Splitting Methods for Differential Equations

Sergio Blanes, Fernando Casas, Ander Murua, Acta Numerica 2024, vol. 33, pp. 1-161

Supplementary Material accompanying the paper

Contents 1 Introduction 2 High order splitting and composition methods 3 Splitting methods for special problems 4 Qualitative properties of splitting methods 5 Highly oscillatory problems 6 Splitting methods for PDEs 7 Splitting methods with complex coefficients 8 A collection of splitting methods 9 Splitting everywhere: some relevant applications References A Testing splitting methods on matrices

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Page Last Updated: 6 April, 2025

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1- MATLAB files containing most of the splitting and composition methods gathered in Section 8

• (a) Methods of Table 8.1: Symmetric compositions of second-order symmetric schemes of order r = 4, 6, 8, 10

• Non-processed schemes:
coefsSS.m
• Processed schemes:
coefsSSproc.m

• (b) Methods of Table 8.2: Splitting methods of order r = 2, 3, 4, 6

• Non-processed schemes:
coefsS.m
• Processed schemes:
coefsSproc.m

• (c) Methods of Table 8.3: RKN splitting methods of order r = 4, 5, 6, 8

• Non-processed schemes:
coefsRKN.m
• Processed schemes:
coefsRKNproc.m

• (d) Methods of Table 8.4: RKN splitting methods of order r = 4, 6, 8 with modified potentials / double commutators

• Non-processed schemes:
coefsRKNmod.m
• Processed schemes:
coefsRKNmod_proc.m

• (e) Methods of Table 8.5: Splitting methods for near-integrable systems

• Non-processed schemes:
coefsNI.m
• Processed schemes:
coefsNIproc.m

• (a) Figure A.1: the compressed file Figure_A1.zip constains all the programs and functions required to generate Figure A.1: one must run Fig_EXP_SS.m, which calls the functions phiSSnEXP and phiSSnEXPproc with the methods whose coefficients are gathered in coefsSS and coefsSSproc.

• (a) Figure 1.3(a): trajectories of six-body system modelling the outer Solar System. Notice that this also corresponds to the illustration in the front page of the volume. All the files required are contained in the compressed file Trajectories_SolarSystem.zip. One must run FigSistSolar_Jacobi_Traj.m, and this calls functions phiKepler.m, phiNbodyKep.m and phiNbodyVI.m. The generated figure can also be conveniently rotated at will.
  

• (b) Figure 9.1: Efficiency diagrams showing the relative error in energy (a) and positions (b) at the final time t=200000 days obtained with different schemes (extrapolation, Matlab routines and splitting methods) when integrating the outer Solar System. The files to run are: FigSistSolar_EXTRAP.m (extrapolation), FigSistSolar_Jacobi.m (splitting) and FigSistSolar_ODE.m (Matlab routines). All the required functions are contained in the compressed file Sec9_SolarSystem.zip.
  

•  File containing the errata we have identified (so far) in the paper: ListOfErrata.pdf