Increment formulations for rounding error reduction in the numerical solution of structured differential systems

Raising the order of geometric numerical integrators by composition and extrapolation

“…h/2 (26) can be used. As mentioned, methods up to order ten obtained by composition of (26) are available in the literature (see [14,21] and references therein).…”

“…As mentioned, methods up to order ten obtained by composition of (26) are available in the literature (see [14,21] and references therein). This method is essentially equivalent to the basic method proposed by Gragg (see [15, p. 294]) to be used with extrapolation for this problem.…”

“…However, this problem has a very specific structure (not fully exploited by (26)) which could allow to build more efficient integrators. …”

“…For the standard extrapolation we consider as the basic second order symmetric integrator the composition (26).…”

“…In addition, each full time step by composition as well as by extrapolation takes the form y n+1 = y n + y n and compensated summation can be used. This corresponds to the following algorithm for N steps (see [26] and references therein): y err = 0 for n = 1 to N y = y n + y err y new = y n + y y err = (y n − y new ) + y y n+1 = y new .…”