go to zero. Further, we explore selection of appropriate moving frames that result in improvement in accuracy of invariant numerical schemes based on modified PDEs. The proposed method is tested using the linear advection equation (in one- and two-dimensions) and the inviscid Burgers' equation. Results obtained for these tests cases indicate that numerical schemes derived from the proposed method perform significantly better than existing schemes not only by virtue of improvement in numerical accuracy but also due to preservation of qualitative properties or symmetries of the underlying differential equations.
在本文中，我们提出了一种通过考虑偏微分方程（PDE）的修正形式来构建用于求解 PDE 的不变有限差分格式的方法。不变格式保留了李对称，是基于等变运动框架方法得到的。虽然由于与繁琐的离散变量变换相关的复杂对称群，通常难以为 PDE 构建不变数值格式，但我们注意到，通过适当修改原始 PDE，可以获得与更方便变换相关的对称性。在某些情况下，对原始 PDE 的修改也有助于避免可能由特定运动框架选择引起的平凡解。在我们提出的方法中，可以通过向原始 PDE 添加微扰项或通过缺陷校正过程来获得 PDE 的修正形式。这些附加项的主要目的是使具有更方便变换的对称性成为可能，然后通过考虑这些特定项趋于零的运动框架从系统中删除这些项。此外，我们探索了选择适当的运动框架，以基于修正的 PDE 提高不变数值格式的准确性。该方法通过线性对流方程（一维和二维）和无粘 Burgers 方程进行了测试。这些测试案例的结果表明，从所提出的方法得到的数值格式不仅由于数值精度的提高，而且由于对基础微分方程的定性性质或对称性的保持，性能明显优于现有格式。
