Strang splitting

For finite matrices $A,B$, Strang splitting approximates the evolution $e^{h(A+B)}$ byIt advances the $A$ dynamics for a half step, the $B$ dynamics for a full step, and the $A$ dynamics for a half step. Expanding the three matrix exponentials givesso the one-step local truncation error is $O(h^3)$. Bounded propagation on a fixed time interval gives global error $O(h^2)$. The identity $S_{-h}=S_h^{-1}$ expresses time symmetry. Commuting matrices make the product exact.^[ref]

The finite-matrix assumptions are relevant when the method is applied after a spatial semidiscretization. Error constants can depend on the spatial mesh through commutators. A fixed-matrix second-order argument alone does not prove an error estimate uniform over increasingly fine spatial grids.

If real matrices satisfy $A^T=A\le 0$ and $B^T=-B$, then $e^{tA}$ is a contraction and $e^{tB}$ is an orthogonal matrix for $t\ge 0$. Hence $\Vert e^{kA/2}{e}^{kB}{e}^{kA/2}{\Vert }_2\le 1$ for every $k\ge 0$. The bound is independent of commutativity and gives unconditional stability for a centered convection-diffusion semidiscretization with homogeneous Dirichlet boundary conditions. It does not itself bound the mesh-dependent commutators that enter a time-error estimate.

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]