 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 compositionis 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.^[ref]