Families of efficient low order processed composition methods

TL;DR

This work targets efficient numerical integration of ODEs and Hamiltonian systems that split into three or more explicitly solvable parts. It develops processed composition methods with kernels and processors to achieve effective orders $4$ and $6$, using Lie-algebraic order theory to minimize leading and higher-order errors. Through analysis and extensive numerical experiments on problems like Lorentz-force dynamics and particle motion around a Reissner–Nordström black hole, the authors demonstrate that processed low-order schemes can outperform state-of-the-art two-part splitting methods at similar computational cost, with improved long-time behavior due to time symmetry. The study also provides practical design principles for constructing kernels and processors, including optimization targets and ordering considerations, and offers accessible code for reproducing the results.

Abstract

New families of composition methods with processing of order 4 and 6 are presented and analyzed. They are specifically designed to be used for the numerical integration of differential equations whose vector field is separated into three or more parts which are explicitly solvable. The new schemes are shown to be more efficient than previous state-of-the-art splitting methods.

Figures (8)

• Figure 1: Error in the trace obtained by kernels of effective order 4 collected in Table \ref{['table.4']}, together with method BM$_6^{[4]}$ and the kernel of BCM$_6^{[4]}$ when they are applied to the linear system \ref{['3.2a']} at final time $t_f=10$. (a): basic scheme $\chi_h$ taken as \ref{['exc']}; (b): $\chi_h$ given by \ref{['foa']}.
• Figure 2: Motion of a charged particle under Lorentz force. (a) Maximum position error for different values of the parameter $\alpha$ and for various 4th-order methods at the final time $t_f=200$, evaluated with the same computational cost $s/h=40$. (b) Efficiency diagram for different 4th-order methods with $\alpha=0.07$ and $t_f=200$.
• Figure 3: Motion of a charged particle under Lorentz force. (a) Maximum position error for different values of the parameter $\alpha$ and for various 6th-order methods at the final time $t_f=200$, evaluated with the same computational cost $s/h=40$. (b) Efficiency diagram for different 6th-order methods with $\alpha=0.04$ and $t_f=200$.
• Figure 4: Motion of a charged particle under Lorentz force. Maximum position error for the method $\hat{\psi}^{[8,4]}$ for different values of the parameter $\alpha$ with all possible orderings of the first-order method $\chi_h$. The simulation was conducted with $s/h=40$ and $t_f=200$.
• Figure 5: Particle around Reissner--Nodström black hole. (a) Maximum position error for different values of the coordinate $r_0$ and for various 4th-order methods at the final time $t_f=10^4$, evaluated with the same computational cost $s/h=0.4$ (b) Efficiency diagram for different 4th-order methods with $r_0=18$ and the same $t_f$.