Splitting methods. (English) Zbl 1105.65341

McLachlan, Robert I.; Quispel, G. Reinout W.

Acta Numerica 11, 341-434 (2002).

Summary: We survey splitting methods for the numerical integration of ordinary differential equations (ODEs). Splitting methods arise when a vector field can be split into a sum of two or more parts that are each simpler to integrate than the original (in a sense to be made precise). One of the main applications of splitting methods is in geometric integration, that is, the integration of vector fields that possess a certain geometric property (e.g., being Hamiltonian, or divergence-free, or possessing a symmetry or first integral) that one wants to preserve. We first survey the classification of geometric properties of dynamical systems, before considering the theory and applications of splitting in each case. Once a splitting is constructed, the pieces are composed to form the integrator; we discuss the theory of such ‘composition methods’ and summarize the best currently known methods. Finally, we survey applications from celestial mechanics, quantum mechanics, accelerator physics, molecular dynamics, and fluid dynamics, and examples from dynamical systems, biology and reaction-diffusion systems.

MSC:

65L10	Numerical solution of boundary value problems involving ordinary differential equations
34A26	Geometric methods in ordinary differential equations
65P10	Numerical methods for Hamiltonian systems including symplectic integrators
37M15	Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems

Keywords:

splitting methods; geometric integration; Hamiltonian; dynamical systems; composition methods; celestial mechanics; quantum mechanics; accelerator physics; molecular dynamics; fluid dynamics; reaction-diffusion systems