# American Institute of Mathematical Sciences

2007, 2007(Special): 495-505. doi: 10.3934/proc.2007.2007.495

## Lie group study of finite difference schemes

 1 Onera, Computational Fluid Dynamics and Aeroacoustics Department (DSNA), BP 72, 20 avenue de la Division Leclerc, 92322 Châtillon Cedex, France 2 Université Pierre et Marie Curie-Paris 6, Institut Jean Le Rond d'Alembert, UMR CNRS 71900, Boîte courrier $n^0$ 162, 4 place Jussieu, 75252 Paris, cedex 05. France 3 ONERA, Computational Fluid Dynamics and Aeroacoustics Department (DSNA), BP 72, 20 avenue de la Division Leclerc, 92322 Châtillon Cedex, France

Received  September 2006 Revised  August 2007 Published  September 2007

Differential equations arising in fluid mechanics are usually derived from the intrinsic properties of mechanical systems, in the form of conservation laws, and bear symmetries, which are not generally preserved by a finite difference approximation, and lead to inaccurate numerical results. This paper deals with the analysis of symmetry group of finite difference equations, which is based on the differential approximation. We develop a new scheme, the related differential approximation of which is invariant under the symmetries of the original differential equations. A comparison of numerical performance of this scheme, with standard ones and a higher order one has been realized for the Burgers equation.
Citation: Emma Hoarau, Claire david@lmm.jussieu.fr David, Pierre Sagaut, Thiên-Hiêp Lê. Lie group study of finite difference schemes. Conference Publications, 2007, 2007 (Special) : 495-505. doi: 10.3934/proc.2007.2007.495
 [1] Ronald E. Mickens. A nonstandard finite difference scheme for the drift-diffusion system. Conference Publications, 2009, 2009 (Special) : 558-563. doi: 10.3934/proc.2009.2009.558 [2] Wei Qu, Siu-Long Lei, Seak-Weng Vong. A note on the stability of a second order finite difference scheme for space fractional diffusion equations. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 317-325. doi: 10.3934/naco.2014.4.317 [3] Wen Li, Song Wang. Pricing American options under proportional transaction costs using a penalty approach and a finite difference scheme. Journal of Industrial & Management Optimization, 2013, 9 (2) : 365-389. doi: 10.3934/jimo.2013.9.365 [4] Tetsuya Ishiwata, Kota Kumazaki. Structure preserving finite difference scheme for the Landau-Lifshitz equation with applied magnetic field. Conference Publications, 2015, 2015 (special) : 644-651. doi: 10.3934/proc.2015.0644 [5] Navnit Jha. Nonpolynomial spline finite difference scheme for nonlinear singuiar boundary value problems with singular perturbation and its mechanization. Conference Publications, 2013, 2013 (special) : 355-363. doi: 10.3934/proc.2013.2013.355 [6] Hawraa Alsayed, Hussein Fakih, Alain Miranville, Ali Wehbe. Finite difference scheme for 2D parabolic problem modelling electrostatic Micro-Electromechanical Systems. Electronic Research Announcements, 2019, 26: 54-71. doi: 10.3934/era.2019.26.005 [7] Monika Eisenmann, Etienne Emmrich, Volker Mehrmann. Convergence of the backward Euler scheme for the operator-valued Riccati differential equation with semi-definite data. Evolution Equations & Control Theory, 2019, 8 (2) : 315-342. doi: 10.3934/eect.2019017 [8] Francesco C. De Vecchi, Andrea Romano, Stefania Ugolini. A symmetry-adapted numerical scheme for SDEs. Journal of Geometric Mechanics, 2019, 11 (3) : 325-359. doi: 10.3934/jgm.2019018 [9] Gabriela Marinoschi. Well posedness of a time-difference scheme for a degenerate fast diffusion problem. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 435-454. doi: 10.3934/dcdsb.2010.13.435 [10] Janosch Rieger. The Euler scheme for state constrained ordinary differential inclusions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2729-2744. doi: 10.3934/dcdsb.2016070 [11] Nora Aïssiouene, Marie-Odile Bristeau, Edwige Godlewski, Jacques Sainte-Marie. A combined finite volume - finite element scheme for a dispersive shallow water system. Networks & Heterogeneous Media, 2016, 11 (1) : 1-27. doi: 10.3934/nhm.2016.11.1 [12] Gary Froyland, Philip K. Pollett, Robyn M. Stuart. A closing scheme for finding almost-invariant sets in open dynamical systems. Journal of Computational Dynamics, 2014, 1 (1) : 135-162. doi: 10.3934/jcd.2014.1.135 [13] Imed Kacem, Eugene Levner. An improved approximation scheme for scheduling a maintenance and proportional deteriorating jobs. Journal of Industrial & Management Optimization, 2016, 12 (3) : 811-817. doi: 10.3934/jimo.2016.12.811 [14] Azmy S. Ackleh, Kazufumi Ito. An approximation scheme for a nonlinear size-dependent population model. Conference Publications, 1998, 1998 (Special) : 1-6. doi: 10.3934/proc.1998.1998.1 [15] Michele Coti Zelati. Remarks on the approximation of the Navier-Stokes equations via the implicit Euler scheme. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2829-2838. doi: 10.3934/cpaa.2013.12.2829 [16] Azmy S. Ackleh, Mark L. Delcambre, Karyn L. Sutton, Don G. Ennis. A structured model for the spread of Mycobacterium marinum: Foundations for a numerical approximation scheme. Mathematical Biosciences & Engineering, 2014, 11 (4) : 679-721. doi: 10.3934/mbe.2014.11.679 [17] Yongchao Liu, Hailin Sun, Huifu Xu. An approximation scheme for stochastic programs with second order dominance constraints. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 473-490. doi: 10.3934/naco.2016021 [18] Blaine Keetch, Yves Van Gennip. A Max-Cut approximation using a graph based MBO scheme. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6091-6139. doi: 10.3934/dcdsb.2019132 [19] Pavol Kútik, Karol Mikula. Diamond--cell finite volume scheme for the Heston model. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 913-931. doi: 10.3934/dcdss.2015.8.913 [20] François Alouges. A new finite element scheme for Landau-Lifchitz equations. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 187-196. doi: 10.3934/dcdss.2008.1.187

Impact Factor: