Operator splitting and splitting error

For a finite-dimensional autonomous linear system y′=(A+B)y with fixed matrices, the exact step is exp(time step times(A+B)). A Lie step applying A first and B second is exp(hB)exp(hA); the rightmost matrix acts first. Expanding both products through second order shows a local defect proportional to BA−AB. If the matrices commute, their exponentials combine exactly and this splitting error vanishes. The sign of the leading commutator reverses when the subflow order is reversed.

The symmetric Strang step applies a half A step, a full B step and a half A step. For fixed bounded matrices it has a third-order local defect and, on a fixed finite time interval with stable propagation, second-order global error. Lie splitting has second-order local and first-order global error under the corresponding finite-dimensional assumptions. These are different uses of the word order: a one-step defect has one extra power of h relative to the accumulated global error over order 1/h steps.

A useful example uses one nilpotent shear matrix for A and its transpose for B. Their individual exponentials are identity plus h times the matrix, so the substeps are exact and elementary. Their sum generates hyperbolic sine and cosine, showing that the combined motion contains interactions not present in one isolated shear. The Lie result misses a second-order component, while the symmetric composition cancels that leading defect but leaves higher-order terms.

This example separates splitting error from inaccurate subsolves. In an implementation, approximate subflows add their own errors and can destroy the intended composition order if solved too loosely. A second-order splitting with a first-order subsolver does not automatically stay second order. Also, stability of each isolated physical process is not a substitute for analyzing the composed approximation in the relevant norm, particularly when the split operators are nonnormal or when negative substeps are introduced.

After spatial discretization, A and B become matrices depending on the mesh. Constants in a formal splitting expansion can grow as that mesh is refined because spatial derivatives and commutators become large. A finite-dimensional order claim at fixed mesh therefore does not establish a PDE estimate uniform in spatial resolution. For unbounded differential operators one must control operator domains, solution regularity, boundary compatibility and the stability of each propagated subproblem.

Boundary forcing can be especially delicate: an intermediate split state may fail the compatibility conditions required by the next subflow, producing order reduction. Nonautonomous coefficients require the correct time placement or an augmented formulation, rather than copying an autonomous composition blindly. The examples here use fixed matrices and homogeneous algebraic initial data, where all expansions are legitimate. They illustrate the mechanism of splitting error while keeping broader PDE convergence conditional on the additional analytic estimates that make the formal commutators meaningful.