Numerical solution of modified differential equations based on symmetry preservation

Ersin Ozbenli and Prakash Vedula
Phys. Rev. E 96, 063304 – Published 4 December 2017

Abstract

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.

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  • Received 26 January 2017
  • Revised 16 June 2017

DOI:https://doi.org/10.1103/PhysRevE.96.063304

©2017 American Physical Society

Physics Subject Headings (PhySH)

  1. Research Areas
Interdisciplinary PhysicsFluid Dynamics

Authors & Affiliations

Ersin Ozbenli* and Prakash Vedula

  • School of Aerospace and Mechanical Engineering, University of Oklahoma, Norman, Oklahoma 73019, USA

  • *ozbenli@ou.edu.
  • pvedula@ou.edu.

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Issue

Vol. 96, Iss. 6 — December 2017

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