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Expand the three matrix exponentials and retain terms through degree two. Their product iswhich agrees with $e^{t(A+B)}$ to that order. Keeping the cubic terms gives the more informative defectwhere $[A,B]=AB-BA$ is the commutator. For example, the coefficients of $A^{2}B,ABA,B{A}^2$ in the product are $1/8,1/4,1/8$, instead of the exact $1/6,1/6,1/6$. The cubic defect is generically nonzero. Therefore Strang splitting has order $p=2$. The symmetry $E(-t)=E(t{)}^{-1}$ is consistent with this cancellation of the quadratic defect; for commuting $A,B$ the composition is exact rather than merely second order.

Split the generator as $iA-iV$. One Strang splitting step isThe two potential half-steps are componentwise multiplications. The circulant matrix $A$ is diagonalized by the discrete Fourier transform; its eigenvalues areThus the middle step consists of a Fast Fourier transform, multiplication of Fourier coefficient $j$ by $e^{ih{\lambda }_j}$, and an inverse transform. Its cost is $O(M\log M)$ and every factor is unitary, so the implementation preserves the discrete norm exactly up to roundoff.

For the Strang splittingreversing $t$ reverses all three factors, soIt is therefore a time-symmetric numerical method. Multiplication of the three exponential series shows agreement with $e^{t(A+B)}$ through degree two. Equivalently, the Baker--Campbell--Hausdorff formula gives an odd modified generatorfor a matrix $D$ made from nested commutators. Exponentiating givesfor a matrix $C$ depending on $A$ and $B$.

The centered spatial second difference has error $O(\Delta x^2)$. The symmetric compositionis Strang splitting, so its local splitting error is $O(\Delta t^3)$ and its global time order is two. Because both semidiscrete subproblems are solved exactly, the total global error is

The periodic centered-difference kinetic matrix $T_h$ is real symmetric, and the sampled real potential $V_h$ is a real diagonal matrix. Therefore $-i{T}_h$ and $-i{V}_h$ are skew-Hermitian, so each matrix exponential in the Strang splitting is a unitary matrix. Their product is unitary as well. Hence

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.