A Group Preserving Scheme for Burgers Equation with Very Large Reynolds Number
Chein-Shan Liu

doi:10.3970/cmes.2006.012.197
Source CMES: Computer Modeling in Engineering & Sciences, Vol. 12, No. 3, pp. 197-212, 2006
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Keywords Burgers equation, Lie algebra, Lorentz Group, Group preserving scheme, Spatial rescale.
Abstract In this paper we numerically solve the Burgers equation by semi-discretizing it at the$n$ interior spatial grid points into a set of ordinary differential equations:$\mathaccentV {dot}05F{\bf u}={\bf f}({\bf u},t)$,${\bf u} \in {\@mathbb R}^n$. Then, we take the dissipative behavior of Burgers equation into account by considering the magnitude$\delimiter "026B30D {\bf u}\delimiter "026B30D$ as another component; hence, an augmented quasilinear differential equations system$\mathaccentV {dot}05F{\bf X}={\bf A}{\bf X}$ with${\bf X}:=({\bf u}^{\unhbox \voidb@x \hbox {\relax \fontsize {8}{9.5}\selectfont T}}, \delimiter "026B30D {\bf u}\delimiter "026B30D )^{\unhbox \voidb@x \hbox {\relax \fontsize {8}{9.5}\selectfont T}} \in {\@mathbb M}^{n+1}$ is derived. According to a Lie algebra property of${\bf A} \in so (n,1)$ we thus develop a new numerical scheme with the transformation matrix${\bf G}\in SO_o(n,1)$ being an element of the proper orthochronous Lorentz group. The numerical results were in good agreement with exact solutions, and it can be seen that the group preserving scheme is better than other numerical methods. Even for very large Reynolds number the group preserving scheme supplemented with a spatial rescaling technique also provides a reliable result without inducing numerical instability.
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