Abstract
In this paper, based on the continuous collocation polynomial approximations, we derive and analyse a class of trigonometric collocation integrators for solving the highly oscillatory hyperbolic system. The symmetry, convergence and energy conservation of the continuous collocation polynomial approximations are rigorously analysed in details. Moreover, we also proved that the continuous collocation polynomial approximations could achieve at superconvergence by choosing suitable collocation points. Numerical experiments verify our theoretical analysis results, and demonstrate the remarkable superiority in comparison with the traditional temporal integration methods in the literature.
Introduction
The recent growth in extended RungeKuttaNyström (ERKN) methods (see, e.g. [12, 16, 33,34,35, 38,39,40, 42]) for secondorder multifrequency highly oscillatory systems and the collocation methods for stiff differential equations (see, e.g. [16, 18, 32, 36]) has let to the development of numerical integrators which systematically incorporate qualitative information transmitted from the underlying problem into their structure. In this paper, we focus on the design and analysis of temporal approximations through continuous trigonometric collocation polynomials for the semilinear highly oscillatory hyperbolic system:
where \(A\in \mathbb {R}^{d\times d}\) is supposed to be a symmetric positive definite or skewHermitian timeinvariant matrix with the decomposition \(A=\Omega ^2\), \(f : [t_0,T]\times \mathbb {R}^d\rightarrow \mathbb {R}^d\) is a nonlinear function, \(\varphi \) and \(\psi \) are the given initial values of the position u(t) and velocity \(\dot{u}(t)\), respectively. This hyperbolic system plays an important role in a wide variety of practical application areas in science and engineering, including nonlinear optics, solid state physics and quantum field theory (see, e.g. [1, 5, 11, 29]). As is known, with suitable spatial discretisation strategies, such as the finite difference method, the finite element method and the spectral method (see, e.g. [3, 14, 17, 20, 21, 26,27,28, 30, 31]), the semilinear wave equations can be converted into the highly oscillatory system (1.1). Therefore, it will be significant to further develop new approaches for efficiently solving the semilinear highly oscillatory system.
In recent works, based on the operator spectrum theory and the Duhamel’s principle, the evolutionary partial differential equations (PDEs) have been formulated as abstract ordinary differential equations (ODEs) in suitable Banach spaces (see, e.g. [4, 19, 22,23,24]). Therefore, in this paper, we will consider both wave equations in highdimensions and secondorder highly oscillatory ordinary differential equations as an semilinear hyperbolic system (1.1) in a suitable Banach space \((\mathcal {X}, \Vert \cdot \Vert )\), and present a new temporal integration strategy and related theoretical analyses. If the nonlinear function \(f: [t_0,T]\times \mathcal {X}\rightarrow \mathcal {X}\) is continuous, then the solution of the semilinear hyperbolic system (1.1) and its derivative satisfy the following variationofconstants formula (see, e.g. [4, 22,23,24, 38, 40, 42])
Since \(A=\Omega ^2\), the operators \(\phi _0\big ((tt_0)^2A\big )\) and \(\phi _1\big ((tt_0)^2A\big )\) are defined as
respectively. Furthermore, for any \(z\in [0,1]\), the solution of the hyperbolic system (1.1) and its derivative at time \(t=t_n+zh\in [t_n,t_{n}+h]\subseteq [t_0,T], n=0,1,2,\ldots \) can be expressed in
where \(V=h^2A\) and \(g\big (t_n+\tau h\big )=f\big (t_n+\tau h,u(t_n+\tau h)\big )\). A considerable amount of attention has been paid to the variationofconstants formula (1.2) or (1.4) to design effective and efficient numerical integrators for solving the highly oscillatory system (1.1). Approximating the nonlinear integrals appearing in formula (1.2) or (1.4) by suitable numerical quadrature formulas, a variety of numerical integrators have been proposed and analysed, including the Gautschitype method of order two (see, e.g. [8,9,10, 12, 15]), the adapted RungeKuttaNyström (ARKN) method (see, e.g. [7, 41]), the extended RungeKuttaNyström (ERKN) method (see, e.g. [6, 25, 34, 35, 38, 40, 42]), the arbitrarily highorder BirkhoffHermite (BH) method [22], and the trigonometric Fourier collocation (TFC) method [32, 36]. The integrators derived by using the variationofconstants formula (1.2) or (1.4), such as Gautschitype method, ARKN method, ERKN method, BH method and TFC method etc., are also termed trigonometric integrators. Therefore, the trigonometric integrators can exactly integrate the multifrequency unperturbed highly oscillatory system \(\ddot{u}(t)+Au(t)=0\) associated with (1.1). In other words, the trigonometric integrators can preserve the oscillatory structure [39] of the hyperbolic system (1.1), since the highly oscillatory behaviour of (1.1) is brought by the linear term Au.
Moreover, it is well known that the collocation integrators not only provide a discrete set of approximations, but also a continuous approximation to the original solution (see [16]). By choosing suitable collocation nodes, the collocation methods could achieve a highest convergence order. In the recent research work (see [32, 36]), Wang et al. took advantage of the shifted Legendre polynomials and the Lagrange polynomials to derive two kinds of trigonometric collocation methods for solving the system (1.1). However, these two kinds of trigonometric collocation methods are discrete approximations to original solutions. In addition, the local error bounds and the longterm convergence analysis are insufficient. Therefore, in this paper, using the continuous polynomial approximation, we will focus on combing the superior performance of the collocation integrators with the trigonometric integrators to design the continuous trigonometric collocation polynomial approximation for solving the highly oscillatory hyperbolic system (1.1), which could continuously approximate the original solution and preserve the oscillatory structure of the underlying system. Furthermore, the structurepreserving behaviour, single step error bounds, the longterm convergence and the superconvergence of the continuous trigonometric collocation polynomial approximations will be rigourously investigated in this work.
This paper is organised as follows. In Sect. 2, we formulate the trigonometric collocation integrators based on the continuous polynomial approximation, and investigate the symmetry of the trigonometric collocation integrator. Section 3 is concerned with the rigorous analysis of single step error bounds, the longterm convergence, and the superconvergence of the continuous trigonometric collocation polynomial approximations. By choosing suitable collocation nodes, the proposed trigonometric collocation integrators can achieve a highest order of convergence. Furthermore, the longterm energy conservation is also analysed in this section. In Sect. 4, we display preliminary numerical results which demonstrate the advantages and efficiency of our new algorithms in comparison with the existing numerical methods in the literature. The last section is devoted to brief conclusions.
Formulation of the continuous collocation polynomial approximation
An essential feature of the collocation integrator is that we not only obtain a discrete numerical solution, but also a continuous polynomial approximation to the solution of the original system (Hairer et. al. [16]). In this section, using continuous collocation polynomial approximations, we will derive a class of trigonometric collocation integrators for solving the highly oscillatory hyperbolic system (1.1) and analyse its symmetry. Our analysis for the collocation integrator will be based on the abstract formulation of the hyperbolic equation (1.1) as an evolution equation in a Banach space \((\mathcal {X}, \Vert \cdot \Vert )\).
Definition 2.1
Let \(0\le c_1<c_2<\cdots <c_s\le 1\) be distinct nodes. The continuous collocation polynomial approximation is to find a polynomial y(t) of degree \(s+1\) in the Banach space \((\mathcal {X}, \Vert \cdot \Vert )\) such that,
Then the numerical solution of the hyperbolic system (1.1) is defined by \(u^{n+1}=y(t_n+h)\) and \(\dot{u}^{n+1}=\dot{y}(t_n+h)\).
At first glance, it is difficult to seek such a polynomial in practice. Fortunately, following the Lagrange interpolation formula, the continuous collocation polynomial y(t) satisfies
Here and in what follows, we always assume that z is in [0, 1]. Applying the variationofconstants formula to (2.2), we obtain
This seems like that the numerical solution of the continuous collocation polynomial approximations determined by Definition 2.1 could coincide with a class of ERKN integrators (see, e.g.[38, 40, 42]). The details will be confirmed in the following theorem.
Theorem 2.1
The continuous collocation polynomial approximations determined by Definition2.1are equivalent to the ERKN methods of the form:
with the weights
and
for \(i,j=1,2,\ldots ,s\), and \(l_i(z)\) is the ith Lagrange basis polynomial.
Proof
It straightforwardly follows from the formula (2.3) that
where the weights \(b_i(v)\), \(\bar{b}_i(v)\) and \(a_{ij}(V)\) are given by (2.5) and (2.6), respectively. The statement of the theorem is proved. \(\square \)
Hairer et al. [16] have pointed out that all second order differential equation \(\ddot{u}={\tilde{f}}(u)\) written as \(\dot{u}=v\) and \(\dot{v}={\tilde{f}}(u)\) are time reversible. The system (1.1) could be expressed as
Therefore, the system (1.1) is clearly time reversible. Moreover, as is known, symmetric methods have excellent longtime behavior when solving reversible differential systems (see Hairer et al. [16]). The design and analysis of the symmetric integrators will be significant in the spirit of geometric integration. Thus, a most welcome feature of the continuous collocation polynomial approaches (2.1) is that it preserves the temporal symmetry by choosing suitable collocation nodes. We are now in a position to prove the symmetry of the trigonometric collocation integrators determined by Definition 2.1.
Theorem 2.2
If the collocation nodes \(c_1,\dots ,c_s\) are symmetric nodes, i.e., \(c_i+c_{si+1}=1\) for \( i=1,2,\dots ,s\), then the numerical solutions of the continuous collocation polynomial approximations (2.1) or the extended RungeKuttaNyström integrators (2.4) are symmetric.
Proof
We only need to prove that the ERKN methods determined by (2.4) are symmetric. Exchanging \(u^{n+1}\leftrightarrow u^n, \dot{u}^{n+1}\leftrightarrow \dot{u}^n\) and replacing h by \(h\) yields
It follows from combining \(\phi _0({V})\times \)(2.8\(+h\phi _1({V})\times \)(2.9) that
Similarly, the transformation \(h{A}\phi _1({V})\times \)(2.8) \(+\phi _0({V})\times \)(2.9) yields
The symmetric nodes \(c_1,c_2,\dots ,c_s\) demonstrate an interesting fact that
Therefore, according to the relations (1.3) and the definition of \(b_i({V})\) and \(\bar{b}_i({V})\), it can be confirmed that
and
The Eqs. (2.11) and (2.12) can be simplified as
and
Inserting the Eqs. (2.13) and (2.14) into (2.10) with a careful calculation gives
It then can be obtained after a series of complicated calculations that
and
Therefore, (2.15) is identical to the following
Replacing all indicates i and j in (2.13), (2.14) and (2.16) by \(si+1\) and \(sj+1\), respectively, we find that the ERKN method (2.4) with weights (2.5) and (2.6) is identical to its adjoint method consisted of (2.13), (2.14) and (2.16). Therefore, the ERKN methods determined by (2.4) are symmetric. This completes the proof of the theorem. \(\square \)
Main theoretical results
In this section, we state the error bounds of our continuous trigonometric collocation polynomial approximations for solving the hyperbolic system (1.1). The single step error bounds and the longterm convergence of continuous trigonometric collocation polynomial approximations (2.1) will be rigorously analysed. Our analysis will be based on the variationofconstants formula (1.2) or (1.4) provided for the hyperbolic system (1.1). Before presenting our theoretical analysis, we need the following hypotheses on the regularity of u and the nonlinearity of f.
Assumption 3.1
Suppose that the exact solution u of the hyperbolic system (1.1) satisfies \(u\in C^{s}\big ([t_0,T];\mathcal {X}\big )\) and the nonlinear function \(f: [t_0,T]\times \mathcal {X}\rightarrow \mathcal {X}\) is Fréchet differentiable along a strip of the exact solution u. We also assume that all occurring derivatives are uniformly bounded.
Assumption 3.2
Let f(t, u) be Lipschitzcontinuous with respect to u, i.e., there exists a real number L such that
for all \(t\in [t_0,T]\) and \(\max \big (\Vert v(t)u(t)\Vert , \Vert w(t)u(t)\Vert \big )\le Const\).
Remark 3.1
Under Assumptions 3.1, it should be noted that the composition:
is also Fréchet differentiable in a strip along the exact solution u(t). Therefore, when \(u\in C^{l}\big ([t_0,T],\mathcal {X}\big )\) and \(g^{(l)}\in L^{\infty }([t_0,T];\mathcal {X})\) for \( l=s,s+1,\dots ,2s\), the following assumptions are valid
where \(K_m,\,m=0,1,2,\ldots , s\) are constants and independent of h and A.
Remark 3.2
Here, we remark that the essence of the continuous collocation polynomial approximation is to find the polynomial y(t) in the Banach space \((\mathcal {X}, \Vert \cdot \Vert )\). The polynomial y(t) is always bounded over the interval \([t_0,T]\), and this implies that there exists a constant \(M>0\) such that
Therefore, Assumption 3.2 is reasonable.
Error estimates
Theorem 3.1
(Single step error bounds I) Suppose that Assumptions 3.1  3.2are satisfied and \(g^{(l)}\in L^{\infty }([t_0,T];\mathcal {X})\) for \( l=s,s+1,\dots ,2s\). Then, when the time step h satisfies the limitation \(0<h\le \sqrt{\frac{1}{2sL}}\), we have the error estimations for a single step:

(i)
Under the local assumptions of \(y(t_n)=u(t_n),\ \dot{y}(t_n)=\dot{u}(t_n)\), the single step error bounds of the continuous trigonometric collocation polynomial y(t) defined by Definition 2.1satisfy:
$$\begin{aligned} \Vert u(t)y(t)\Vert \le C_1h^{s+2}\quad {\hbox {and}}\quad \nonumber \\ \Vert \dot{u}(t)\dot{y}(t)\Vert \le C_1h^{s+1},\quad \forall t\in [t_n,t_n+h]. \end{aligned}$$(3.2) 
(ii)
Furthermore, if the collocation polynomial y(t) satisfies \(y^{(m)}(t_n)=u^{(m)}(t_n)\) for \(m=1,\ldots ,s\), then the derivatives of y(t) satisfy the following estimations:
$$\begin{aligned} &\Vert u^{(m)}(t)y^{(m)}(t)\Vert \le C_2h^{s+2m}\quad {\hbox {and}}\quad \nonumber \\ &\Vert u^{(m+1)}(t)y^{(m+1)}(t)\Vert \le C_2h^{s+1m},\quad \forall t\in [t_n,t_n+h]. \end{aligned}$$(3.3)
Here, we point out that the constants \(C_1\) and \(C_2\) satisfy
and are obviously independent of h and A.
Proof
According to the definition of the continuous trigonometric collocation polynomial defined by Definition 2.1, it is clear that the polynomial y(t) satisfies
Moreover, it is evident that the exact solution of (1.1) satisfies
where the interpolation error \(E_n(z,h)=\frac{1}{s!}w_s(z)g^{(s)}(t_n+\xi ^nh)\) for some \(\xi ^n\in \left( \min \{z,c_1,\dots ,c_s\}, \max \{z,c_1,\dots ,c_s\}\right) \) and \(w_s(z)=\prod \limits _{i=1}^s(zc_i)\). In the light of Assumption 3.1, \(E_n(z,h)\) and its derivatives satisfy the estimations
Then, subtracting (3.4) from (3.5) leads to
where \(e(t_n+zh)=u(t_n+zh)y(t_n+zh)\) and \(\Delta f^{ni}=f\big (t_n+c_ih,u(t_n+c_ih)\big )f\big (t_n+c_ih,y(t_n+c_ih)\big )\).

(i)
Applying the variationofconstants formula to the Eq. (3.6), we obtain
$$\begin{aligned} \left\{ \begin{array}{ll} &{}e(t_n+zh)=\phi _0(z^2V)e(t_n)+zh\phi _1(z^2V)\dot{e}(t_n)\\ &{}\quad +h^2\sum \limits _{i=1}^{s}\int _{0}^{z}(z\tau )\phi _1\big ((z\tau )^2V\big )l_i(\tau ) \mathrm{d}\tau \Delta f^{ni}\\ &{}\quad +h^{s+2}\int _{0}^{z}(z\tau )\phi _1\big ((z\tau )^2V\big )E_n(\tau ,h)\mathrm{d}\tau ,\\ &{}\dot{e}(t_n+zh)=zhA\phi _1(z^2V)e(t_n)+\phi _0(z^2V)\dot{e}(t_n)\\ &{}\quad +h\sum \limits _{i=1}^{s}\int _{0}^{z}\phi _0\big ((z\tau )^2V\big )l_i(\tau )\mathrm{d} \tau \Delta f^{ni}\\ &{}\quad +h^{s+1}\int _{0}^{z}\phi _0\big ((z\tau )^2V\big )E_n(\tau ,h)\mathrm{d}\tau . \end{array} \right. \end{aligned}$$(3.7)Taking norms on both sides of Eqs. (3.7) and using Assumption 3.2 and the local assumptions \(y(t_n)=u(t_n),\ \dot{y}(t_n)=\dot{u}(t_n)\), we obtain
$$\begin{aligned} \Vert e(t_n+zh)\Vert \le h^2sL\max \limits _{0\le z\le 1}\Vert e(t_n+zh)\Vert +K_0h^{s+2} \end{aligned}$$(3.8)and
$$\begin{aligned} \Vert \dot{e}(t_n+zh)\Vert \le h sL\max \limits _{0\le z\le 1}\Vert e(t_n+zh)\Vert +K_0h^{s+1}. \end{aligned}$$(3.9)If the time stepsize h satisfies \(h^2sL\le \frac{1}{2}\), namely, \(h\le \sqrt{\frac{1}{2sL}}\), then we have
$$\begin{aligned} \max \limits _{0\le z\le 1}\Vert e(t_n+zh)\Vert \le 2K_0h^{s+2}. \end{aligned}$$(3.10)Inserting the result (3.10) into the inequality (3.9) leads to
$$\begin{aligned} \Vert \dot{e}(t_n+zh)\Vert \le h^{s+1}\left( 2K_0h^2sL+K_0\right) \le 2K_0h^{s+1}. \end{aligned}$$Thus, let \(C_1\ge 2K_0\) in the above results, and then the proof of the first statement is complete.

(ii)
The proof of the second statement follows from applying the variationofconstants formula to
$$\begin{aligned} h^me^{(m+2)}(t_n+zh)=Ah^me^{(m)}(t_n+zh)+\sum \limits _{i=1}^{s}l_i^{(m)}(z)\Delta f^{ni}+h^sE^{(m)}(z,h), \end{aligned}$$i.e.,
$$\begin{aligned} \left\{ \begin{array}{ll} &{}h^me^{(m)}(t_n+zh)=h^m\phi _0(z^2V)e^{(m)}(t_n)\\ &{}\quad +zh^{m+1}\phi _1(z^2V)e^{(m+1)}(t_n)+h^2\sum \limits _{i=1}^{s}\int _{0}^{z} (z\tau )\phi _1\big ((z\tau )^2V\big )l_i^{(m)}(\tau )\mathrm{d}\tau \Delta f^{ni}\\ &{}\quad +h^{s+2}\int _{0}^{z}(z\tau )\phi _1\big ((z\tau )^2V\big )E^{(m)}(\tau ,h)\mathrm{d}\tau ,\\ &{}h^me^{(m+1)}(t_n+zh)=zh^{m+1}A\phi _1(z^2V)e^{(m)}(t_n)\\ &{}\quad +h^m\phi _0(z^2V)e^{(m+1)}(t_n)+h\sum \limits _{i=1}^{s}\int _{0}^{z}\phi _0 \big ((z\tau )^2V\big )l_i^{(m)}(\tau )\mathrm{d}\tau \Delta f^{ni}\\ &{}\quad +h^{s+1}\int _{0}^{z}\phi _0\big ((z\tau )^2V\big )E^{(m)}(\tau ,h)\mathrm{d}\tau . \end{array} \right. \end{aligned}$$Thus, the local assumptions \(y^{(m)}(t_n)=u^{(m)}(t_n)\) for \(m=1,2,\ldots ,s\) lead to
$$\begin{aligned}&h^m\Vert e^{(m)}(t_n+zh)\Vert \le h^2L\sum \limits _{i=1}^{s}\max \limits _{0\le z\le 1}l_i^{(m)}(z)\nonumber \\&\quad \times \max \limits _{0\le z\le 1}\Vert e(t_n+zh)\Vert +K_mh^{s+2} \end{aligned}$$(3.11)and
$$\begin{aligned}&h^m\Vert e^{(m+1)}(t_n+zh)\Vert \le hL\sum \limits _{i=1}^{s}\max \limits _{0\le z\le 1}l_i^{(m)}(z)\nonumber \\&\quad \times \max \limits _{0\le z\le 1}\Vert e(t_n+zh)\Vert +K_mh^{s+1}. \end{aligned}$$(3.12)Inserting the results (3.2) into the left sides of (3.11) and (3.12) and choosing the constant \(C_2\) to satisfy
$$\begin{aligned} \max \limits _{1\le m\le s}\left( 2K_0h^2L\sum \limits _{i=1}^{s}\max \limits _{0\le z\le 1}l_i^{(m)}(z)+K_m\right) \le C_2, \end{aligned}$$we obtain
$$\begin{aligned} \Vert e^{(m)}(t_n+zh)\Vert&\le C_2h^{s+2m}\quad {\hbox {and}}\quad \Vert \dot{e}^{(m+1)}(t_n+zh)\Vert \le C_2h^{s+1m},\\&\qquad m=1,2,\dots ,s. \end{aligned}$$This completes the proof.
\(\square \)
Actually, Theorem 3.1 demonstrate the local error bounds for the continuous collocation polynomial approximations (2.1). In what follows, we will investigate the longterm convergence of the continuous collocation polynomial approximations. For this purpose, we rewritten the Eq. (3.7) as the following compact form:
where \(\Psi (z,\tau ,V)\) is defined as:
that is, \(\Psi (z,\tau ,V)\) is the rotation operator
It is easy to see that the norm of \(\Psi (z,\tau ,V)\) satisfies:
This will be repeatedly used in our following theoretical analysis. For simplicity of analysis, we introduce the energy norm \(\big (u(t),\dot{u}(t)\big )\) for the solutions \(\big (u(t),\dot{u}(t)\big )\in L^{\infty }([t_0,T];\mathcal {X})\), i.e.,
Moreover, we quote the following Gronwall inequality, which will play an important role in the convergence analysis.
Lemma 3.1
(See, e.g. [22, 23]) Let \(\lambda \) be positive, \(a_k, b_k\), \(k\in \mathbb {N}\), be nonnegative and assume further that
Then
Now, we will show that the continuous collocation polynomial approximation defined by Definition 2.1 with nodes \(\{c_i\}_{i=1}^s\) is of order s at least.
Theorem 3.2
(Longterm convergence I) Suppose that the Assumptions 3.1–3.2are satisfied and \(g^{(s)}\in L^{\infty }([t_0,T];\mathcal {X})\). Then, under the limitation of the time stepsize \(0<h\le \sqrt{\frac{1}{2sL}}\), we obtain the error estimations:
where y(t) is the collocation polynomial of degree \(s+1\) and the constant \(C_4\) independent of n, h and A.
Proof
For any \(t\in [t_0,T]\), there exists a nonnegative integer \(n\in \mathbb {N}\) such that \(t=t_n+zh, \forall z\in [0,1]\). It follows from taking norms on both sides of Eq. (3.13) and the first equation in (3.7) that:
and
Let the time stepsize \(h_0\) be sufficiently small and meet the condition \(h_0^2sL\le \frac{1}{2}\), i.e., \(h_0\le \sqrt{\frac{1}{2sL}}\). Then we have
Inserting the result (3.18) into the inequality obtained by adding (3.16) and (3.17) leads to
Letting \(z=1\) in (3.19) and using the Gronwall’s inequality (see Lemma 3.1) to it, we have
Therefore, there exist a constant \(C_3\ge 4TK_0\exp \big ((1+4sL)T\big )\) such that
Furthermore, inserting the results (3.20) into (3.19) yields
Obviously, the constant \(C_4\) is independent on n, h and A. Therefore, the theorem is confirmed. \(\square \)
Superconvergence of continuous trigonometric collocation polynomial approximations
In the light of the results shown in Theorem 3.1 and Theorem 3.2, the term \(\max \limits _{0\le z\le 1}w_s(z)=\max \limits _{0\le z\le 1}\prod \limits _{i=1}^szc_i\) appears in the error bounds. This observation implies that by choosing suitable collocation nodes \(c_i\), such as the zeros of the shifted Legendre polynomial, can minimize the error bounds to achieve a higher convergence order. In our numerical experiments, we will choose the collocation nodes \(c_i\) as the zeros of the shifted Legendre polynomials
to generate the Gausstype and Lobattotype time integrators, respectively. The numerical results in Sect. 4 will show that the continuous trigonometric collocation polynomial approximation can be of superconvergence. In this subsection, we will present the rigorous theoretical analysis of this superconvergence. To this end, we consider \(y(t_n+zh)\) as the solution of the following perturbed differential equation:
where the remainder \(\delta (t_n+zh)\) will vanish at the chosen collocation points, i.e., \(\delta (t_n+c_ih)=0\) for \(i=1,2,\dots ,s.\) Subtracting the perturbed differential equation (3.21) from the Eq. (1.1) and applying the variationofconstant formula to this result, we obtain after the linearisation that
where the remainder \(r(t_n+zh)\) is of magnitude \(\mathcal {O}(\Vert u(t_n+zh)y(t_n+zh)\Vert ^2)\). Since the defect \(\delta (t_n+zh)\) vanishes at the collocation nodes \(c_1,\dots ,c_s\), the integral related to \(\Vert \delta (t_n+zh)\Vert \) is equal to its quadrature error, i.e.,
where p is the algebraic order of the quadrature formula corresponding to the nodes \(c_1,\dots ,c_s\). In what follows, we will show that the precision of the trigonometric collocation integrators defined by Definition 2.1 could be improved by choosing suitable collocation nodes.
Theorem 3.3
(Single step error bounds II) Suppose that the exact solution u(t) of the system (1.1) satisfies the Assumption 3.1. Then, there exists a sufficiently small \(h_0>0\), such that when the time step h meets the limitation \(0<h\le h_0\), the single step error bounds of the collocation polynomial y(t) further satisfy
where \(s\le p\le 2s\) is the algebraic order of the quadrature formula corresponding to the nodes \(c_1,\dots ,c_s\).
Proof
It follows from taking norms on both sides of (3.22) that
and
Let the stepsize h satisfy \(h\le \sqrt{\frac{1}{2K_0}}\). The inequality (3.25) leads to
Inserting (3.27) into (3.26) yields
Moreover, the statements of Theorem 3.2 yield that the remainder \(r(t_n+zh)\) is of the magnitude \(\mathcal {O}(u(t_n+zh)\Vert y(t_n+zh)\Vert ^2)=\mathcal {O}(h^{2s})\). Therefore, if the order of the quadrature formula satisfies \(s\le p\le 2s\), we have
By taking \(h_0\le \min \big \{\sqrt{\frac{1}{2K_0}}, \sqrt{\frac{1}{2sL}}\big \}\), the conclusion of the theorem is proved. \(\square \)
Theorem 3.4
(Longterm convergence II) Suppose that the collocation polynomial y(t) of degree \(s+1\) satisfies (2.1) and the exact solution u(t) of the system (1.1) satisfies Assumption 3.1. Then, under the limitation of the time stepsize \(h\le \min \big \{\sqrt{\frac{1}{2K_0}}, \sqrt{\frac{1}{2sL}}\big \}\), we obtain the following superconvergence results
where \(s\le p\le 2s\) is the algebraic order of the quadrature formula corresponding to the nodes \(c_1,\dots ,c_s\).
Proof
It follows from rewriting the Eq. (3.22) as the following compact form that
Taking norms on both sides of Eq. (3.30) gives
Similarly, by taking norms on both sides of the first equation in (3.30), we have
It then follows from choosing the stepsize h satisfies \(h\le \sqrt{\frac{1}{2K_0}}\) that
Inserting (3.33) into the results obtained by adding up (3.31) and (3.32) yields that
Taking \(z=1\) in (3.34) and using the Gronwall’s inequality (see Lemma 3.1) to the obtained result leads to
According to Theorem 3.2 we have \(\Vert r(t_n+zh)\Vert =\mathcal {O}\big (\Vert u(t_n+zh)y(t_n+zh)\Vert ^2\big )=\mathcal {O}(h^{2s})\). Therefore, if the order of the quadrature formula satisfies \(s\le p\le 2s\), we obtain
Inserting (3.35) into (3.34) yields
The statement of the theorem is confirmed. \(\square \)
Remark 3.3
If the nodes \(\{c_i\}_{i=1}^s\) are chosen, from the analysis of Theorem 3.3 and Theorem 3.4, we have
This implies that the continuous trigonometric collocation polynomial approximations have the same order as the underlying quadrature formula. For instance, we take \(\{c_i\}_{i=1}^s\) as zeros of the shifted Legendre polynomials
the accuracy of the continuous trigonometric collocation polynomial approximations can achieve order 2s or \(2s2\), respectively. The fact will be demonstrated again in our numerical experiments in Sect. 4.
Longterm behaviour of energy conservation
In this section, we turn to another essential property of the conservative hyperbolic system (1.1). It is easy to see that the hyperbolic system (1.1) has the following important conservative quantity:
where the inner product \((\cdot ,\cdot )\) is corresponding to \(\Vert \cdot \Vert \), and F is the primitive function of f defined as:
In what follows, we will focus on analysing the longterm energy conservation of the continuous trigonometric collocation polynomial approximations for solving the hyperbolic system (1.1).
Theorem 3.5
Let \(0\le c_1<c_2<\cdots <c_s\le 1\) be the collocation nodes and y(t) defined by Definition 2.1be the collocation polynomial corresponding to the nodes. Then the energy defined in (3.38) satisfies the estimation
where \(t\in [t_0,T]\) and \(s\le p\le 2s\) is the algebraic order of the quadrature formula corresponding to the nodes.
Proof
For any \(t_0\le t\le T\), there exists a number \(n\in \mathbb {N}\) such that \(t=t_n+\mu h,\ \forall \mu \in [0,1]\). Therefore, the energy error can be expressed as
In order to obtain the estimation for the energy (3.38) over a long time interval \([t_0,T]\), we need to estimate \(H\big (y(t_k+\mu h),\dot{y}(t_k+\mu h)\big )H\big (y(t_k),\dot{y}(t_k)\big )\) for \(k=0,1,\dots ,n\). Clearly, the collocation polynomial \(y(t_k+\mu h)\) satisfies
For any \(k=0,1,\ldots ,n\), by taking the inner product of both sides of (3.41) with \(y(t_k+\mu h)\), we have
Therefore, integrating the Eq. (3.42) for \(\mu \) from 0 to z and using the Hölder’s inequality, we obtain
Here, \(\Vert \cdot \Vert _{*}\) is the dual norm of \(\Vert \cdot \Vert \) and M is the bound of the collocation polynomial y(t) over the interval \([t_0,T]\). Inserting the above results into (3.41) leads to
Since the defect \(\delta (t_k+\mu h)\) vanishes at the collocation nodes \(c_1,\dots ,c_s\), the integral related to \(\Vert \delta (t_n+\mu h)\Vert _{*}\) is also equal to its quadrature error, namely,
Therefore, we obtain the estimate of energy error as follows
The conclusion of the theorem is confirmed. \(\square \)
Numerical experiments
In this section, we will derive four practical continuous trigonometric collocation polynomial approximations and illustrate the numerical results for solving the KleinGordon equation, twodimensional sineGordon equation, and Duffing equation. We report numerical results to support our error estimates and demonstrate the superiority of the continuous collocation polynomial approximation. The numerical results are really promising, which are powerful to verify our theoretical analysis results for our continuous trigonometric collocation polynomial approximations.

Trigonometric collocation polynomial approximation with Gauss nodes: By taking the nodes \(c_1,\cdots ,c_s\) as the zeros of the sth shifted GaussLegendre polynomial
$$\begin{aligned} \frac{\mathrm{d}^s}{\mathrm{d}x^s}\Big (x^s(x1)^s\Big ), \end{aligned}$$then the Gauss quadrature formulas means that the integral (3.23) has order \(p=2s\). In this work, we choose the twopoint GaussLegendre nodes
$$\begin{aligned} c_1=\frac{3\sqrt{3}}{6},\qquad c_2=\frac{3+\sqrt{3}}{6}, \end{aligned}$$and the threepoint GaussLegendre nodes
$$\begin{aligned} c_1=\frac{5\sqrt{15}}{10},\qquad c_2=\frac{1}{2},\qquad c_3=\frac{5+\sqrt{15}}{10}, \end{aligned}$$to construct the fourthorder and sixthorder trigonometric collocation time integrators, which are denoted by GTC2s4 and GTC3s6, respectively.

Trigonometric collocation polynomial approximation with Lobatto nodes: If we take the nodes \(c_1,\cdots ,c_s\) as the zeros of the sth Legendre polynomial
$$\begin{aligned} \frac{\mathrm{d}^{s2}}{\mathrm{d}x^{s2}}\Big (x^{s1}(x1)^{s1}\Big ), \end{aligned}$$then the Lobatto quadrature formulas lead the integral (3.23) could achieve the highest possible order \(p=2s2\). Similarly, by taking the threepoint Lobatto nodes
$$\begin{aligned} c_1=0,\qquad c_2=\frac{1}{2},\qquad c_3=1, \end{aligned}$$and the fourpoint Lobatto nodes
$$\begin{aligned} c_1=0,\qquad c_2=\frac{5\sqrt{5}}{10},\qquad c_3=\frac{5+\sqrt{5}}{10},\qquad c_4=1, \end{aligned}$$we can derive the fourthorder and sixthorder trigonometric collocation time integrators, and denoted as LTC2s4 and LTC3s6, respectively.
In order to demonstrate the superiority of the proposed integrators, we select the following time integrators for comparison:

BH1: the symmetric BirkhoffHermite time integrator of order four derived in [22]

BH2: the symmetric BirkhoffHermite time integrator of order six derived in [22]

GAS2s4: the twostage Gauss time integration method of order four presented in [16];

GAS3s6: the threestage Gauss time integration method of order six presented in [16];

LIIIA3s4: the Labatto IIIA method of order four presented in [16];

LIIIA4s6: the Labatto IIIA method of order six presented in [16].

ERKN3s4: the threestage explicit ERKN method of order four derived in [38];

ERKN7s6: the sevenstage explicit ERKN method of order six presented in [33];
During the numerical experiments, it should be noted that the fixedpoint iteration are used for all the implicit integrators. The iteration procedure will be stopped once the \(l^{\infty }\) norm of the difference between two successive approximations is smaller than the fixed error tolerance \(10^{15}\). Here, we also point out that if the error produced by a method is too large for some time stepsize h, then the corresponding point will not be plotted in the figure. Moreover, we should point out that the ERKN3s4 and ERKN7s6 methods are quite different with our proposed methods in this paper. The chosen ERKN3s4 and ERKN7s6 methods are derived by solving the order conditions (see [33, 38]). Our derived time integrators are yielded by using the continuous collocation polynomial approximation.
All computations in the numerical experiments are carried out by using MATLAB 2016b on the computer Lenovo ThinkCentre M8300t (CPU: Intel (R) Core (TM) i72400 CPU @ 3.10 GHz, Memory: 8 GB, Os: Microsoft Windows 10 with 64 bit).
Problem 4.1
Consider the dimensionless relativistic KleinGordon equation which is highly oscillatory in time (see, e.g. [2, 34, 37]):
over the region \((x,t) \in [L,L] \times [0,T]\) with the initial functions
and the cubic nonlinearity, i.e. \(f(u)=4u^3\). Here, the dimensionless parameter \(\varepsilon \) is chosen as \(\varepsilon =0.5\) and 0.1, respectively. Moreover, we suppose that the KleinGordon equation (4.1) is equipped with the periodic boundary conditions. Therefore, the Fourier pseudospectral method will be used to discretise the spatial derivatives, and the KleinGordon equation (4.1) can be converted into the following form
where \(U(t)=\big (U_1(t),\dots ,U_M(t)\big )^{\top }\) with \(U_i(t)\approx u(x_i,t)\), and \(A=D_2+I/\varepsilon ^2\) is an \(M\times M\) matrix used to approximate the operator \(\Delta +1/\varepsilon ^2\). Here, \(D_2\) is the secondorder symmetric semidefinite spectral differential matrix (see [14, 17, 26, 27]), and I is the unit matrix. The discrete energy is given by
where the discrete \(l^p\) norms \(\Vert \cdot \Vert _{p}\) for \(p=2,4\) are defined as
the discrete inner product (u, v) is defined by
Here, \(\Delta x=2L/M\) is the spatial stepsize. In order to test the longterm numerical behaviour of the proposed time integrators, we set \(L=30,\ T=100\) and fixed the spatial mesh size \(M=1024\) for this problem. In addition, as is known, the exact solution of the Eq. (4.1) cannot be represented explicitly. Therefore, we use the posterior error, i.e., \(RE=\Vert U(h;T)U(h/2;T)\Vert _2\), to compute the convergence order.
In Tables 1 and 2, we test the numerical precision of the time integrators “GTC2s4”, “GTC3s6”, “LTC3s4” and “LTC4s6” with different \(\varepsilon \). The numerical data indicate that the continuous trigonometric collocation polynomial approximations with Gauss points and Lobatto points can achieve 2sth order and \((2s2)\)th order, respectively. The computational results in Tables 1 and 2 demonstrate that the temporal accuracy is completely consistent with our theoretical analysis. The proposed continuous trigonometric collocation polynomial approximations can be of superconvergence (see Theorem 3.4). The logarithms of the posterior errors, i.e., \(\log _{10}(RE)\), are plotted in Figs. 1 and 2, in comparison with the classical time integrators “BH1”, “BH2”, “ERKN3s4”, “ERKN7s6”, “GAS2s4”, “GAS3s6”, “LIIIA3s4” and “LIIIA4s6”. It can be observed from these two figures that the proposed continuous trigonometric collocation integrators are much more accurate than these traditional methods. The data in Tables 3 and 4 illustrate the precision of the discrete energy conservation, which is consistent with our theoretical analyse results in Theorem 3.5. The proposed temporal integrators have better longterm behaviour of energy conservation.
Problem 4.2
Consider the following sineGordon equations in two dimensions (see, e.g. [22, 23]):
where \(\kappa ^2\) is a dimensionless parameter. We also suppose that the two dimensional sineGordon equation (4.3) is subjected to periodic boundary conditions. The spatial derivatives are approximated by the two dimensional Fourier pseudospectral method. Therefore, the Eq. (4.3) can be converted to the following matrix form:
where \(D_2^x=\mathcal {F}^{1}_{M_x}\,\Lambda _{2}^x\,\mathcal {F}_{M_x}\) and \(D_2^y=\mathcal {F}^{1}_{M_y}\,\Lambda _{2}^y\,\mathcal {F}_{M_y}\) are the secondorder Fourier spectral matrices (see, e.g. [13]) in the x direction and y direction, respectively. The discrete energy is given by
where \(\Delta x=2/M_x\) and \(\Delta y=2/M_y\) are the spatial stepsizes. More details can be found in [13, 26, 27]. For this problem, we take the dimensionless parameter \(\kappa =1/20\) and set the mesh sizes \(M_x=M_y=256\). In Fig. 3 and 4, we present the simulation results and the corresponding contour plots at the time points \(t=0, 20, 40, 50, 80\) and 100 with \(h=0.01\). The phenomena is termed circular ring solitons and obviously is periodicity. Figure 3 demonstrates the phenomena of the periodic oscillations and radiations of the circular ring solitons as time goes on. The CPU time required to reach \(T=100\) is 439.476393 seconds. The numerical data in Table 5 clearly indicate that the convergence order of the derived time integrators, which are consistent with our theoretical analysis results in Theorem 3.4. Figure 5 demonstrates that the time integrators “GTC2s4”, “GTC3s6”, “LTC3s4” and “LTC4s6” have better longterm computational behaviour than the chosen classical integrators. Table 6 shows the numerical precision of the longterm preservation of the discrete energy (4.4). The numerical results in Table 6 confirm the theoretical result in Theorem 3.5.
Problem 4.3
We consider the Duffing equation (see, e.g. [32])
with \(0\le k<\omega \). This is a Hamiltonian system with the Hamiltonian
The analytic solution is given by
where sn means the Jacobian elliptic function. We choose \(k=0.03\) and different frequencies \(\omega =10\) and 20 which are similar to those in [32]. The problem is investigated over the interval [0, 1000] to verify the convergence order and the precision of the energy conservation for the constructed time integrators with different \(\omega \).
The data in Tables 7 and 8 show the convergence order of the time integrators “GTC2s4”, “GTC3s6”, “LTC3s4” and “LTC4s6” with different \(\omega \). The tabular data implies that the continuous trigonometric collocation polynomial approximations with Gauss points and Lobatto points could achieve 2sth order and \((2s2)\)th order, respectively. The logarithm of the global errors \(GE=\Vert q_Nq(1000)\Vert _2\) against different steps for Problem 4.3 are plotted in Figs. 6 and 7. In comparison with the “GAS2s4”, “GAS3s6”, “LIIIA3s4”, “LIIIA4s6”, “BH1” and “BH2” integrators, the proposed time integrators in this paper have much better accuracy and cost less CPU time. However, the proposed integrators and the chosen ERKN integrators have similar precision under the same CPU time. The numerical results in Tables 9 and 10 indicate the precision of the energy conservation.
In conclusion, the numerical results for Problem 4.1, Problem 4.2 and Problem 4.3 are consistent with our theoretical analysis results. The continuous collocation polynomial approximations can be superconvergence and have better numerical behaviour than the chosen classical integrators. Comparing with the classical implicit methods, i.e., “GAS2s4” method, “GAS3s6” method, “LIIIA3s4” method, “LIIIA4s6” method, “BH1” method and “BH2” method, the proposed time integrators “GTC2s4”, “GTC3s6”, “LTC3s4” and “LTC4s6” have better precision while cost less CPU time. However, comparing with the explicit ERKN3s4 method and ERKN7s6 method, our proposed time integrators have similar precision under the same CPU time. Moreover, the numerical results in [39] indicated that the ERKN3s4 and ERKN7s6 methods have better numerical behaviour than the ARKN3s4, TFCr2 and TFCr3 methods. Therefore, we could confirm that the proposed time integrators in this paper would be better than the ARKN3s4, TFCr2 and TFCr3 methods. The Gautschitype methods are time integrators with second order precision at most. We will not compare with the Gautschitype, ARKN and TFC methods in this paper.
Conclusion
Taking into account the superiority of the continuous collocation methods and the trigonometric integrators, we proposed and analysed the continuous trigonometric collocation polynomial approximations for the highly oscillatory hyperbolic system (1.1) in this paper. The derived trigonometric collocation integrators inherit the superconvergence of the classical continuous collocation methods and also can preserve the oscillatory structure of the underlying highly oscillatory systems. The resulting trigonometric collocation integrators were analysed in details for the local error bounds, longterm convergence, superconvergence, symmetry and longterm energy conservation. Furthermore, the remarkable numerical behaviour of the continuous trigonometric collocation polynomial approximations was demonstrated by the numerical experiments in comparison with the existing numerical methods in the literature.
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Acknowledgments
The research is supported in part by the Natural Science Foundation of China under Grant 11801280 and 12071419, by the Natural Science Foundation of the Jiangsu Higher Education Institutions under Grant 18KJB110020, by the Natural Science Foundation of Jiangsu Province under Grant BK20180780, and by the Scientific Research Foundation of Nanjing University of Information Science & Technology under Grant 2017r072.
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Liu, C., Wu, X. Continuous trigonometric collocation polynomial approximations with geometric and superconvergence analysis for efficiently solving semilinear highly oscillatory hyperbolic systems. Calcolo 58, 6 (2021). https://doi.org/10.1007/s10092020003942
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Keywords
 Hyperbolic systems
 Structurepreserving integrators
 Superconvergence
 Trigonometric polynomial approximations
 Highly oscillatory systems
Mathematics Subject Classification
 65L07
 65L20
 65L70
 65M12
 65M15
 65M22