The preservation of qualitative properties of a dynamical system under discretisation has found much interest lately. This article studies as an important example of such a qualitative property the invariance under a symmetry group. In many cases it is either not possible to find a numerical method preserving the symmetry or such methods are prohibitively expensive. The authors propose an approach, bases on previous work of one of them [cf. R. I. McLachlan, G. R. W. Quispel and G. S. Turner, SIAM J. Numer. Anal. 35, No. 2, 586-599 (1998; 912.34015)], that permits to preserve the symmetry up to any desired order in the stepsize.
The basic idea is to construct to each numerical method \(\phi\) and to each symmetry \(h\) the adjoint method \(h^{-1}\phi h\). Then at each time step either the original method or its adjoint is applied based on the pattern of the Thue-Morse sequence. The more steps of this sequence are used, the higher the order of preservation of the symmetry will be.
As concrete examples some matrix flows on O\((n)\) and SL\((n)\), resp., and splitting methods applied to equations of the form \(\dot y=f(y)+g(y)\) are considered. The results confirm the theoretical considerations.