= Higher-order composition of a symmetric splitting
Let $S_h$ be a smooth symmetric second-order approximation to an autonomous flow, with all required positive and negative substeps defined. Its three-stage composition
$$
S_{\gamma h}S_{\delta h}S_{\gamma h},\qquad \gamma=(2-2^{1/3})^{-1},\quad\delta=-2^{1/3}(2-2^{1/3})^{-1},
$$
is fourth order in the usual finite-dimensional setting. The identities $2\gamma+\delta=1$ and $2\gamma^3+\delta^3=0$ preserve elapsed time and cancel the cubic defect. Symmetry removes the next even power from the formal logarithm, leaving a fifth-order local defect.
Here $\delta<0$: the middle stage runs backward. For finite <matrix exponentials> it exists, although it can amplify decaying modes. A backward substep of an unbounded diffusion generator need not define a bounded evolution at all. Thus these coefficients do not automatically give a usable fourth-order method for every <heat equation> discretization or continuum problem.<Splitting methods with complex coefficients by Sergio Blanes and others>[§1, equation(11) and negative-coefficient discussion, printed pp3–4]{ref}
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