Splitting Methods for differential equations

TL;DR

This paper surveys operator-splitting methods for differential equations, detailing how a system x' = f(x) can be decomposed as f = f1 + ... + fm and solved by composing exact or approximate flows of the simpler subsystems. It develops high-order schemes via composition (e.g., combining Strang maps) and analyzes order conditions using BCH formulas, Lyndon words, and integral representations, including the role of negative time steps. The work highlights geometric numerical integration aspects, showing how splitting preserves invariants like symplectic structure or unitarity, and discusses stability, processing, and long-time behavior through backward error analysis and modified equations. It then specializes these ideas to highly oscillatory problems and to PDEs (including Schrödinger and parabolic equations), covering RKN-type splittings, commutator-based schemes, IMEX formulations, and ADI/LOD approaches, with practical guidance on adapting splittings to non-autonomous problems and boundary conditions. The paper culminates in extensive method cataloging, theoretical insights into order and stability, and demonstrations across classical Hamiltonian systems, celestial mechanics, quantum dynamics, and PDE contexts, underscoring the broad utility of splitting methods in preserving system structure and enabling efficient, long-time simulations.

Abstract

This overview is devoted to splitting methods, a class of numerical integrators intended for differential equations that can be subdivided into different problems easier to solve than the original system. Closely connected with this class of integrators are composition methods, in which one or several low-order schemes are composed to construct higher-order numerical approximations to the exact solution. We analyze in detail the order conditions that have to be satisfied by these classes of methods to achieve a given order, and provide some insight about their qualitative properties in connection with geometric numerical integration and the treatment of highly oscillatory problems. Since splitting methods have received considerable attention in the realm of partial differential equations, we also cover this subject in the present survey, with special attention to parabolic equations and their problems. An exhaustive list of methods of different orders is collected and tested on simple examples. Finally, some applications of splitting methods in different areas, ranging from celestial mechanics to statistics, are also provided.

Figures (16)

• Figure 1.1: Simple pendulum. Top: phase space and three trajectories with initial conditions $(q_0,p_0)=(-5,\frac{5}{2}),(1,1),(\frac{1}{10},0)$. Bottom: relative error in energy committed by different splitting methods along the solution with initial condition $(q_0,p_0)=(\frac{1}{10},0)$ in the interval $t\in[0,500]$ with (left) 1200 evaluations and (right) 2400 evaluations of the potential.
• Figure 1.2: Pendulum. Relative error in phase space for different splitting methods along the solution with initial condition $(q_0,p_0)=(\frac{1}{10},0)$ in the interval $t\in[0,500]$ and time step $h=\frac{5}{12}$.
• Figure 1.3: Top: Trajectories of the six-body system modeling the outer Solar System. Bottom: relative error in energy as a function of time for an interval of 200000 days obtained with different splitting methods with (left) 1200 evaluations and (right) 2400 evaluations of the force.
• Figure 1.4: Outer Solar System. Relative error in position as a function of $t$ for a time interval $[0,t_f=200000]$ days with $h=t_f/1200$ obtained with different splitting methods.
• Figure 1.5: Time-dependent Schrödinger equation with a double-well potential. Left: $V(x)$, initial and final wave function with $M=256$ discretization points. Right: relative error in energy at the final time vs. number of FFTs for different values of the time step obtained with the Strang method $S_2$, and two 4th-order splitting schemes: one involving 6 evaluations of $V$ (RKN$_64$) and another with 4 evaluations of $V$, and incorporating in addition the double commutator $[V,[T,V]]$ (RKNm$_44$).