Embedded (4, 5) pairs of explicit 7-stage Runge-Kutta methods with FSAL property

TL;DR

The paper addresses the classification of embedded $(4,5)$ pairs of explicit 7-stage Runge--Kutta methods with the First Same As Last (FSAL) property. It rewrites the order conditions in a compact, multi-vector form and reduces the degrees of freedom to a six-parameter node description $(c_2,c_3,c_4,c_5,c_6,c'_3)$, then identifies five 4-dimensional families of embedded pairs (types A, B, C and primed variants A', B'). It provides explicit relations for remaining coefficients and demonstrates how the last-order conditions constrain the pairs, with new, efficient examples compared against existing schemes. The work highlights the geometric structure of the solution space, notes the close relationship between primed and unprimed families, and contributes practical 7-stage FSAL pairs suitable for adaptive-step ODE solvers, along with appendices supplying detailed formulas for the A- and B-type constructions.

Abstract

The general case of embedded (4, 5) pairs of explicit 7-stage Runge--Kutta methods with FSAL property (a_7j = b_j, 1 <= j <= 7, c_7 = 1) is considered. Besides exceptional cases, the pairs form five 4-dimensional families. The pairs within two (already known) families satisfy the simplifying assumption sum_j a_ij c_j = c_i^2 / 2, i >= 3.

Figures (3)

• Figure 1: Schematic depiction of the five families. The left half contains non-FSAL pairs of $6$-stage methods, on the right are pairs of $7$-stage methods with FSAL property.
• Figure 2: A two-dimensional cut through the six-dimensional space $(c_2, c_3, c_4, c_5, c_6, c'_3)$. Here $c_2 = 1 / 5$ and $c_5 = 4 / 5$. The nodes $c_4$ and $c_6$ are set according to the eqs. (\ref{['c4_eq_something']}) and (\ref{['c6_eq_1']}), respectively. The dashed, dotted, and solid curves correspond to pairs of type A, B, and C, respectively. The equations for the curves are (A) $c'_3 = c_3^2 / 2$, (B) $c'_3 = 3 (5 c_3 - 1) (1 + c_3) / 50$, and (C) $c'_3 = c_3 (5 c_3 - 1) \bigl[ 13 - 12 c_3 \pm (73 - 208 c_3 + 144 c_3^2)^{1 / 2} \bigr] / 20$. All the three curves intersect at $c_3 = \bigl( 6 \pm 6^{1 / 2} \bigr) / 10$, or when $3 - 12 c_3 + 10 c_3^2 = 0$. The type C curve intersects twice with the ones of type A and B at $(c_3, c'_3) = (0, 0)$ and $(c_3, c'_3) = (c_2, 0)$, respectively. (At these four points some of the matrix elements of ${\pmb{A}}$ are infinite, so they do not correspond to any embedded pairs.) The structure of intersections stays the same even when only one of the eqs. (\ref{['c4_eq_something']}) and (\ref{['c6_eq_1']}) is satisfied.
• Figure 3: Efficiency curves for problems A3, A4 DETEST, D5 DETEST, and PLEI HNW93: the pair in Table \ref{['type_B_pair']} (dashed curve), Table \ref{['sqrt4054']} (thin dashed curve), Table \ref{['method_similar_to_Tsit5']} (solid curve), and Tsi11 (thin solid curve), DoPr80 (dotted curve), and BoSh96 (thin dotted curve) pairs. The adaptive step size scheme $h \leftarrow 0.9 h \bigl( \hbox{ATOL} / E \bigr){}^{1 / 5}$ was used. (The starting step size $h_0 = 10^{-6}$ was swiftly corrected by the adaptive step size control.) Here $\hbox{ATOL}$ is the absolute error tolerance, and $E$ is the $\ell^2$-norm of the difference vector between the two solutions within a pair. The steps with $E > \hbox{ATOL}$ were rejected, but they were still contributing to the number of the r.h.s. evaluations. For A3, A4, and D5 problems the maximal value of the $\ell^2$-norm of the error $\Vert {\pmb{\tilde{x}}}(t) - {\pmb{x}}(t) \Vert_2$ along the whole trajectory $0 \le t \le 20$ is plotted. For PLEI the $\ell^2$-norm of the error was measured at the end of the integration interval $t = 3$, using only $14$ components of ${\pmb{x}}$ that correspond to the coordinates of the stars.