The Second Lie-Group $SO_o(n,1)$ Used to Solve Ordinary Differential Equations

  •  Chein-Shan Liu    
  •  Wun-Sin Jhao    


Liu (2001) derived the first augmented Lie-group $SO_o(n,1)$ symmetry for the nonlinear ordinary differential equations (ODEs): $\dot{\bf x}={\bf f}({\bf x},t)$, and developed the corresponding group-preserving scheme (GPS). However, the earlier formulation did not consider the rotational effect of nonlinear ODEs. In this paper, we derive the second augmented Lie-group $SO_o(n,1)$ symmetry by taking the rotational effect into account. The numerical algorithm exhibits two solutions of the Lie-group ${\bf G} \in SO_o(n,1)$, depending on the sign of $\|{\bf f}\|^2 \|{\bf x}\|^2-2({\bf f}\cdot{\bf x})^2$, which means that the algorithm may be switched between two states, depending on ${\bf x}$. We give numerical examples to assess the new algorithm GPS2, which upon comparing with the GPS can raise the accuracy about three orders. It is interesting that for the chaotic system the signum function sign$(\|{\bf f}\|^2 \|{\bf x}\|^2-2({\bf f}\cdot{\bf x})^2)$ is frequently switched between $+1$ and $-1$ in time.

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