Strang splitting

Strang splitting approximates $e^{h(A+B)}$ by $e^{hA/2}{e}^{hB}{e}^{hA/2}$. Its time symmetry cancels the quadratic local commutator defect, giving local error $O(h^3)$ and global order two.

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.

If $S_h$ is a symmetric second-order splitting, then $S_{\gamma h}{S}_{\delta h}{S}_{\gamma h}$ is fourth order for $\gamma =(2-2^{1/3}{)}^{-1}$ and $\delta =-2^{1/3}(2-2^{1/3}{)}^{-1}$. The identities $2\gamma +\delta =1$ and $2{\gamma }^3+{\delta }^3=0$ preserve the total step and cancel the leading odd defect.