On symmetric-conjugate composition methods in the numerical integration of differential equations

Abstract

We analyze composition methods with complex coefficients exhibiting the so-called “symmetry-conjugate” pattern in their distribution. In particular, we study their behavior with respect to preservation of qualitative properties when projected on the real axis and we compare them with the usual left-right palindromic compositions. New schemes within this family up to order 8 are proposed and their efficiency is tested on several examples. Our analysis shows that higher-order schemes are more efficient even when time step sizes are relatively large.

Applying splitting methods with complex coefficients to the numerical integration of unitary problems

“…Splitting and composition methods with complex coefficients, although computationally between 2 and 4 times more costly than their real counterparts when applied to ODEs involving real vector fields, possess however some remarkable properties: their truncation errors with the minimum number of stages are typically very small, and their stability threshold is comparatively large. Moreover, when the numerical solution is projected at each time step, they lead to approximations that still preserve important qualitative features (such as symplecticity and time-symmetry) up to an order much higher than the order of the method itself [7,9,4].…”

“…leading to a time-symmetric composition scheme, usually referred to as Yoshida's method, here denoted as S [4] (h). Notice, however, that there are four more complex solutions.…”

“…leads again to two time-symmetric methods, denoted as Ψ [4] P,c (h), whereas the second one, denoted as Ψ [4] SC,c (h),…”

“…(here the bar indicates the complex conjugate), corresponds to a so-called symmetric-conjugate composition method [4]: it is symmetric in the real part of the coefficients and skew-symmetric in the imaginary part. Here and in the sequel, the first sub-index in a method (either P or SC) refers to its type (either palindromic or symmetric-conjugate, respectively), whereas the second sub-index (either r or c) indicates that the a i coefficients in the splitting are real or complex, respectively.…”

“…One of them, ω 5,1 = 3 j=1 γ 5 j , has been typically used to measure the relative error of methods of the same class. If one defines the error as E = |ω 5,1 |, then one has for the previous methods the following values of E: S [4] (h) E = 5.29 . .…”