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Put $\delta =\Delta t$ and $L=A+B$, keeping the spatial grid fixed for the temporal error assertion. The matrix exponential expansions giveHence the Lie-Trotter splitting commutator error isThe energy estimate from part 1 makes $e^{tA}$, $e^{tB}$ and $e^{tL}$ contractions, so both $S_{\delta }$ and $E_{\delta }$ have norm at most one. Telescope their $n$th powers:For $n\delta \le T$, this givesThus the Lie-Trotter splitting has global temporal error $O(\Delta t)$. The constant here may depend on the fixed spatial operators; a mesh-uniform joint-limit error estimate needs additional regularity and commutator bounds. If $A$ and $B$ commute, this particular exponential splitting is exact, but variable diffusion coefficients generally remove that commutativity.

Suppose a semidiscrete linear PDE is $u_t=(A+B)u$. Direct evaluation of $e^{h(A+B)}$ may be expensive, while $e^{hA}$ and $e^{hB}$ can be cheap, parallelizable, or exactly structure-preserving. Splitting is especially effective when $A$ and $B$ represent different spatial directions, kinetic and potential energy, diffusion and reaction, or linear and nonlinear pieces.

The Lie-Trotter splitting commutator error follows from multiplying the exponential series:It therefore has local error $O(h^2)$ and global order one. Reversing the factors changes the sign of the leading commutator. The symmetric average in composition form is Strang splitting,whose symmetry $S_{-h}=S_h^{-1}$ removes even powers from the modified generator. Its local error is $O(h^3)$, involving nested commutators such as $[A,[A,B]]$ and $[B,[A,B]]$, and its global order is two.

For multidimensional diffusion, directional splitting replaces one large elliptic solve by successive one-dimensional tridiagonal solves; alternating-direction implicit methods are standard examples. For the Schrödinger equation, take $A=i{\partial }_x^2$ and $B=-iV(x)$. The kinetic subflow is diagonal in Fourier space, the potential subflow is pointwise multiplication, and Strang splitting is both second order and exactly norm-preserving.

A symmetric method has even global order. If its leading modified-flow defect is $h^{2p+1}E$, a symmetric composition $S_{a_{1}h}\cdots S_{a_{s}h}$ raises the order when $\sum a_i=1$ and $\sum a_i^{2p+1}=0$, together with the required higher commutator conditions. In particular, the higher-order composition of a symmetric splittingis fourth order because $2\gamma +\delta =1$ and $2{\gamma }^3+{\delta }^3=0$. Its negative middle step is harmless for reversible unitary problems but can be ill-posed or strongly unstable for parabolic semigroups. Higher-order parabolic splittings therefore use more specialized complex coefficients, commutator corrections, or extrapolation.

The matrix exponential is the absolutely convergent seriesMultiplying the two expansions givesOn the other hand,Subtracting yieldsThis is the leading Lie-Trotter splitting commutator error.