Operator splitting approximates evolution generated by a sum of operators through separate evolutions of its parts. For bounded matrices , one step of the Lie product approximation to isExpanding both matrix exponentials gives . Thus noncommutativity creates the leading local error; if and commute, the step is exact.
The symmetric composition cancels the quadratic local error and has local error for bounded matrices. With suitable stability over a fixed time interval this gives second-order global accuracy. In a reaction-diffusion equation, separate reaction and diffusion substeps can each use a method suited to that process. For unbounded differential operators, boundary conditions and regularity require additional analysis rather than a formal transfer of the matrix calculation.[ref]
Splitting methods with complex coefficients by Sergio Blanes and others. Introduction, Lie-Trotter and Strang compositions
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