In this paper, we focus on the following sstage modified implicit RKN method for the secondorder ODEs (1):
$$ \textstyle\begin{cases} Y_{i}=y_{0}+c_{i}\gamma_{i}h{y'_{0}}+h^{2}\sum_{j=1}^{s}a_{ij}f(t_{0}+c_{j}h,Y_{j}),\quad i=1,\ldots,s, \\ y_{1}=y_{0}+g_{2}h{y'_{0}}+h^{2}\sum_{i=1}^{s}\bar{b}_{i}f(t_{0}+c_{i}h,Y_{i}), \\ y'_{1}={y'_{0}}+h\sum_{i=1}^{s}b_{i}f(t_{0}+c_{i}h,Y_{i}). \end{cases} $$
(3)
This modified RKN method can be expressed by the Butcher tableau
c
 e 
γ
 A 
 1 
\(g_{2}\)

\(\bar{b}^{T}\)

  1 
\(b^{T}\)

=
\(c_{1}\)
 1 
\(\gamma_{1}\)

\(a_{11}\)
 ⋯ 
\(a_{1s}\)

⋮  ⋮  ⋮  ⋮  ⋱  ⋮ 
\(c_{s}\)
 1 
\(\gamma_{s}\)

\(a_{s1}\)
 ⋯ 
\(a_{ss}\)

 1 
\(g_{2}\)

\(\bar{b}_{1}\)
 ⋯ 
\(\bar{b}_{s}\)

  1 
\(b_{1}\)
 ⋯ 
\(b_{s}\)

Note that scheme (3) coincides with the classical sstage RKN formula when the coefficients \(g_{2}=1\), \(\gamma_{i}=1\), \(i=1,\ldots,s\), and the remaining coefficients are constant. The objective of this section is to find out when the modified RKN scheme (3) is symmetric, symplectic, exponentially fitted respectively. In the following subsections, we put forward these three important properties one by one.
2.1 Symmetric conditions
The concept of adjoint method is the hinge of symmetry. First, let us give the definition of adjoint method. We denote a onestep method for secondorder ODEs (1) as \(\varPhi_{h}: (y_{0},y'_{0})^{\mathrm{T}}\mapsto(y_{1},y'_{1})^{\mathrm{T}}\).
Definition 2.1
The adjoint method \(\varPhi_{h}^{*}\) of a onestep method \(\varPhi_{h}\) is the inverse map of the original method with reversed time step −h, i.e., \(\varPhi_{h}^{*}:=\varPhi_{h}^{1}\). In other words, \(y_{1}=\varPhi_{h}^{*}(y_{0})\) is implicitly defined by \(\varPhi_{h}(y_{1})=y_{0}\). A method for which \(\varPhi_{h}^{*}=\varPhi_{h}\) is called symmetric (see [11]).
In the case of sstage RKN methods (3), a set of sufficient conditions for the methods to be symmetric are given by
$$ \begin{aligned} &c_{i}=1c_{s+1i}, \qquad b_{i}=b_{s+1i}, \\ &\bar{b}_{i}=g_{2}b_{s+1i}\bar{b}_{s+1i}, \qquad c_{i}\gamma_{i}=g_{2}c_{s+1i} \gamma_{s+1i}, \\ &a_{ij}=a_{s+1i,s+1j}c_{s+1i}\gamma _{s+1i}b_{s+1j}+g_{2}b_{s+1j} \bar{b}_{s+1j}. \end{aligned} $$
(4)
This can be obtained following the procedure in [32] and many other papers. In this paper we consider methods (3) whose coefficients are zdependent, as we do for EF type methods [28, 29] or adapted type methods [25, 26]. Then the conditions for the methods to be symmetric are given by
$$\begin{aligned} &c_{i}(z)=1c_{s+1i}(z),\qquad b_{i}(z)=b_{s+1i}(z), \\ &\bar{b}_{i}(z)=g_{2}(z)b_{s+1i}(z) \bar{b}_{s+1i}(z), \\ &c_{i}(z)\gamma_{i}(z)=g_{2}(z)c_{s+1i}(z) \gamma_{s+1i}(z), \\ & a_{ij}(z)=a_{s+1i,s+1j}(z)c_{s+1i}(z) \gamma_{s+1i}(z)b_{s+1j}(z) +g_{2}(z)b_{s+1j}(z)\bar{b}_{s+1j}(z), \end{aligned}$$
where \(z=i \omega h\), ω is the principal frequency of the problem. We assume that the coefficients of methods (3) are even functions of h, as we frequently encounter in the case of EFRKN methods, so that these conditions reduce to (4).
2.2 Symplectic conditions
The second important property used in this paper is symplecticity. Now, we turn to the symplectic conditions for scheme (3). Symplecticity is defined for a Hamiltonian system. On many occasions, the problem under consideration takes the form of a Hamiltonian system
$$ \dot{p}=\frac{\partial}{\partial q}U(t,q), \qquad\dot{q}=M^{1}p, $$
with the Hamiltonian \(H(t,p,q)=\frac{1}{2}p^{\mathrm{T}}M^{1}p + U(t,q)\), where M is a symmetric positive definite constant matrix. This system is equivalent to the secondorder equation (1) with \(f(t,q)=M^{1}\frac{\partial}{\partial q}U(t,q)\). Now we can give the definition of symplecticity which can be found in [10].
Definition 2.2
A onestep method is symplectic if, for every smooth Hamiltonian function H and for every step size h, the corresponding flow preserves the differential 2form
$$ \mathrm{d}p\wedge\mathrm{d}q=\sum_{i=1}^{d} \mathrm{d}p_{i}\wedge\mathrm{d}q_{i}, $$
where the oneforms \(\mathrm{d}p_{i}\), respectively \(\mathrm{d}q_{i}\), map a tangent vector ξ to its ith, respectively \((n+i)\)th, component. Here, we assume that p and q all have n components. Furthermore, \(\mathrm{d}p_{i}\wedge\mathrm {d}q_{i}\) is a bilinear map acting on a pair of vectors
$$\begin{aligned} \mathrm{d}p_{i}\wedge\mathrm{d}q_{i}(\xi_{1}, \xi_{2})&=\det \begin{pmatrix} \mathrm{d}p_{i}(\xi_{1}) & \mathrm{d}p_{i}(\xi_{2}) \\ \mathrm{d}q_{i}(\xi_{1}) & \mathrm{d}q_{i}(\xi_{2}) \end{pmatrix} \\ &=\mathrm{d}p_{i}(\xi_{1})\,\mathrm{d}q_{i}(\xi_{2})  \mathrm{d}p_{i}(\xi_{2})\,\mathrm{d}q_{i}(\xi_{1}) \end{aligned}$$
and satisfies Grassmann’s rules for exterior multiplication
$$\begin{aligned} \mathrm{d}p_{i}\wedge\mathrm{d}p_{j} =  \mathrm{d}p_{j}\wedge\mathrm{d}p_{i}, \qquad \mathrm{d}p_{i}\wedge\mathrm{d}p_{i} = 0. \end{aligned}$$
Accordingly, scheme (3) is symplectic if
$$ \mathrm{d}y_{1}\wedge\mathrm{d}y'_{1}= \mathrm{d}y_{0}\wedge\mathrm{d}y'_{0}. $$
Using the expressions of \(y_{1}\) and \(y'_{1}\) in (3), we have
$$\begin{aligned} \mathrm{d}y_{1}\wedge\mathrm{d}y'_{1} =& \mathrm{d}y_{0}\wedge\mathrm{d}y'_{0}+h\sum _{i=1}^{s}b_{i}\,\mathrm{d}y_{0} \wedge\mathrm{d}f(Y_{i}) +hg_{2}\,\mathrm{d}y'_{0}\wedge \mathrm{d}y'_{0}+h^{2}\sum _{i=1}^{s}g_{2}b_{i}\, \mathrm{d}y'_{0}\wedge\mathrm{d}f(Y_{i}) \\ &{}+h^{2}\sum_{i=1}^{s} \bar{b}_{i}\,\mathrm{d}f(Y_{i})\wedge\mathrm{d}y'_{0}+h^{3} \sum_{i,j=1}^{s}\bar{b}_{j}b_{i}\, \mathrm{d}f(Y_{j})\wedge\mathrm{d}f(Y_{i}). \end{aligned}$$
Eliminating \(\mathrm{d}y_{0}\) in the second term by inserting the first equation in (3), we obtain
$$\begin{aligned} \mathrm{d}y_{1}\wedge\mathrm{d}y'_{1} =& \mathrm{d}y_{0}\wedge\mathrm{d}y'_{0}+h^{2} \sum_{i=1}^{s}(g_{2}b_{i}b_{i}c_{i} \gamma_{i}\bar{b}_{i})\,\mathrm{d}y'_{0} \wedge\mathrm{d}f(Y_{i}) \\ &{}+\frac{1}{2}h^{3}\sum_{i,j=1}^{s}( \bar{b}_{j}b_{i}b_{i}a_{ij})\, \mathrm{d}f(Y_{j})\wedge\mathrm{d}f(Y_{i}) \\ &{}+\frac{1}{2}h^{3}\sum_{i,j=1}^{s}( \bar{b}_{i}b_{j}b_{j}a_{ji})\, \mathrm{d}f(Y_{i})\wedge\mathrm{d}f(Y_{j}). \end{aligned}$$
Therefore, (3) is symplectic if the following conditions are satisfied:
$$ \begin{aligned} &\bar{b}_{i}+(c_{i} \gamma_{i}g_{2})b_{i}=0,\quad i=1,\ldots,s, \\ &b_{i}(\bar{b}_{j}a_{ij})=b_{j}( \bar{b}_{i}a_{ji}),\quad i,j=1,\ldots,s. \end{aligned} $$
(5)
2.3 Exponential fitting conditions
Following Albrecht’s approach [2, 3], each stage of scheme (3) can be viewed as a linear multistep method on a nonequidistant grid. With each stage one can associate a linear function as follows:

for the internal stages,
$$ \varphi_{i} \bigl[y(t);h;{\mathbf {a}} \bigr]=y(t+c_{i}h)y(t)c_{i} \gamma_{i}hy'(t)h^{2}\sum _{j=1}^{s}a_{ij}y''(t+c_{j}h), \quad i=1,2,\ldots,s; $$

for the final stages,
$$ \begin{aligned} &\varphi \bigl[y(t);h;{\bar{\mathbf {b}}} \bigr]=y(t+h)y(t)hg_{2}y'(t)h^{2} \sum _{i=1}^{s}\bar{b}_{i}y''(t+c_{i}h), \\ &\varphi \bigl[y(x);h;{\mathbf {b}} \bigr]=y'(t+h)y'(t)h \sum_{i=1}^{s}b_{i}y''(t+c_{i}h). \end{aligned} $$
The following equations can be obtained by requiring that these functions vanish for the functions from the set \(\{\exp(\pm\lambda t) \lambda\in\mathbb{R} \mbox{ or } \lambda \in i\mathbb{R}\}\) (here i is the imaginary unit)
$$ \textstyle\begin{cases} e^{\pm c_{i}z}=1\pm c_{i}\gamma_{i}(z)z+ z^{2}\sum_{j=1}^{s}a_{ij}(z)e^{\pm c_{j}z}\quad (\text{for internal stage ${\bar{\mathbf{a}}}$}), \\e^{\pm z}=1\pm g_{2}z+z^{2}\sum_{i=1}^{s}\bar{b}_{i}(z)e^{\pm c_{i}z}\quad (\text{for the final stage ${\bar{\mathbf{b}}}$}), \\e^{\pm z}=1\pm z\sum_{i=1}^{s}b_{i}(z)e^{\pm c_{i}z},\quad z=\lambda h\ (\text{for the final stage ${\mathbf{b}}$}). \end{cases} $$
(6)
By definitions of \(\cosh(z)=(e^{z} + e^{z})/2\) and \(\sinh (z)=(e^{z}e^{z})/2\), equation (6) implies that
$$\begin{aligned} & \textstyle\begin{cases} \sum_{j=1}^{s}a_{ij}(z)\cosh(c_{j}z)=\frac{\cosh(c_{i}z)1}{z^{2}}, \\ \sum_{j=1}^{s}a_{ij}(z)\sinh(c_{j}z)=\frac{\sinh(c_{i}z)c_{i}\gamma _{i}(z)z}{z^{2}},\quad i=1,\ldots,s, \end{cases}\displaystyle \end{aligned}$$
(7)
$$\begin{aligned} & \textstyle\begin{cases} \sum_{i=1}^{s}\bar{b}_{i}(z)\cosh(c_{i}z)=\frac{\cosh(z)1}{z^{2}}, \qquad \sum_{i=1}^{s}\bar{b}_{i}(z)\sinh(c_{i}z)=\frac{\sinh(z)g_{2}z}{z^{2}}, \\ \sum_{i=1}^{s}b_{i}(z)\sinh(c_{i}z)=\frac{\cosh(z)1}{z}, \qquad \sum_{i=1}^{s}b_{i}(z)\cosh(c_{i}z)=\frac{\sinh(z)}{z}. \end{cases}\displaystyle \end{aligned}$$
(8)
Equations (7) and (8) are the exponentially fitted conditions for scheme (3). Until now, we have obtained the three pivotal properties used in our method in Sect. 3.
2.4 Algebraic order conditions
In this subsection, we will present the algebraic order conditions for exponentially fitted Runge–Kutta–Nystöm (EFRKN) methods. As it occurs in the case of a classical RKN method, for an EFRKN method the local truncation errors in the approximations of the solution and its derivative can be expressed as
$$\begin{aligned} &e_{n+1}=y(t_{0}+h)y_{1}=\sum _{j=1}^{p1}h^{j+1} \Biggl(\sum _{i=1}^{k_{j}}{d_{i}^{(j+1)}F^{(j)}(y_{0})} \Biggr)+O \bigl(h^{p+1} \bigr), \\ &e'_{n+1}=y'(t_{0}+h)y'_{1}= \sum_{j=1}^{p}h^{j} \Biggl(\sum _{i=1}^{k_{j}}{{d'}_{i}^{(j)}F^{(j)}(y_{0})} \Biggr)+O \bigl(h^{p+1} \bigr), \end{aligned}$$
where \(F^{(j)}(y_{0})\) denotes an elementary differential and the terms \(d_{i}^{(j+1)}\) and \({d'}_{i}^{(j)}\) depend on the coefficients of the EFRKN method. Method (3) is of order p if, for every sufficiently smooth IVP(1) and for every small step size h, the local truncation errors of the numerical solutions satisfy
$$\begin{aligned} &e_{1}=y(t_{0}+h)y_{1}=O \bigl(h^{p+1} \bigr), \\ &e'_{1}=y'(t_{0}+h)y'_{1}=O \bigl(h^{p+1} \bigr), \end{aligned}$$
or equivalently,
$$\begin{aligned} &d_{i}^{(j+1)}=0,\quad i=1,\ldots,k_{j},j=1, \ldots,p1, \\ &{d'}_{i}^{(j)}=0,\quad i=1, \ldots,k_{j},j=1,\ldots,p. \end{aligned}$$
In order to obtain the order conditions, we assume the following expansions:
$$\begin{aligned} &\bar{b}_{i}(z)=\bar{b}_{i}^{(0)}+ \bar{b}_{i}^{(2)}z^{2}+\bar{b}_{i}^{(4)}z^{4}+ \cdots,\qquad b_{i}(z)=b_{i}^{(0)}+b_{i}^{(2)}z^{2}+b_{i}^{(4)}z^{4}+ \cdots, \\ &\gamma_{i}(z)=1+\gamma_{i}^{(2)}z^{2}+ \gamma_{i}^{(4)}z^{4}+\cdots,\qquad a_{ij}(z)=a_{ij}^{(0)} a_{ij}^{(2)}z^{2}+a_{ij}^{(4)}z^{4}+ \cdots, \\ &g_{2}(z)=1+g_{2}^{(2)}z^{2}+g_{2}^{(4)}z^{4}+ \cdots. \end{aligned}$$
Then the order conditions up to fourth order for (3) are presented as follows.
Order 1 requires:
$$ {d'}_{1}^{(1)}:=\sum _{i}b_{i}^{(0)}1=0. $$
Order 2 requires in addition:
$$ {d'}_{1}^{(2)}:=\sum _{i}b_{i}^{(0)}c_{i} \frac{1}{2}=0,\qquad d_{1}^{(2)}:=\sum _{i}\bar{b}_{i}^{(0)} \frac{1}{2}=0. $$
Order 3 requires in addition:
$$\begin{aligned} &{d'}_{1}^{(3)}:=\frac{1}{2} \biggl( \sum_{i}b_{i}^{(0)}c_{i}^{2} \frac{1}{3} \biggr)=0,\qquad{d'}_{2}^{(3)}:= \sum_{i}b_{i}^{(0)}\sum _{k}a_{ik}^{(0)} \frac{1}{6}=0, \\ &{d'}_{3}^{(3)}:=\sum _{i}b_{i}^{(2)}=0,\qquad d_{1}^{(3)}:=\sum_{i} \bar{b}_{i}^{(0)}c_{i}\frac{1}{6}=0, \\ &d_{2}^{(3)}:=g_{2}^{(2)}=0. \end{aligned}$$
Order 4 requires in addition:
$$\begin{aligned} &{d'}_{1}^{(4)}:=\frac{1}{6} \biggl( \sum_{i}b_{i}^{(0)}c_{i}^{3} \frac{1}{4} \biggr)=0, \qquad{d'}_{2}^{(4)}:= \sum_{i}b_{i}^{(0)}\sum _{k}c_{i}a_{ik}^{(0)} \frac{1}{8}=0, \\ &{d'}_{3}^{(3)}:=\sum _{i}b_{i}^{(0)}\sum _{k}a_{ik}^{(0)}c_{k} \frac{1}{24}=0, \qquad{d'}_{4}^{(4)}:=\sum _{i}b_{i}^{(0)}c_{i} \gamma_{i}^{(2)}=0, \\ &{d'}_{5}^{(4)}:=\sum _{i}b_{i}^{(2)}c_{i}=0, \qquad d_{1}^{(4)}:=\frac{1}{2} \biggl(\sum _{i}\bar{b}_{i}^{(0)}c_{i}^{2} \frac{1}{12} \biggr)=0, \\ &{d}_{2}^{(4)}:=\sum_{i} \bar{b}_{i}^{(0)}\sum_{k}a_{ik}^{(0)} \frac{1}{24}=0, \qquad{d}_{3}^{(4)}:=\sum _{i}\bar{b}_{i}^{(2)}=0. \end{aligned}$$
From Theorem 2.1 in [8], we know that the modified RKN method (3) has algebraic order at least 2.