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  "body": "\n## Statement\n\nLet $A$ and $B$ be complex $d \\times d$ matrices and let $e^{M}$ denote the [[matrix exponential|020M04FN1H]].\n\n1. As $h \\to 0$,\n$$e^{hA/2}\\, e^{hB}\\, e^{hA/2} - e^{h(A + B)} = O(h^3).$$\n2. For each $t \u003e 0$, as $n \\to \\infty$,\n$$\\left( e^{tA/(2n)}\\, e^{tB/n}\\, e^{tA/(2n)} \\right)^n - e^{t(A + B)} = O(n^{-2}).$$\n\n## Remarks\n\nFor clause 1, expand both sides in powers of $h$: the terms of order $0$, $1$ and $2$ agree, because the symmetric product gives $\\frac{h^2}{2}\\left(\\frac{A^2}{4} + \\frac{A^2}{4} + B^2 + AB + BA + \\frac{A^2}{2}\\right) = \\frac{h^2}{2}(A + B)^2$.\nFor clause 2, write the difference of $n$-th powers as a telescoping sum of $n$ terms.\nEach term is a local error $O(n^{-3})$ multiplied by matrices of norm at most $e^{t(\\norm{A} + \\norm{B})}$.\nThe [[Lie–Trotter product formula|KH7EPKDCH3]] uses $e^{tA/n} e^{tB/n}$ instead, and its error is only $O(n^{-1})$; the symmetric arrangement gains one order.\nIf $AB = BA$, both splittings are exact.\nIn applications, $e^{hA}$ and $e^{hB}$ are replaced by the flows $\\varphi^A$ and $\\varphi^B$ of $u' = A(u)$ and $u' = B(u)$ for nonlinear operators.\nThe composition $\\varphi^A_{h/2} \\circ \\varphi^B_h \\circ \\varphi^A_{h/2}$ is still a method of order $2$ for smooth problems.\n\n## Explanations\n\n### L1 human\n\nSome evolution equations are a sum of two parts, each easy to solve alone.\nStrang splitting advances the first part half a step, the second part a full step, and the first part another half step.\nDoing the halves on both sides makes the error much smaller than doing one part after the other.\n\n### L2 human\n\nStrang splitting is used for reaction-diffusion equations, for the Schrödinger equation (kinetic and potential parts, solved by FFT and pointwise), and for advection in several dimensions.\nConsecutive half steps of $A$ merge, so the cost is nearly that of the first-order Lie–Trotter splitting.\nThe leapfrog method for Hamiltonian systems is the Strang splitting of the kinetic and potential flows.\n",
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