Runge–Kutta Schemes for Addressing Left-Endpoint Singularities

Abstract

In classical explicit Runge–Kutta methods for solving initial value problems (IVPs) of the form $y^{\prime }\left(x\right)=f(x,y),y\left(x_0\right)=y_0$, the first stage is typically given by evaluating the right-hand side at the initial point, i.e., $f_1=f(x_0,y_0)$. However, this approach becomes inefficient or even ill-posed when the $f(x_0,y_0)$ exhibits a singularity at $x_0$, as is common in many physically motivated problems such as the Lane–Emden equation or Thomas–Fermi model. To address this issue, we propose an alternative approach that was originally introduced by Oliver for low-order methods. In this formulation, the first stage is shifted away from the singular point and is instead evaluated at a shifted location: $f_1=f(x_0+c_1\tau ,y_0)$, where $\tau$ is the step size and $c_1\ne 0$ is a nonzero coefficient. This allows the method to bypass the singularity while preserving consistency with the IVP. We derive the corresponding order conditions for algebraic order six and construct an eight-stage scheme satisfying these constraints. The resulting method demonstrates significantly improved efficiency when applied to problems with initial-point singularities, outperforming classical Runge–Kutta pairs of orders $6\left(5\right)$ and even $8\left(7\right)$.

1. Introduction

The first-order initial value problem stands as the fundamental model for Ordinary Differential Equations (ODEs), and is given by:

$$y^{\prime }\left(x\right)=f(x,y\left(x\right)),y\left(x_0\right)=y_0,$$

(1)

where the function $f:\mathbb{R}\times {\mathbb{R}}^m\to {\mathbb{R}}^m$.

Runge–Kutta (RK) schemes are among the most widely utilized single-step numerical techniques for approximating solutions to (1). These approaches do not rely on previously computed solution values. If $y_k$ denotes an approximation to $y\left(x_k\right)$, then an RK method updates the approximation at the next point $x_{k+1}=x_k+{\tau }_k$ by the rule:

$$y_{k+1}=y_k+{\tau }_k\left(\sum _{i=1}^s{b}_i{f}_i\right),$$

(2)

where

$$f_i=f\left(x_k+c_i{\tau }_k,y_k+{\tau }_k\sum _{j=1}^s{a}_{ij}{f}_j\right),i=1,2,\dots ,s.$$

This formulation describes an s-stage Runge–Kutta method. We focus on explicit methods where $a_{ij}=0$ whenever $j\ge i$. The associated coefficients are conveniently organized in the Butcher tableau [1,2]:

$$\begin{array}{ccccc}{c}_1& & & & \\ c_2& a_{21}& & & \\ ⋮& & & & \\ c_s& a_{s1}& a_{s2}& \cdots & a_{ss}\\ & b_1& b_2& \cdots & b_s\end{array},$$

and equivalently in compact matrix notation as

$$\begin{array}{cc}c& A\\ & b\end{array},$$

where $A\in {\mathbb{R}}^{s\times s}$, $c\in {\mathbb{R}}^s$, and $b^T\in {\mathbb{R}}^s$.

This structure allows the successive and explicit computation of the function evaluations $f_i$, for $i=1,2,\dots ,s$.

Runge–Kutta (RK) methods represent a fundamental family of numerical schemes for solving initial value problems (IVPs) of ordinary differential equations (ODEs). A practical extension of these methods involves the construction of so-called RK pairs. These pairs consist of two methods of the same stage number but differing algebraic orders—typically denoted as RKp(q), where p is the order of the primary method and $q<p$ is the order of the embedded method. Both methods share the same set of intermediate function evaluations (or “stages”), which makes the additional computation cost negligible while providing valuable error estimation for adaptive step-size control [3,4].

Such embedded pairs are widely used in practice due to their efficiency and robustness. The idea is to obtain two approximations of the solution at each step: one of higher order and one of lower order. The difference between these two approximations provides a local estimate of the truncation error. Based on this estimate, the algorithm can dynamically adjust the step size to meet a user-defined accuracy tolerance. If the estimated error is too large, the step is rejected and retried with a smaller step size; if the error is sufficiently small, the step can be accepted and potentially followed by a larger one.

Some of the most widely adopted embedded RK pairs include RK5(4) [5], RK6(5) [6,7], RK8(7) [8,9], and RK9(8) [10,11] or even RK7(5) [12]. In these pairs the first number indicates the order of the primary method and the second the order of the embedded one. These methods strike a balance between computational cost and accuracy, making them suitable for stiff and non-stiff problems alike, as well as for problems with rapidly changing solutions where fine step control is crucial.

In applications where high precision is required—such as in orbital mechanics, computational physics, and control theory—higher order RK pairs like RK8(7) and RK9(8) are particularly advantageous. They allow for larger step sizes while still satisfying stringent accuracy requirements, significantly reducing the total number of steps and function evaluations compared to lower-order methods.

While standard numerical integrators such as ode78 and ode15s exhibit robust convergence characteristics for a wide class of well-posed initial value problems, their application to singular differential equations is fundamentally constrained at the origin. The necessity of initiating these solvers at a displaced point $x_0+ϵ$ to avoid the singularity introduces a non-trivial initialization error and can lead to a significant increase in the number of function evaluations during the first few steps. Our analysis does not suggest a general deficiency in these established methods but rather highlights a specific failure to resolve the solution directly from $x_0=0$ without auxiliary analytical intervention or artificial shifting.

The continued development and refinement of RK pairs remain a significant area of research in numerical analysis, especially in conjunction with modern error estimation techniques and high-performance computing environments.

This study explores an alternative approach to explicit Runge–Kutta schemes where the initial derivative value, specifically $f_1$, is defined as $f(x_0+c_1\tau ,y_0)$ for a non-zero constant $c_1\ne 0$ and step size $\tau$. This deviates from the conventional Runge–Kutta (RK) method, where the trivial row simplifying condition ($A·e=c$) is typically enforced. The methods developed in this paper are termed Runge–Kutta-Oliver (RKO) methods, revisiting an underdeveloped framework where this condition is not imposed ($A·e\ne c$).

Specifically, this paper focuses on developing a novel sixth-order RKO scheme that utilizes eight stages per step. This new method, designated as Oliver6, introduces certain conditions, such as satisfying the first-row simplifying condition ($A·e=c$) only for indices beyond the first (due to $c_1\ne 0$), demanding $A·c={\displaystyle \frac{1}{2}}{c}^2$ for those same indices, and imposing $b_1=b_2=0$ to streamline the order conditions. Additionally, it incorporates the column simplification condition given by $b·(A+C-I_s)=0$. This novel method demonstrates superior efficiency compared to traditional Runge–Kutta pairs of orders 6(5) and even 8(7), especially for applications involving initial singularities, such as the Lane–Emden equation.

The proposed eighth-stage method is designed as a specialized startup integrator specifically tailored to address the ill-posedness of the first step in singular initial value problems. By focusing on the rigorous resolution of the solution at the origin, this work provides the end-to-end validation necessary for a consistent initialization phase, while the current framework establishes the theoretical and numerical viability of the scheme for fixed-step startup; it serves as the requisite foundation for future extensions into fully adaptive, long-range integration suites, where the removal of the initial singularity is a prerequisite for global stability.

2. Runge–Kutta Order Conditions

The derivation of Runge–Kutta (RK) methods typically relies on satisfying the so-called trivial row simplifying condition:

$$c_i=\sum _{j=1}^s{a}_{ij},i=1,2,\dots ,s,$$

which, in matrix notation, becomes $c=A·e$, where $e={[1,1,\dots ,1]}^T\in {\mathbb{R}}^s$.

Oliver, in his foundational study [13] investigated RK schemes without enforcing this simplification. His findings demonstrated that no four-stage method of order four can be built under this relaxed setting. Since then, this particular avenue of RK theory has remained largely unexplored, especially concerning higher-order methods that omit the above condition.

In this study, we revisit this underdeveloped framework and explicitly consider the case:

$$A·e\ne c.$$

We will show that such methods become viable and even competitive when focusing on constructing fifth-order schemes using six stages. To distinguish this generalized framework from the classical RK formulation, we introduce the terminology Runge–Kutta–Oliver (RKO) methods. Clearly, standard RK methods form a special subset of the RKO family in which the row simplification is imposed.

Especially in the present work we will consider sixth order methods with Butcher tableau as shown in

Table 1. RK methods with eight stages considered here.

$\frac{1}{2}{c}_2$
$c_2$	$c_2$
$2c_2$	0	$2c_2$
$c_4$	0	$a_{42}$	$a_{43}$
$c_5$	0	$a_{52}$	$a_{53}$	$a_{54}$
$c_6$	0	$a_{62}$	$a_{63}$	$a_{64}$	$a_{65}$
$c_7$	0	$a_{72}$	$a_{73}$	$a_{74}$	$a_{75}$	$a_{76}$
1	0	$a_{82}$	$a_{83}$	$a_{84}$	$a_{85}$	$a_{86}$	$a_{87}$
b	0	0	$b_3$	$b_4$	$b_5$	$b_6$	$b_7$	$b_8$

:

Applying an RKO method to the initial value problem (1), the analytical Taylor expansion of the solution around $x_0$ yields [4]:

$$y(x_0+\tau )=y_0+\tau f+{\displaystyle \frac{{\tau }^2}{2}}\left(f_{y}f+f_x\right)+{\displaystyle \frac{{\tau }^3}{6}}\left(f_{yy}{f}^2+2f_{xy}f+f_{xx}+f_y^{2}f+f_y{f}_x\right)+\cdots ,$$

(3)

where the various partial derivatives are given by

$$f_x={\displaystyle \frac{\partial f}{\partial x}},f_y={\displaystyle \frac{\partial f}{\partial y}},f_{xx}={\displaystyle \frac{{\partial }^{2}f}{\partial x^2}},f_{xy}={\displaystyle \frac{{\partial }^{2}f}{\partial x\partial y}},,etc.,$$

and all of them are evaluated at the point $(x_0,y_0)$. It is important to emphasize that when $y_0\in {\mathbb{R}}^m$ for $m>1$, these expressions involve Fréchet derivatives rather than simple scalar differentials.

Simultaneously, the internal stages $f_i$ of the RKO scheme are also expanded around $(x_0,y_0)$ using Taylor series. For instance,

$$f_1=f(x_0+c_1\tau ,y_0)=f+c_1\tau f_x+{\displaystyle \frac{1}{2}}{c}_1^2{\tau }^2{f}_{xx}+\cdots ,$$

with all quantities again evaluated at $(x_0,y_0)$.

Accordingly, we obtain the following representation:

$$y_1=y_0+\tau g_{11}f+{\tau }^2\left(g_{21}{f}_{y}f+g_{22}{f}_x\right)+{\tau }^3\left(g_{31}{f}_{yy}{f}^2+g_{32}{f}_{xy}f+g_{33}{f}_{xx}+g_{34}{f}_y^{2}f+g_{35}{f}_y{f}_x\right)+\cdots$$

(4)

where the constants $g_{ij}$ are entirely determined by the coefficients $A,b,c$.

Subtracting Equation (4) from the Taylor expansion in (3), we obtain the local discretization error:

$$y\left(x_1\right)-y_1=\tau \left(g_{11}-1\right)f+{\tau }^2\left(g_{21}-{\displaystyle \frac{1}{2}}\right)·f_{y}f+{\tau }^2\left(g_{22}-{\displaystyle \frac{1}{2}}\right)·f_x+\cdots$$

(5)

A method is said to possess algebraic order p when its local error is of order $O\left({\tau }^{p+1}\right)$. This implies that the coefficients in the series expansion above must vanish for all powers of $\tau$ less than or equal to p. Therefore, achieving order p necessitates satisfying a collection of algebraic conditions, commonly referred to as order conditions. For instance, demanding

$$T_1^{\left(1\right)}=g_{11}-1=b·e-1=0,$$

yields a first-order accurate scheme.

To achieve second-order accuracy, the following constraints must be met:

$$T_1^{\left(2\right)}=g_{21}-{\displaystyle \frac{1}{2}}=b·A·e-{\displaystyle \frac{1}{2}}=0\mathrm{and}{T}_2^{\left(2\right)}=g_{22}-{\displaystyle \frac{1}{2}}=b·c-{\displaystyle \frac{1}{2}}=0.$$

In general, for a method of order p, the coefficient of ${\tau }^{p+1}$ is referred to as its principal local error term.

In traditional Runge–Kutta theory (where $Ae=c$), the concept of rooted trees was employed to encode the order conditions. For the broader class of RKO methods, we extend this idea using rooted trees with two types of leaves, marked by two distinct symbols—white (∘) and black (•). A rooted tree of order n consists of n interconnected nodes with these two colors, structured such that the root (a white node) initiates the branching and no reconnections are allowed. Importantly, only white nodes (∘) are permitted to have offspring. Under this framework, the root corresponds to the coefficient vector b, the white nodes represent matrix A, and the black nodes are associated with the vector c.

Each rooted tree with two-colored terminal nodes corresponds to a unique order condition, a relationship that becomes evident when the tree is dissected. Specifically, we separate its branches at the root. The following naming convention is adopted: a tree is denoted by ${\rho }_{ij}$, where the first index i indicates the total number of vertices, and the second index j labels the individual trees within the family of trees containing i nodes. The simplest case, the single-node tree •, is assigned ${\rho }_{11}$, while the isolated leaf ∘ corresponds to ${\rho }_{12}$. The complete collection of trees with two nodes is as follows:

Notice that the single prominent upper vertex is consistently represented as white. Below are the five distinct rooted trees composed of three nodes:

Observe again that the upper large nodes are white. Only ${\rho }_{31}$ and ${\rho }_{34}$ remain after assumption $A·e=c$.

A rooted tree containing p vertices can be constructed by attaching, to a newly introduced root node, a collection of smaller trees whose total order is $p-1$. Consequently, we can represent such a structure as $\rho =[{\rho }_{ij},{\rho }_{kl},\dots ,{\rho }_{mn}]$, where $i+k+\dots +m=p-1$ and ${\rho }_{ij},{\rho }_{kl},\dots ,{\rho }_{mn}$ denote the constituent subtrees joined at the new root. In cases where an identical subtree (such as ${\rho }_{mn}$) appears g times in this attachment process, we can abbreviate the repetition using the notation ${\rho }_{kl}^g$. For instance, ${\rho }_{33}=\left[{\rho }_{21}^2\right]$. It is also important to note that the sequence in which the subtrees are listed has no bearing on the overall structure. That is; for example, ${\rho }_{32}=[{\rho }_{12},{\rho }_{11}]=[{\rho }_{11},{\rho }_{12}]$.

Referring back to the diagram mentioned earlier, the corresponding tree can be described as $\rho =[{\rho }_{12},{\rho }_{11},{\rho }_{22},{\rho }_{32}]$, where all components are collectively joined (attached) to form a new root node.

Several mappings are introduced over the trees examined in this context. Following the approach outlined in ([14], p. 164), we may likewise define:

⋄

Order r($\rho$):

$$r\left(\left[{\rho }_{ij}^{{\psi }_1},{\rho }_{kl}^{{\psi }_2},\cdots ,{\rho }_{mn}^{{\psi }_p}\right]\right)=1+{\psi }_{1}r\left({\rho }_{ij}\right)+\dots +{\psi }_{p}r\left({\rho }_{mn}\right)$$

⋄

Symmetry $\sigma$($\rho$):

$$\sigma \left(\left[{\rho }_{ij}^{{\psi }_1},{\rho }_{kl}^{{\psi }_2},\cdots ,{\rho }_{mn}^{{\psi }_p}\right]\right)={\psi }_1!\cdots {\psi }_k!\sigma {\left({\rho }_{ij}\right)}^{{\psi }_1}\cdots \sigma {\left({\rho }_{mn}\right)}^{{\psi }_p}$$

⋄

Density $\gamma$($\rho$):

$$\gamma \left(\left[{\rho }_{ij}^{{\psi }_1},{\rho }_{kl}^{{\psi }_2},\cdots ,{\rho }_{mn}^{{\psi }_p}\right]\right)=r\left(\left[{\rho }_{ij}^{{\psi }_1},{\rho }_{kl}^{{\psi }_2},\cdots ,{\rho }_{mn}^{{\psi }_p}\right]\right)\gamma {\left({\rho }_{ij}\right)}^{{\psi }_1}\cdots \gamma {\left({\rho }_{mn}\right)}^{{\psi }_p}$$

⋄

Elementary weights $\mathrm{\Theta }\left(\rho \right)$:

$$\mathrm{\Theta }\left(\left[{\rho }_{ij}^{{\psi }_1},{\rho }_{kl}^{{\psi }_2},\cdots ,{\rho }_{mn}^{{\psi }_p}\right]\right)=\left(A{\left(\mathrm{\Theta }\left({\rho }_{ij}\right)\right)}^{{\psi }_1}\right)\odot \left(A{\left(\mathrm{\Theta }\left({\rho }_{kl}\right)\right)}^{{\psi }_2}\right)\odot \cdots \odot \left(A{\left(\mathrm{\Theta }\left({\rho }_{mn}\right)\right)}^{{\psi }_p}\right)$$

⋄

Elementary differentials $F\left(\rho \right)$:

$$F\left(\left[{\rho }_{ij}^{{\psi }_1},{\rho }_{kl}^{{\psi }_2},\cdots ,{\rho }_{mn}^{{\psi }_p}\right]\right)=f^{\left({\psi }_1+\cdots +{\psi }_p\right)}\left(\underset{{\psi }_{1}times}{\underbrace{F\left({\rho }_{ij}\right),\cdots ,F\left({\rho }_{ij}\right)}},\cdots ,\underset{{\psi }_{p}times}{\underbrace{F\left({\rho }_{mn}\right),\cdots ,F\left({\rho }_{mn}\right)}}\right)$$

We define $\sigma \left(\circ \right)=1$, $r\left(\circ \right)=1$, $\gamma \left(\circ \right)=1$, $F\left(\circ \right)=f$, and $\mathrm{\Theta }\left(\circ \right)=e$, where the symbol ⊙ signifies the element-wise multiplication of vectors. Namely, Hadamard product. That is,

$${[w_1{w}_2\cdots w_n]}^T\odot {[u_1{u}_2\cdots u_n]}^T={[w_1{u}_1{w}_2{u}_2\cdots w_n{u}_n]}^T.$$

This type of product has lower precedence than standard operations such as parentheses, exponentiation, and dot products. For instance, we observe that

$$c^2=c\odot c={[c_1^2,c_2^2,\dots ,c_s^2]}^T.$$

Expression (5) is structured as

$$y\left(x_1\right)-y_1=\sum _{i=1}^{\infty }\sum _{\rho \in T^{\left(i\right)}}{\tau }^i{\displaystyle \frac{1}{\sigma \left(\rho \right)}}\left(b·\mathrm{\Theta }\left(\rho \right)-{\displaystyle \frac{1}{\gamma \left(\rho \right)}}\right)F\left(\rho \right)$$

where $T^{\left(i\right)}$ denotes the collection of rooted trees with two-colored leaves of total order i. The mappings $\gamma$ and $\sigma$ assign integer values to each tree $\rho$. The function $\mathrm{\Theta }\in {\mathbb{R}}^s$ is derived entirely from $A,b,c$ and is uniquely determined by the structure of $\rho$. The term $F$ represents the corresponding elementary differential.

Accordingly, a Runge–Kutta–Oliver (RKO) method is said to achieve accuracy of order p if and only if

$$\mathrm{\Psi }\left(\rho \right)={\displaystyle \frac{1}{\sigma \left(\rho \right)}}\left(b·\mathrm{\Theta }\left(\rho \right)-{\displaystyle \frac{1}{\gamma \left(\rho \right)}}\right)=0,$$

for every $\rho \in T^{\left(i\right)}$, with $i=1,2,\cdots ,p$. This equation characterizes the order conditions, linking them directly to rooted trees bearing two types of colored leaves.

We now proceed to derive the order conditions corresponding to different algebraic accuracies.

1st order (${\rho }_{11}$)

$$\mathrm{\Psi }\left(\circ \right)=\mathrm{\Psi }\left({\rho }_{11}\right)={\displaystyle \frac{1}{\sigma \left(\circ \right)}}\left(b·\mathrm{\Theta }\left(\circ \right)-{\displaystyle \frac{1}{\gamma \left(\circ \right)}}\right)=b·e-1=0$$

It is worth noting that no order constraint is linked to ${\rho }_{12}$, namely the tree ∘. This is the sole configuration that is exempt from any associated condition. This exclusion is necessary, as the trivial tree ∘ cannot possess any successors. As a result, all trees of higher complexity can appropriately be treated as offspring of those arising at second order.

2nd order ${\rho }_{21}=\left[{\rho }_{11}\right]$

2nd order ${\rho }_{22}=\left[{\rho }_{12}\right]$

At the third order, a total of five distinct trees can be identified. The process can be systematically extended to derive the complete set of order constraints required for a RKO scheme to attain accuracy of order p.

Table 2. Correspondence of trees with 1-st to 3-rd order conditions.

Error Coefficient	Tree
$\mathrm{\Psi }\left({\rho }_{11}\right)=b·e-1$
$\mathrm{\Psi }\left({\rho }_{21}\right)=b·A·e-{\displaystyle \frac{1}{2}}$
$\mathrm{\Psi }\left({\rho }_{22}\right)=b·c-{\displaystyle \frac{1}{2}}$
$\mathrm{\Psi }\left({\rho }_{31}\right)={\displaystyle \frac{1}{2}}b·c^2-{\displaystyle \frac{1}{6}}$
$\mathrm{\Psi }\left({\rho }_{32}\right)=b·(c\odot A·e)-{\displaystyle \frac{1}{3}}$
$\mathrm{\Psi }\left({\rho }_{33}\right)={\displaystyle \frac{1}{2}}b·{(A·e)}^2-{\displaystyle \frac{1}{6}}$
$\mathrm{\Psi }\left({\rho }_{34}\right)=b·A·c-{\displaystyle \frac{1}{6}}$
$\mathrm{\Psi }\left({\rho }_{35}\right)=b·A·A·e-{\displaystyle \frac{1}{6}}$

presents a summary of the order conditions up to third order, alongside their linkage to the respective rooted trees. The tree $\rho =[{\rho }_{12},{\rho }_{11},{\rho }_{22},{\rho }_{32}]$ illustrated earlier corresponds to the error expression

$$b·(c\odot (A·e)\odot (A·c)\odot (A·(c\odot A·e)))=b·(c\odot A·e\odot A·c\odot (A·(c\odot A·e)))={\displaystyle \frac{1}{108}}.$$

Thus, the upper large white circle corresponds to b, the line between two white circles corresponds to A, the line from white to black circle corresponds to c and the lower-most white circles correspond to e.

We may proceed following the approach outlined in [15]. Beginning with the two second-order trees (${\rho }_{21}$ and ${\rho }_{22}$), the remaining structure proceeds in much the same fashion. Characteristics such as the order, symmetry, density, integer decompositions, generating functions, and overall tree proliferation continue to obey similar combinatorial patterns.

Table 3. Number of algebraic conditions required for achieving accuracy of order p.

Order p	1	2	3	4	5	6	7	8	9	10
No. of conditions for RK	1	1	2	4	9	20	48	115	286	719
No. of conditions for RKO	1	2	5	13	37	108	332	1042	3360	11019

presents the number of order conditions associated with each value of p. These quantities are consistent with those documented in [16], except for the case $p=1$, as ${\rho }_{12}$ does not correspond to any order condition. An asymptotic approximation for the total number of conditions was provided by V. Kotesovec in [17], and is expressed as:

$${\displaystyle \frac{138402122262389007}{286334285732500481}}·{\left({\displaystyle \frac{13500571437155581}{3508060758296959}}\right)}^p·p^{-1.5}$$

Hence, to construct a RKO method of order four, one must fulfill $1+2+5+13=21$ independent order constraints.

In contrast, a classical Runge–Kutta method of the same order requires only $1+1+2+4=8$ conditions to be satisfied. This highlights a notable disadvantage of RKO schemes. In [18], we also presented the order conditions for algebraic orders four and five in Mathematica [19] notation. Here, in

Table 4. $\mathrm{\Psi }\left({\rho }_{6j}\right)$ for $j=1,2,3,\cdots ,108$.

$-\frac{1}{720}+b·A·A·A·A·c$	$-\frac{1}{720}+b·A·A·A·A·A·e$	$\frac{1}{2}\left(-\frac{1}{360}+b·A·A·A·c^2\right)$
$-{\displaystyle \frac{1}{360}}+b·A·A·A·(c\odot A·e)$	${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{360}}+b·A·A·A·{(A·e)}^2\right)$	$-{\displaystyle \frac{1}{240}}+b·A·A·(c\odot A·c)$
$-{\displaystyle \frac{1}{240}}+b·A·A·(A·c\odot A·e)$	$-{\displaystyle \frac{1}{240}}+b·A·A·(c\odot A·A·e)$	$-{\displaystyle \frac{1}{240}}+b·A·A·(A·e\odot A·A·e)$
${\displaystyle \frac{1}{6}}\left(-{\displaystyle \frac{1}{120}}+b·A·A·c^3\right)$	${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{120}}+b·A·A·(c^2\odot A·e)\right)$	${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{120}}+b·A·A·(c\odot {(A·e)}^2)\right)$
${\displaystyle \frac{1}{6}}\left(-{\displaystyle \frac{1}{120}}+b·A·A·{(A·e)}^3\right)$	$-{\displaystyle \frac{1}{180}}+b·A·(c\odot A·A·c)$	$-{\displaystyle \frac{1}{180}}+b·A·(A·e\odot A·A·c)$
$-{\displaystyle \frac{1}{180}}+b·A·(c\odot A·A·A·e)$	$-{\displaystyle \frac{1}{180}}+b·A·(A·e\odot A·A·A·e)$	${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{90}}+b·A·(c\odot A·c^2)\right)$
${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{90}}+b·A·(A·c^2\odot A·e)\right)$	$-{\displaystyle \frac{1}{90}}+b·A·(c\odot A·(c\odot A·e))$	$-{\displaystyle \frac{1}{90}}+b·A·(A·e\odot A·(c\odot A·e))$
${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{90}}+b·A·(c\odot A·{(A·e)}^2)\right)$	${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{90}}+b·A·(A·e\odot A·{(A·e)}^2)\right)$	${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{120}}+b·A·{(A·c)}^2\right)$
$-{\displaystyle \frac{1}{120}}+b·A·(A·c\odot A·A·e)$	${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{120}}+b·A·{(A·A·e)}^2\right)$	${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{60}}+b·A·(c^2\odot A·c)\right)$
$-{\displaystyle \frac{1}{60}}+b·A·(c\odot A·c\odot A·e)$	${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{60}}+b·A·(A·c\odot {(A·e)}^2)\right)$	${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{60}}+b·A·(c^2\odot A·A·e)\right)$
$-{\displaystyle \frac{1}{60}}+b·A·(c\odot A·e\odot A·A·e)$	${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{60}}+b·A·({(A·e)}^2\odot A·A·e)\right)$	${\displaystyle \frac{1}{24}}\left(-{\displaystyle \frac{1}{30}}+b·A·c^4\right)$
${\displaystyle \frac{1}{6}}\left(-{\displaystyle \frac{1}{30}}+b·A·(c^3\odot A·e)\right)$	${\displaystyle \frac{1}{4}}\left(-{\displaystyle \frac{1}{30}}+b·A·(c^2\odot {(A·e)}^2)\right)$	${\displaystyle \frac{1}{6}}\left(-{\displaystyle \frac{1}{30}}+b·A·(c\odot {(A·e)}^3)\right)$
${\displaystyle \frac{1}{24}}\left(-{\displaystyle \frac{1}{30}}+b·A·{(A·e)}^4\right)$	$-{\displaystyle \frac{1}{144}}+b·(c\odot A·A·A·c)$	$-{\displaystyle \frac{1}{144}}+b·(A·e\odot A·A·A·c)$
$-{\displaystyle \frac{1}{144}}+b·(c\odot A·A·A·A·e)$	$-{\displaystyle \frac{1}{144}}+b·(A·e\odot A·A·A·A·e)$	${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{72}}+b·(c\odot A·A·c^2)\right)$
${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{72}}+b·(A·e\odot A·A·c^2)\right)$	$-{\displaystyle \frac{1}{72}}+b·(c\odot A·A·(c\odot A·e))$	$-{\displaystyle \frac{1}{72}}+b·(A·e\odot A·A·(c\odot A·e))$
${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{72}}+b·(c\odot A·A·{(A·e)}^2)\right)$	${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{72}}+b·(A·e\odot A·A·{(A·e)}^2)\right)$	$-{\displaystyle \frac{1}{48}}+b·(c\odot A·(c\odot A·c))$
$-{\displaystyle \frac{1}{48}}+b·(A·e\odot A·(c\odot A·c))$	$-{\displaystyle \frac{1}{48}}+b·(c\odot A·(A·c\odot A·e))$	$-{\displaystyle \frac{1}{48}}+b·(A·e\odot A·(A·c\odot A·e))$
$-{\displaystyle \frac{1}{48}}+b·(c\odot A·(c\odot A·A·e))$	$-{\displaystyle \frac{1}{48}}+b·(A·e\odot A·(c\odot A·A·e))$	$-{\displaystyle \frac{1}{48}}+b·(c\odot A·(A·e\odot A·A·e))$
$-{\displaystyle \frac{1}{48}}+b·(A·e\odot A·(A·e\odot A·A·e))$	${\displaystyle \frac{1}{6}}\left(-{\displaystyle \frac{1}{24}}+b·(c\odot A·c^3)\right)$	${\displaystyle \frac{1}{6}}\left(-{\displaystyle \frac{1}{24}}+b·(A·c^3\odot A·e)\right)$
${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{24}}+b·(c\odot A·(c^2\odot A·e))\right)$	${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{24}}+b·(A·e\odot A·(c^2\odot A·e))\right)$	${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{24}}+b·(c\odot A·(c\odot {(A·e)}^2))\right)$
${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{24}}+b·(A·e\odot A·(c\odot {(A·e)}^2))\right)$	${\displaystyle \frac{1}{6}}\left(-{\displaystyle \frac{1}{24}}+b·(c\odot A·{(A·e)}^3)\right)$	${\displaystyle \frac{1}{6}}\left(-{\displaystyle \frac{1}{24}}+b·(A·e\odot A·{(A·e)}^3)\right)$
$-{\displaystyle \frac{1}{72}}+b·(A·c\odot A·A·c)$	$-{\displaystyle \frac{1}{72}}+b·(A·A·c\odot A·A·e)$	$-{\displaystyle \frac{1}{72}}+b·(A·c\odot A·A·A·e)$
$-{\displaystyle \frac{1}{72}}+b·(A·A·e\odot A·A·A·e)$	${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{36}}+b·(A·c\odot A·c^2)\right)$	${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{36}}+b·(A·c^2\odot A·A·e)\right)$
$-{\displaystyle \frac{1}{36}}+b·(A·c\odot A·(c\odot A·e))$	$-{\displaystyle \frac{1}{36}}+b·(A·(c\odot A·e)\odot A·A·e)$	${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{36}}+b·(A·c\odot A·{(A·e)}^2)\right)$
${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{36}}+b·(A·{(A·e)}^2\odot A·A·e)\right)$	${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{36}}+b·(c^2\odot A·A·c)\right)$	$-{\displaystyle \frac{1}{36}}+b·(c\odot A·e\odot A·A·c)$
${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{36}}+b·({(A·e)}^2\odot A·A·c)\right)$	${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{36}}+b·(c^2\odot A·A·A·e)\right)$	$-{\displaystyle \frac{1}{36}}+b·(c\odot A·e\odot A·A·A·e)$
${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{36}}+b·({(A·e)}^2\odot A·A·A·e)\right)$	${\displaystyle \frac{1}{4}}\left(-{\displaystyle \frac{1}{18}}+b·(c^2\odot A·c^2)\right)$	${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{18}}+b·(c\odot A·c^2\odot A·e)\right)$
${\displaystyle \frac{1}{4}}\left(-{\displaystyle \frac{1}{18}}+b·(A·c^2\odot {(A·e)}^2)\right)$	${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{18}}+b·(c^2\odot A·(c\odot A·e))\right)$	$-{\displaystyle \frac{1}{18}}+b·(c\odot A·e\odot A·(c\odot A·e))$
${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{18}}+b·({(A·e)}^2\odot A·(c\odot A·e))\right)$	${\displaystyle \frac{1}{4}}\left(-{\displaystyle \frac{1}{18}}+b·(c^2\odot A·{(A·e)}^2)\right)$	${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{18}}+b·(c\odot A·e\odot A·{(A·e)}^2)\right)$
${\displaystyle \frac{1}{4}}\left(-{\displaystyle \frac{1}{18}}+b·({(A·e)}^2\odot A·{(A·e)}^2)\right)$	${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{24}}+b·(c\odot {(A·c)}^2)\right)$	${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{24}}+b·({(A·c)}^2\odot A·e)\right)$
$-{\displaystyle \frac{1}{24}}+b·(c\odot A·c\odot A·A·e)$	$-{\displaystyle \frac{1}{24}}+b·(A·c\odot A·e\odot A·A·e)$	${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{24}}+b·(c\odot {(A·A·e)}^2)\right)$
${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{24}}+b·(A·e\odot {(A·A·e)}^2)\right)$	${\displaystyle \frac{1}{6}}\left(-{\displaystyle \frac{1}{12}}+b·(c^3\odot A·c)\right)$	${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{12}}+b·(c^2\odot A·c\odot A·e)\right)$
${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{12}}+b·(c\odot A·c\odot {(A·e)}^2)\right)$	${\displaystyle \frac{1}{6}}\left(-{\displaystyle \frac{1}{12}}+b·(A·c\odot {(A·e)}^3)\right)$	${\displaystyle \frac{1}{6}}\left(-{\displaystyle \frac{1}{12}}+b·(c^3\odot A·A·e)\right)$
${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{12}}+b·(c^2\odot A·e\odot A·A·e)\right)$	${\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{12}}+b·(c\odot {(A·e)}^2\odot A·A·e)\right)$	${\displaystyle \frac{1}{6}}\left(-{\displaystyle \frac{1}{12}}+b·({(A·e)}^3\odot A·A·e)\right)$
${\displaystyle \frac{1}{120}}\left(-{\displaystyle \frac{1}{6}}+b·c^5\right)$	${\displaystyle \frac{1}{24}}\left(-{\displaystyle \frac{1}{6}}+b·(c^4\odot A·e)\right)$	${\displaystyle \frac{1}{12}}\left(-{\displaystyle \frac{1}{6}}+b·(c^3\odot {(A·e)}^2)\right)$
${\displaystyle \frac{1}{12}}\left(-{\displaystyle \frac{1}{6}}+b·(c^2\odot {(A·e)}^3)\right)$	${\displaystyle \frac{1}{24}}\left(-{\displaystyle \frac{1}{6}}+b·(c\odot {(A·e)}^4)\right)$	${\displaystyle \frac{1}{120}}\left(-{\displaystyle \frac{1}{6}}+b·{(A·e)}^5\right)$

, we present the sixth order conditions in the same notation. The package introduced in [18] was used for delivering them.

There are various publications focusing on symbolic tools for order condition derivation along with contemporary progress in Runge–Kutta pair development, see [20].

3. The New Method

We continue by developing methods of order six utilizing eight stages. This corresponds to the typical count of function evaluations required to construct classical Runge–Kutta pairs of orders $6\left(5\right)$ [6,9]. The complete system of order conditions comprises 166 equations.

The first-row simplifying condition

$$A·e=c,$$

(6)

is satisfied only for indices beyond the first, due to the fact that $c_1\ne 0$. Here, we also demand

$$A·c={\displaystyle \frac{1}{2}}{c}^2,$$

(7)

for the same indices. By imposing $b_1=b_2=0$, we are able to fully leverage the advantages offered by these assumptions—specifically, we bypass the need to resolve any order conditions involving the term $A·e$. Additionally, we adopt the column simplification condition given by

$$b·(A+C-I_s)=0,$$

(8)

where $C=$ diag$\left(c\right)$ represents the diagonal matrix formed from the vector c, and $I_s\in {\mathbb{R}}^{s\times s}$ denotes the identity matrix of dimension s. We may also take in account the assumptions visible from Table 1. i.e.,

$$b_1=b_2=0,a_{31}=a_{41}=a_{51}=a_{61}=a_{71}=a_{81}=0,c_8=1,c_1={\displaystyle \frac{1}{2}}{c}_2,\mathrm{and}{c}_3=2c_2.$$

Thus, it suffices to simultaneously satisfy only the subset of order conditions enumerated in

Table 5. Conditions of orders 1–6, remaining after assumptions (6)–(8).

$b·e=1$,	$b·c=\frac{1}{2}$,	$b·c^2=\frac{1}{3}$,	$b·c^3=\frac{1}{4}$
$b·c^4={\displaystyle \frac{1}{5}}$,	$b·c^5={\displaystyle \frac{1}{6}}$,	$b·\left(c\odot A^{3}c\right)={\displaystyle \frac{1}{144}}$,	$b·\left(c\odot A^2{c}^2\right)={\displaystyle \frac{1}{72}},$
$b·\left(c\odot A·\left(c\odot Ac\right)\right)={\displaystyle \frac{1}{48}}$,	$b·\left(c\odot A{c}^3\right)={\displaystyle \frac{1}{24}}$,	$b·\left(Ac\odot A^{2}c\right)={\displaystyle \frac{1}{72}}$,	$b·\left(Ac\odot A{c}^2\right)={\displaystyle \frac{1}{36}},$
$b·\left(c^2\odot A^{2}c\right)={\displaystyle \frac{1}{36}}$,	$b·\left(c^2\odot A{c}^2\right)={\displaystyle \frac{1}{18}}$,	$b·\left(c\odot {\left(Ac\right)}^2\right)={\displaystyle \frac{1}{24}}$,	$b·\left(c^3\odot Ac\right)={\displaystyle \frac{1}{12}},$
$b·\left(c\odot A^{2}c\right)={\displaystyle \frac{1}{30}}$,	$b·\left(c\odot A{c}^2\right)={\displaystyle \frac{1}{15}}$,	$b·{\left(Ac\right)}^2={\displaystyle \frac{1}{20}}$,	$b·\left(c^2\odot Ac\right)={\displaystyle \frac{1}{10}},$
$b·A·\left(c\odot A^{3}e\right)={\displaystyle \frac{1}{180}}$.	$b·A^{5}e={\displaystyle \frac{1}{720}}$,	$b·\left(c\odot Ac\right)={\displaystyle \frac{1}{8}}$,

along with (6)–(8) for indices $2,3,\cdots ,8$. Whenever an operation is omitted there, we consider it as dot product. We also remark that $A^2=A·A$ and $A^3=A^2·A$ in that Table.

The newly developed method is designated as Oliver6, and all coefficients obtained through this construction process are presented in

Table 6. Oliver6 method with eight stages considered here.

${\displaystyle \frac{1144432963}{6643492122}}$
${\displaystyle \frac{5559514421}{16136633360}}$	$c_2$
${\displaystyle \frac{7848380347}{11390062741}}$	0	${\displaystyle \frac{7848380347}{11390062741}}$
${\displaystyle \frac{584707217}{6263009916}}$	0	${\displaystyle \frac{2038489149}{11710836283}}$	$-{\displaystyle \frac{979254389}{12133029106}}$
${\displaystyle \frac{7336443377}{14331032729}}$	0	${\displaystyle \frac{1218317615}{11868495787}}$	${\displaystyle \frac{926723527}{9607657285}}$	${\displaystyle \frac{2591978082}{8285874317}}$
${\displaystyle \frac{3366171338}{14345148073}}$	0	$-{\displaystyle \frac{2068185857}{3111019654}}$	${\displaystyle \frac{113735432}{10738351501}}$	${\displaystyle \frac{1535838404}{3124329735}}$	${\displaystyle \frac{2966794647}{7467690986}}$
${\displaystyle \frac{12514972339}{14503282389}}$	0	${\displaystyle \frac{532060426}{6995522011}}$	${\displaystyle \frac{534518399}{7459440327}}$	$-{\displaystyle \frac{7520958647}{16381048698}}$	${\displaystyle \frac{3969100659}{17188243175}}$	${\displaystyle \frac{6741483509}{7145956343}}$
1	0	$-{\displaystyle \frac{2996179351}{5024492031}}$	$-{\displaystyle \frac{8618258207}{3723209651}}$	${\displaystyle \frac{37501796467}{6250912484}}$	${\displaystyle \frac{50887946307}{11796949565}}$	$-{\displaystyle \frac{61337130011}{7621064844}}$	${\displaystyle \frac{25273357857}{15351082598}}$
b	0	0	$-{\displaystyle \frac{705037407}{23374354019}}$	${\displaystyle \frac{6763940556}{32112921161}}$	${\displaystyle \frac{3182558507}{7612909937}}$	${\displaystyle \frac{211210562}{2122521419}}$	${\displaystyle \frac{12318266848}{44188992159}}$	${\displaystyle \frac{247357763}{10656047973}}$

, accurate to 18 decimal places, making it suitable for double-precision computations.

It is also observed that the Oliver6 method possesses an interval of absolute stability of $(-5.45,0)$, which is quite satisfactory for a sixth-order explicit scheme, especially when compared to the stability intervals of $(-4.2,0)$ for DLMP6(5) and $(-3.9,0)$ for PD6(5).

4. Numerical Experiments

To implement the Oliver6 scheme, the user evaluates the first stage at $f_1=f(x_0+c_1\tau ,y_0)$ with $c_1={\displaystyle \frac{1144432963}{6643492122}}$, effectively bypassing the singularity at $x_0$. The remaining stages $f_i$ for $i=2,\dots ,8$ are computed via the standard explicit relation:

$$f_i=f\left(x_k+c_i\tau ,y_k+\tau \sum _{j=1}^{i-1}{a}_{ij}{f}_j\right)$$

The solution is updated using the formula:

$$y_1=y_0+\tau \sum _{i=3}^8{b}_i{f}_i$$

where $b_1=b_2=0$ as per the method’s construction.

4.1. Lane–Emden Problem

The Lane–Emden equation [21,22], describes the behavior of gases governed by classical thermodynamic principles. For this reason, we selected this single-parameter family of problems to evaluate the performance of the newly proposed method. The differential equation is defined as:

$$y^{″}\left(x\right)=-y{\left(x\right)}^{\lambda }-{\displaystyle \frac{2}{x}}{y}^{\prime }\left(x\right),$$

(9)

subject to the initial conditions $y\left(0\right)=1$ and $y^{\prime }\left(0\right)=0$.

It is well known that traditional explicit Runge–Kutta methods encounter difficulties at the starting point of integration, particularly at $x=0$. A practical workaround involves initiating the numerical solution slightly to the right of the origin. Although this technique proves effective—as we will later demonstrate—it still falls short compared to our proposed strategy.

Conventionally, the numerical integration of singular initial value problems requires the analytical determination of $y^{″}\left(0\right)$ via L’Hôpital’s rule to resolve the indeterminate form ${\displaystyle \frac{2}{x}}{y}^{\prime }\left(x\right)$ as $x\to 0$. For Equation (9), this substitution yields the relation $y^{″}\left(0\right)+2y^{″}\left(0\right)=-y{\left(0\right)}^{\lambda }$, from which $y^{″}\left(0\right)$ is derived to construct a starting Taylor series. This expansion is typically utilized to approximate the solution at a small displacement $ϵ>0$, effectively shifting the initial point to $y\left(ϵ\right)\approx y\left(0\right)+{\displaystyle \frac{1}{2}}{y}^{″}\left(0\right){ϵ}^2$ to avoid the $0/0$ indeterminacy at the origin. In contrast, the Oliver6 method bypasses this analytical preprocessing by evaluating the first stage at $x_0+c_1\tau$. Since $c_1\ne 0$, the denominator in the governing equation remains non-vanishing, allowing the solver to proceed directly from $x_0=0$ without the need for manual Taylor series initialization or the explicit calculation of higher-order derivatives.

We implemented the new scheme in MATLAB R2022b [23] available at Nat. Kapodistrian University of Athens. For benchmarking, we used the well-known Runge–Kutta pair of orders 6(5), known as PD6(5), introduced by Prince and Dormand [9]. This method also utilizes eight stages per step, consistent with the structure adopted by Oliver6. In addition, we considered a higher-accuracy Runge–Kutta pair denoted $8\left(7\right)$, implemented as ode78 in MATLAB—a widely adopted solver within contemporary Runge–Kutta frameworks [10]. This scheme requires 13 stages per integration step.

Since our investigation centers on the initial phase of the integration process, we evaluated the results after just a single step. The step sizes examined were $\tau ={\displaystyle \frac{1}{10}}$, $\tau ={\displaystyle \frac{1}{20}}$, and $\tau ={\displaystyle \frac{1}{100}}$. Simultaneously, different integer values for $\lambda$ were considered. Given that only $\lambda \in [0,5]$ is of physical relevance, we chose $\lambda =0,1,2,3,4,5$.

Closed-form solutions are available for only three cases, namely $\lambda =0,1$, and 5:

$$y\left(x\right)=1-{\displaystyle \frac{1}{6}}{x}^2,y\left(x\right)={\displaystyle \frac{1}{x}}sinx,\mathrm{and}y\left(x\right)={\left({\displaystyle \frac{x^2}{3}}+1\right)}^{-1/2},$$

respectively. For $\lambda =2,3,4$, we generated reference values using highly accurate numerical solvers within Mathematica. These outcomes are summarized in

Table 7. Double precision approximations of $y\left(\tau \right)$ for $\lambda =2,3,4$.

	$\tau =0.1$	$\tau =0.05$	$\tau =0.01$
$\lambda =2$	${\displaystyle \frac{58254745}{58351901}}$	${\displaystyle \frac{50394000}{50415001}}$	${\displaystyle \frac{26048406343}{26048840486}}$
$\lambda =3$	${\displaystyle \frac{41995999}{42066004}}$	${\displaystyle \frac{134463997}{134520026}}$	${\displaystyle \frac{599999}{600009}}$
$\lambda =4$	${\displaystyle \frac{84127009}{84267174}}$	${\displaystyle \frac{89630667}{89668010}}$	${\displaystyle \frac{6403581345}{6403688071}}$

.

It is worth noting that sixth-order methods tend to offer optimal performance when moderate precision is sufficient—typically in the range of 6 to 8 significant digits. For problems requiring higher accuracy, methods of elevated order are more appropriate. Conversely, for rough tolerances, lower-order integrators are preferable. This explains the motivation behind the development of a variety of schemes and method pairs across different orders.

The outcomes for all schemes following a single integration step are presented in

Table 8. Accurate digits after one step $\tau$ for problem (9) and $\lambda =0$.

$\lambda =0$	$\tau =0.1$	$\tau =0.05$	$\tau =0.01$
Oliver6	$15.7$	$15.7$	$15.7$
PD$6\left(5\right)$	$1.9$	$2.5$	$3.9$
ode78	$2.5$	$3.1$	$4.5$

,

Table 9. Accurate digits after one step $\tau$ for problem (9) and $\lambda =1$.

$\lambda =1$	$\tau =0.1$	$\tau =0.05$	$\tau =0.01$
Oliver6	$9.0$	$10.0$	$12.6$
PD$6\left(5\right)$	$1.9$	$2.5$	$3.9$
ode78	$2.5$	$3.1$	$4.5$

,

Table 10. Accurate digits after one step $\tau$ for problem (9) and $\lambda =2$.

$\lambda =2$	$\tau =0.1$	$\tau =0.05$	$\tau =0.01$
Oliver6	$7.0$	$8.7$	$12.2$
PD$6\left(5\right)$	$1.9$	$2.5$	$3.9$
ode78	$2.5$	$3.1$	$4.5$

,

Table 11. Accurate digits after one step $\tau$ for problem (9) and $\lambda =3$.

$\lambda =3$	$\tau =0.1$	$\tau =0.05$	$\tau =0.01$
Oliver6	$6.5$	$8.3$	$12.0$
PD$6\left(5\right)$	$1.9$	$2.5$	$3.9$
ode78	$2.5$	$3.1$	$4.5$

,

Table 12. Accurate digits after one step $\tau$ for problem (9) and $\lambda =4$.

$\lambda =4$	$\tau =0.1$	$\tau =0.05$	$\tau =0.01$
Oliver6	$6.2$	$8.0$	$11.8$
PD$6\left(5\right)$	$1.9$	$2.5$	$3.9$
ode78	$2.5$	$3.1$	$4.5$

and

Table 13. Accurate digits after one step $\tau$ for problem (9) and $\lambda =5$.

$\lambda =5$	$\tau =0.1$	$\tau =0.05$	$\tau =0.01$
Oliver6	$6.0$	$7.8$	$11.7$
PD$6\left(5\right)$	$1.9$	$2.5$	$3.9$
ode78	$2.5$	$3.1$	$4.5$

. For PD6(5) and ode78, the integration began at $x_0={10}^{-15}$ to circumvent potential issues associated with evaluating $f_1=f(x_0,y_0)=f(0,y_0)$. In cases where the exact solution is available, employing it directly offers limited insight—particularly in trivial scenarios such as $\lambda =0$, then (using $ϵ={10}^{-15}$)

$$y\left({10}^{-15}\right)\approx 1-{\displaystyle \frac{1}{6}}\times {10}^{-30}=1\mathrm{in}\mathrm{double}\mathrm{precision}.$$

Thus, we may verify the rightmost lower figure from Table 8, writing:

>> sol=ode78(@(x,y) [y(2);-2*y(2)/x-1],[1e-15 .01],[1 0]’, …

odeset(’MaxStep’,1,’Initialstep’,.01,’abstol’,1e30,’reltol’,1e30));

>> sol.stats.nsteps

ans =

     1

>> log10(abs(sol.y(1,end)-(1-0.01^2/6)))

ans =

   -4.4842

• rounded to $4.5$ for the presentation. The corresponding run for the new method can be retrieved from http://users.uoa.gr/~tsitourasc/oliver6.m (accessed on 29 January 2026) in MATLAB format. Also, the coefficients for Oliver6 method can be found there. Observe that if the correct starting point 0 is put in the above run instead of 1e-15, then the method fails completely to start. Conventional Runge–Kutta pairs may give competitive accuracies at a very high cost. Thus, in the following run we may achieve an interesting accuracy at hundredfold cost.

>> sol=ode78(@(x,y) [y(2);-2*y(2)/x-1],[1e-15 .01],[1 0]’, …

odeset(’abstol’,1e-13,’reltol’,1e-13));

>> log10(abs(sol.y(1,end)-(1-0.01^2/6)))

ans =

  -15.9546

>> sol.stats

ans =

  struct with fields:

     nsteps: 28

    nfailed: 32

    nfevals: 860

The proposed scheme delivers an accuracy improvement ranging from $3.5$ to $13.5$ digits when compared to PD6(5) and ode78. On average, it achieves roughly 6 additional correct digits. This constitutes a substantial enhancement, especially given that all methods share the same algebraic order and utilize an equal number of stages. Note that when $\lambda =0$, Oliver6 achieves maximum accuracy, effectively reaching machine precision—approximately $15.7$ according to log10(eps). Higher accuracies are practically of no meaning in double precision and MATLAB.

A practical assessment of the convergence rate via Runge’s rule is omitted for these problems because the Lipschitz condition is violated at the left endpoint, rendering classical algebraic order analysis inapplicable for the initial integration step.

Implicit Runge–Kutta pairs may handle better initial point evaluations and deliver accurate results, but at a very high cost. Implicit Euler (where $c_1=1$) is able to deliver 10–11 digits of accuracy after spending millions of function evaluations. Methods radau5 and radau9 [24] cannot start from $x_0=0$ even if $c_1\ne 0$ for them also. This is due to internal evaluation of $f(x_0,y_0)$ for initial residuals, estimation of the initial Jacobian, step size estimation, and Newton iteration.

Notably, the method also performs well for non-integer values of $\lambda$. For instance, considering $\lambda ={\displaystyle \frac{5}{2}}$, we have the approximations:

$$y\left(0.1\right)\approx {\displaystyle \frac{90736784}{90888075}},y\left(0.05\right)\approx {\displaystyle \frac{53766399}{53788804}},y\left(0.01\right)\approx {\displaystyle \frac{239999}{240003}},$$

with the corresponding numerical outcomes presented in

Table 14. Accurate digits after one step $\tau$ for problem (9) and $\lambda ={\displaystyle \frac{5}{2}}$.

$\lambda =\frac{5}{2}$	$\tau =0.1$	$\tau =0.05$	$\tau =0.01$
Oliver6	$6.7$	$8.5$	$12.1$
PD$6\left(5\right)$	$1.9$	$2.5$	$3.9$
ode78	$3.0$	$3.1$	$4.5$

.

The proposed method appears to surpass the performance of both PD6(5) and ode78, even when applied to alternative formulations of the Lane–Emden problem. One such example is the equation modeling isothermal gas spheres:

$$y^{″}\left(x\right)+{\displaystyle \frac{2}{x}}{y}^{\prime }\left(x\right)+exp\left(y\right(x\left)\right)=0,y\left(0\right)=0,y^{\prime }\left(0\right)=0.$$

(10)

An approximate analytical expression for the solution, as reported in [25], is given by:

$$y\left(x\right)\approx x^2\left(x^2\left(x^2\left(x^2\left(-{\displaystyle \frac{4087x^2}{1796256000}}-{\displaystyle \frac{61}{1632960}}\right)-{\displaystyle \frac{1}{540}}\right)-{\displaystyle \frac{1}{120}}\right)-{\displaystyle \frac{1}{6}}\right).$$

This degree of precision is adequate for our comparative evaluation within the framework of double-precision floating-point arithmetic. The corresponding numerical outcomes are presented in

Table 15. Accurate digits after one step $\tau$ for problem (10).

	$\tau =0.1$	$\tau =0.05$	$\tau =0.01$
Oliver6	$5.8$	$7.0$	$9.8$
PD$6\left(5\right)$	$1.9$	$2.5$	$3.9$
ode78	$2.7$	$3.1$	$4.5$

.

No improved results were observed when using other well-known classical Runge–Kutta pairs. Their performance remained comparable to that of PD6(5) and ode78.

4.2. The Rational Smooth Transition

This problem represents a class of equations where the singularity is “removable” if and only if the initial condition is chosen precisely to suppress the divergent homogeneous solution. It has the form

$$y^{\prime }+{\displaystyle \frac{2}{x}}y=3,y\left(0\right)=0$$

and exact solution: $y\left(x\right)=x$.

In physical systems, this structure often appears in spherical growth models. Consider a substance diffusing into a sphere where the concentration y depends on the radius x. The term ${\displaystyle \frac{2}{x}}{y}^{\prime }$ (or its first-order reduction) accounts for the geometric “crowding” as one moves toward the center ($x\to 0$), while the equation is singular at the origin, the physical reality requires the concentration to be finite and smooth at the center.

The general solution is $y\left(x\right)=x+{\displaystyle \frac{C}{x^2}}$. Any choice of $C\ne 0$ would result in an infinite concentration at the center, which is physically impossible. Thus, nature “chooses” the $C=0$ path.

We again tested various step lengths under the assumption that the exact solution is unknown, setting $y\left({10}^{-15}\right)=0$ for PD6(5) and ode78. The corresponding results are reported in

Table 16. Accurate digits after one step $\tau$ for Rational Smooth Transition.

	$\tau =1$	$\tau =0.5$	$\tau =0.1$
Oliver6	$14.4$	$14.8$	$15.4$
PD$6\left(5\right)$	$-0.7$	$-0.4$	$0.3$
ode78	$0.0$	$0.3$	$1.0$

. Notably, the proposed method achieves accuracy close to machine precision, whereas the other methods fail to produce meaningful results.

4.3. The Exponential Balancer

The Exponential Balancer is a more complex test case where both the coefficient and the forcing term are singular, but their Taylor series expansions perfectly align to allow a bounded solution. It is given by

$$y^{\prime }+{\displaystyle \frac{2}{x}}y={\displaystyle \frac{e^x}{x}},y\left(0\right)={\displaystyle \frac{1}{2}},$$

(11)

with exact solution:

$$y\left(x\right)={\displaystyle \frac{e^x(x-1)+1}{x^2}}.$$

This equation models radiative transport or heat generation within a cylinder or sphere where the source term ($e^x$) is non-linear. The singularity at $x=0$ is a coordinate singularity, not a physical one.

To see why $y\left(0\right)=0.5$, we examine the Taylor expansion of the solution:

$$y\left(x\right)={\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{3}}x+{\displaystyle \frac{1}{8}}{x}^2+{\displaystyle \frac{1}{30}}{x}^3+\dots$$

(12)

As $x\to 0$, $y\left(x\right)\to 0.5$ and $y^{\prime }\left(x\right)\to 1/3$. This demonstrates that despite the $1/x$ terms, the physical quantity y (such as temperature or pressure) transitions smoothly through the origin.

We repeated the experiments setting again $y\left({10}^{-15}\right)=0$ for PD6(5) and ode78 and using different step lengths. The associated results are reported in

Table 17. Accurate digits after one step $\tau$ for Exponential Balancer.

	$\tau =0.1$	$\tau =0.05$	$\tau =0.01$
Oliver6	$6.0$	$6.9$	$9.0$
PD$6\left(5\right)$	$0.7$	$1.0$	$1.7$
ode78	$1.4$	$1.7$	$2.4$

. Notably, the proposed method demonstrates consistently strong performance, while the remaining methods exhibit comparatively poor results.

The conventional methods may furnish better results. Indeed, after using especially

$$x_0={10}^{-10}\mathrm{and}{y}_0={\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{3}}{x}_0,$$

(13)

we may get the results shown in

Table 18. Accurate digits after one step $\tau$ for Exponential Balancer and use of (13).

	$\tau =0.1$	$\tau =0.05$	$\tau =0.01$
PD$6\left(5\right)$	$3.6$	$4.1$	$5.6$
ode78	$4.5$	$5.1$	$6.5$

. Of course, selecting an optimal $x_0$ is not a straightforward task; moreover, using the solution (12) to initiate the procedure presents an additional challenge. Even this constitutes an unfair comparison for Oliver 6; nevertheless, the latter method, which uses standard initials from (11), again yields superior results.

4.4. The Thomas–Fermi Model as a Singular IVP

The Thomas–Fermi model remains one of the most significant semi-classical descriptions of the electron distribution in heavy atoms [26,27]. Mathematically, it is formulated as a second-order, nonlinear ordinary differential equation (ODE) that relates the dimensionless electrostatic potential y to the radial distance x.

The governing equation and boundary conditions for the Thomas–Fermi equation are given by:

$${\displaystyle \frac{d^{2}y}{d{x}^2}}={\displaystyle \frac{1}{\sqrt{x}}}{y}^{3/2},x\in (0,\infty )$$

(14)

The left boundary $x=0$ represents the location of the nucleus. At this point, the screening effect of the electron cloud is zero, leading to the first initial condition: $y\left(0\right)=1$.

To determine the unique solution for a neutral atom, a second condition is imposed at infinity, requiring the potential to vanish: ${lim}_{x\to \infty }y\left(x\right)=0$. This boundary value problem is typically converted into an Initial Value Problem (IVP) by determining the specific initial slope B:

$$y^{\prime }\left(0\right)=B\approx -1.588071022617731$$

Standard Taylor series expansions fail at the origin because the second derivative $y^{″}\left(x\right)$ diverges as $x\to 0$ due to the $x^{-1/2}$ term. Instead, the solution must be expressed as a Puiseux series, commonly referred to as the Baker series. This expansion incorporates fractional powers to accommodate the singular behavior:

$$y\left(x\right)=1+Bx+{\displaystyle \frac{4}{3}}{x}^{3/2}+{\displaystyle \frac{2B}{5}}{x}^{5/2}+{\displaystyle \frac{1}{3}}{x}^3+{\displaystyle \frac{3B^2}{70}}{x}^{7/2}+\dots$$

The presence of the $x^{3/2}$ term indicates that while the function and its first derivative are finite at the origin, the second derivative is non-analytic. This series is essential for initializing numerical solvers, as it allows for an accurate transition from the singular point $x=0$ to a small displacement $ϵ$ where standard integration techniques, such as the Runge–Kutta method, become stable.

The constant B is a fundamental parameter of the model. Physically, it represents the initial rate of screening. If $\left|B\right|$ is smaller than the critical value, the solution eventually diverges, representing a “compressed” atom. Conversely, if $\left|B\right|$ is larger, the solution reaches $y=0$ at a finite radius, which is used to model positive ions. The neutral atom solution exists precisely on the separatrix between these two regimes. We repeated the experiments using different step lengths, and the associated results are reported in

Table 19. Accurate digits after one step $\tau$ for Thomas–Fermi model ($x_0={10}^{-15}$).

	$\tau =0.1$	$\tau =0.01$	$\tau =0.001$
Oliver6	$2.1$	$3.6$	$5.1$
PD$6\left(5\right)$	$-5.0$	$-2.4$	$-0.3$
ode78	$-4.8$	$-2.1$	$-0.1$

. The other methods started at $ϵ={10}^{-15}$. The behavior of PD6(5) and ode78 are clearly out of order. Not only are they exploding for large steps, but they also move to the imaginary axis, furnishing complex results.

In case we admit heavy use of known results,

$$y\left(ϵ\right)\approx 1+Bϵ+{\displaystyle \frac{4}{3}}{ϵ}^{1.5}\mathrm{and}{y}^{\prime }\left(ϵ\right)\approx B+2{ϵ}^{{\displaystyle \frac{1}{2}}}+B{ϵ}^{1.5}+{ϵ}^2,$$

(15)

and move right to $x_0=ϵ={10}^{-6}$, then PD6(5) and ode78 may present acceptable results as we see in

Table 20. Accurate digits for Thomas–Fermi model, after one step $\tau$ with $x_0={10}^{-6}$ and (15).

	$\tau =0.1$	$\tau =0.01$	$\tau =0.001$
PD$6\left(5\right)$	$0.1$	$2.2$	$4.3$
ode78	$0.2$	$2.4$	$4.5$

. Once again, determining an optimal $x_0$ and employing (15) are not tasks that are easy to accomplish. This comparison remains biased against Oliver 6; however, the latter method, once more initialized at standard $x_0=0$, still produces superior results.

5. Discussion

We considered here a class of methods that appears to be a noteworthy contribution when applied to the Lane–Emden equation and similar problems. These methods prove particularly effective in initializing the integration process, as they can successfully handle the singularity at the left endpoint. Nevertheless, their utility is largely limited to this initial phase. Once the integration has advanced beyond the first step, higher-order methods may be employed. Nevertheless, no conventional explicit Runge–Kutta pair can properly start on the problems under consideration, as they fail to address the singularity at the left boundary. A possible workaround is to slightly shift the starting point to the right; for example, by choosing $x_0={10}^{-15}$, since otherwise no integration can begin. This approach, however, introduces a fixed error in the initial derivative, which affects the accuracy of the numerical solution throughout the entire integration interval. A key drawback is that these methods require very small step sizes to maintain low error, which significantly raises their computational cost.

None of the standard MATLAB ODE solvers—ode23, ode23s, ode23t, ode23tb, ode45, ode78, ode89, ode113, ode15s—is capable of initiating integration from the true initial point in the problems examined here. To attain accuracy comparable to that achieved by Oliver6 with just 8 stages, these solvers often require hundreds or even thousands of function evaluations. Other implicit methods, such as radau5 and radau9, also fail to effectively initialize the integration process. All of these methods demand substantial computational effort during the initial stages in order to achieve an acceptable level of accuracy, ultimately rendering their overall performance less efficient than that of the proposed method over the entire integration interval.

6. Conclusions

This study presents a novel approach within the broader class of Runge–Kutta–Oliver (RKO) methods, which generalize classical Runge–Kutta techniques by relaxing standard simplifying assumptions. Specifically, we examined methods where the first row condition $A·e=c$ is not imposed, leading to increased flexibility in method construction. Utilizing a rooted-tree framework with two-colored leaves, we derived a comprehensive set of order conditions up to sixth algebraic order. Based on this foundation, we constructed a new method, denoted Oliver6, which achieves sixth-order accuracy using eight stages.

Our numerical investigations demonstrate that Oliver6 significantly outperforms established methods such as PD6(5) and ode78, especially when applied near initial singularities. In many test cases, the method achieved up to 6 more correct digits of accuracy per step compared to its classical counterparts. Importantly, Oliver6 attains this improvement without increasing the computational burden, maintaining the same number of stages.

The results suggest that RKO methods, despite their increased complexity in order condition derivation, offer notable advantages in accuracy and stability for problems with challenging initial behavior. Future work may focus on adaptive variants and their application to broader classes of singular or stiff initial value problems.