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Ander Murua coauthored complex-coefficient operator splitting methods with Sergio Blanes and Fernando Casas. They analyze how composition coefficients influence order and error propagation. Positive real parts allow complex-time substeps to bypass an obstruction associated with negative real coefficients, particularly relevant to dissipative evolution equations.^[ref]

Fernando Casas studied operator splitting with Sergio Blanes and Ander Murua. Their work constructs higher-order compositions using complex coefficients with positive real parts. The approach addresses the negative-step obstruction of real-coefficient splittings, making high-order integration possible in settings where backward evolution, such as backward diffusion, is problematic.^[ref]

Let $S_h$ be a smooth symmetric second-order approximation to an autonomous flow, with all required positive and negative substeps defined. Its three-stage compositionis fourth order in the usual finite-dimensional setting. The identities $2\gamma +\delta =1$ and $2{\gamma }^3+{\delta }^3=0$ preserve elapsed time and cancel the cubic defect. Symmetry removes the next even power from the formal logarithm, leaving a fifth-order local defect.

Here $\delta <0$: the middle stage runs backward. For finite matrix exponentials it exists, although it can amplify decaying modes. A backward substep of an unbounded diffusion generator need not define a bounded evolution at all. Thus these coefficients do not automatically give a usable fourth-order method for every heat equation discretization or continuum problem.^[ref]

Operator splitting approximates evolution generated by a sum of operators through separate evolutions of its parts. For bounded matrices $A,B$, one step of the Lie product approximation to $u^{\prime }=(A+B)u$ isExpanding both matrix exponentials gives $e^{hA}{e}^{hB}-e^{h(A+B)}=\frac{1}{2}{h}^2(AB-BA)+O(h^3)$. Thus noncommutativity creates the leading local error; if $A$ and $B$ commute, the step is exact.

The symmetric composition $e^{hA/2}{e}^{hB}{e}^{hA/2}$ cancels the quadratic local error and has local error $O(h^3)$ 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]

Sergio Blanes coauthored methods for splitting differential-equation evolution with Fernando Casas and Ander Murua. They studied complex composition coefficients with positive real parts, avoiding the backward real-time steps required by higher-order real splittings. This matters for parabolic partial differential equations, where a backward diffusion substep can be ill posed.^[ref]

For finite matrices $A,B$, Strang splitting approximates the evolution $e^{h(A+B)}$ byIt advances the $A$ dynamics for a half step, the $B$ dynamics for a full step, and the $A$ dynamics for a half step. Expanding the three matrix exponentials givesso the one-step local truncation error is $O(h^3)$. Bounded propagation on a fixed time interval gives global error $O(h^2)$. The identity $S_{-h}=S_h^{-1}$ expresses time symmetry. Commuting matrices make the product exact.^[ref]

The finite-matrix assumptions are relevant when the method is applied after a spatial semidiscretization. Error constants can depend on the spatial mesh through commutators. A fixed-matrix second-order argument alone does not prove an error estimate uniform over increasingly fine spatial grids.