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 Incoming links: Operator splitting

Ander Murua 2026-11-05

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]

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^{'} = (A + B) u

u_{n + 1} = e^{h A} e^{h B} u_{n} .

Expanding both matrix exponentials gives

e^{h A} e^{h B} - e^{h (A + B)} = \frac{1}{2} h^{2} (A B - B A) + O (h^{3})

. Thus noncommutativity creates the leading local error; if

A

and

B

commute, the step is exact.

The symmetric composition

e^{h A /2} e^{h B} e^{h A /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]