A Well-Conditioned Spectral Galerkin–Levin Method for Highly Oscillatory Integrals

Abstract

This paper addresses the numerical evaluation of highly oscillatory integrals by developing a spectral Galerkin–Levin approach that efficiently solves Levin’s differential equation formulation for such integrals. The method employs Legendre polynomials as basis functions to approximate the solution, leveraging their orthogonality and favorable numerical properties. A key finding is that the Galerkin–Levin formulation is invariant with respect to the choice of polynomial basis—be it monomials or classical orthogonal polynomials—although the use of Legendre polynomials leads to a more straightforward derivation of practical quadrature rules. Building on this, this paper derives a simple and efficient numerical quadrature for both scalar and matrix-valued highly oscillatory integrals. The proposed approach is computationally stable and well-conditioned, overcoming the limitations of collocation-based methods. Several numerical examples validate the method’s high accuracy, stability, and computational efficiency.

1. Introduction

Highly oscillatory integrals (HOIs)—that is, integrals involving rapidly oscillating functions—frequently arise in mathematical physics, particularly in the context of dynamic problems in classical and quantum mechanics, electromagnetism, and related fields [1]. These integrals typically do not admit closed-form solutions, necessitating the development of specialized numerical quadrature methods for their evaluation. Classical quadrature techniques, which rely on polynomial approximation of the integrand, generally fail to achieve the required accuracy in such cases.

The development of numerical methods for evaluating highly oscillatory integrals dates back to the work of Filon [2], who was the first to propose an efficient quadrature rule for computing Fourier integrals. His method involves partitioning the integration interval into subintervals and applying a modified Simpson’s rule to each. Since then, numerous variants of Filon-type methods have been proposed [3,4,5], each tailored to address specific characteristics of different classes of HOIs.

The continued development in this area underscores the relevance and challenge of accurately evaluating HOIs. For example, Wu and Sun [6] presented an efficient algorithm for approximating integrals with highly oscillatory Hankel kernels using a Clenshaw–Curtis–Filon rule for fast computation. Gao and Chang [7] developed a generalized bivariate Filon–Clenshaw–Curtis method capable of achieving high accuracy across a wide range of oscillatory parameters while maintaining low computational complexity. Kassid et al. [8] proposed a Nystöm-type method for solving a class of nonlinear highly oscillatory Volterra integral equations with trigonometric kernels. Their approach leads to a nonlinear system involving oscillatory integrals, which is then efficiently handled using a two-point generalized quadrature rule to obtain a fully discretized scheme. Chen et al. [9] derived a Filon-type method for computing finite-part integrals involving hypersingular and highly oscillatory terms. Similarly, Zhao et al. [10] applied a generalized two-point quadrature method to Volterra integral equations with highly oscillatory Bessel kernels. By estimating appropriate quadrature weights, they ensured the solvability of the resulting discretized system.

A fundamentally different approach to evaluating highly oscillatory integrals (HOIs) was introduced by Levin [11], who transformed the integration problem into the solution to an associated ordinary differential equation (ODE). This technique proved to be highly accurate for certain classes of HOIs and has since inspired numerous methods based on Levin’s original idea. For instance, Evans and Webster [12] enhanced Levin’s method using Chebyshev polynomials and Chebyshev nodes to improve numerical stability and accuracy. Khan et al. [13] further advanced the method by employing radial basis functions, resulting in a well-conditioned version of the Levin approach. Olver [14] proposed a hybrid technique that combines Filon-type quadrature with Levin’s method to effectively handle integrals with stationary points. Zaman and Siraj-ul-Islam [15] developed a stable algorithm based on reproducing kernel functions for the numerical evaluation of oscillatory integrals, both with and without a stationary phase. In their framework, reproducing kernel functions—defined on a real Hilbert space—serve as basis functions in the Levin formulation. Ma [16] introduced a modified Levin quadrature to construct a class of composite quadrature rules for evaluating HOIs with weak singularities or stationary points. Wang and Xiang [17] extended the method further by proposing augmented Levin methods for vector-valued HOIs involving exotic oscillators and turning points. In their approach, the original Levin ODE is reformulated into an augmented system, which is then solved using the spectral collocation method.

Since the Levin approach relies on solving differential equations, selecting an appropriate solution technique is crucial for achieving high accuracy in the evaluation of highly oscillatory integrals (HOIs). Among the widely used analytical and semi-analytical methods for solving differential equations, spectral methods, the Variational Iteration Method (VIM) [18,19,20], and the Homotopy Perturbation Method (HPM) [21,22,23] have received considerable attention due to their simplicity and effectiveness in handling a broad class of nonlinear problems. The VIM, originally introduced by He [24], constructs correction functionals based on variational principles, enabling the iterative refinement of approximate solutions with relatively low computational cost. The HPM combines classical perturbation techniques with homotopy theory to produce rapidly converging series solutions without requiring small parameters, making it appealing in various applications in physics and engineering. Despite their advantages, both the VIM and HPM face limitations when applied to problems involving highly oscillatory behavior, such as those governed by Levin’s differential formulation. These methods typically require problem-specific constructions—such as tailored correction functionals in the VIM or embedding parameters in the HPM—limiting their generality and robustness. Moreover, Levin’s equation lacks initial conditions, which makes it an ill-posed problem in traditional formulations and complicates its resolution using standard iterative or perturbative techniques. In contrast, spectral methods naturally handle such formulations through global approximation and yield well-posed algebraic systems. Therefore, for HOIs arising from Levin’s formulation, spectral approaches offer a more robust, general, and highly accurate solution framework.

Spectral approximation methods—particularly Galerkin-type methods—are known for their high convergence rates and numerical stability across a wide range of applications. Chakraborty et al. [25] established superconvergence rates for the multi-Galerkin method without relying on iterated formulations. Yang et al. [26] applied a multistep pseudo-spectral continuous Galerkin method, based on orthogonal Legendre polynomials, to solve first-order Volterra integro-differential equations. Cai [27] proposed a spectral approach for solving a class of second-kind Volterra integral equations with highly oscillatory kernels featuring stationary points. Qin and Cao [28] developed a Legendre–Galerkin spectral collocation least-squares method for approximating Darcy flow problems in both homogeneous and heterogeneous media.

Ma and Liu [29] proposed a well-conditioned spectral Levin method for evaluating HOIs. Their method involves expanding both the known and unknown functions into Chebyshev polynomial series (i.e., a spectral representation on the Chebyshev basis), followed by truncation of the series. However, a notable drawback of this approach is the lack of thorough analysis of the truncation error. To address this limitation, Ma and Liu [30] introduced the Jacobi–Galerkin–Levin method, which employs fractional Jacobi polynomials to construct a Galerkin approximation of the Levin equation. In particular, the method is studied in detail for the case of Chebyshev polynomials. Another well-conditioned technique was proposed by Molabahrami [31], who approximated the unknown function using monomials and then applied the Galerkin method. Although this work includes a detailed convergence analysis, it does not fully address the issue of numerical quadrature. Specifically, the method is formulated for analytic functions rather than their discrete nodal values, which limits its applicability in practical numerical computations.

Motivated by the high accuracy and stability of spectral methods, this paper proposes a spectral Galerkin–Levin approach, which combines Levin’s transformation of the integration problem into an ordinary differential equation with Galerkin orthogonalization using Legendre polynomials. The proposed method addresses key limitations of existing approaches and yields a highly efficient and straightforward quadrature rule for evaluating highly oscillatory integrals.

2. Levin’s Approach to the Evaluation of Highly Oscillatory Integrals

Consider the integral

$$\mathcal{I}\left(\omega \right)={\mathit{\int }}_{-1}^{1}f\left(x\right)exp\left(\mathrm{i}\omega g\left(x\right)\right)dx,$$

(1)

where $f\left(x\right)$ and $g\left(x\right)$ are continuously differentiable functions on the interval $\left[-1,1\right]$, and $\mathrm{i}$ denotes the imaginary unit, ${\mathrm{i}}^2=-1$.

The Levin approach to evaluating the integral (1) is based on finding a function $\varphi \left(x\right)$ that satisfies the differential equation [11]:

$${\displaystyle \frac{d}{dx}}\left(\varphi \left(x\right)exp\left(\mathrm{i}\omega g\right(x\left)\right)\right)=f\left(x\right)exp\left(\mathrm{i}\omega g\left(x\right)\right).$$

(2)

Using this function, the integral (1) can be evaluated as

$$\mathcal{I}\left(\omega \right)=\varphi \left(1\right)exp\left(\mathrm{i}\omega g\right(1\left)\right)-\varphi (-1)exp\left(\mathrm{i}\omega g\right(-1\left)\right).$$

(3)

Since $exp\left(\mathrm{i}g\right(x\left)\right)\ne 0$, Equation (2) can be rewritten as

$$\mathcal{L}\left(\varphi \left(x\right)\right)\equiv {\displaystyle \frac{d\varphi \left(x\right)}{dx}}+\mathrm{i}\omega {\displaystyle \frac{dg\left(x\right)}{dx}}\varphi \left(x\right)-f\left(x\right)=0,$$

(4)

which is the differential equation satisfied by the sought function $\varphi \left(x\right)$ (hereafter referred to as the Levin equation).

Levin [11] proposed representing the function $\varphi \left(x\right)$ as a finite series expansion in terms of a complete system of basis functions $\left\{{\psi }_k\left(x\right)\right\}$ (e.g., monomials or polynomials):

$$\varphi \left(x\right)=\sum _{k=1}^n{a}_k{\psi }_k\left(x\right),$$

(5)

where the coefficients $a_k$ are determined by solving a system of linear algebraic equations obtained from Equation (4) via the collocation technique:

$$\sum _{k=1}^n{a}_k{\displaystyle \frac{d{\psi }_k\left(x_j\right)}{dx}}+\mathrm{i}\omega {\displaystyle \frac{dg\left(x_j\right)}{dx}}\sum _{k=1}^n{a}_k{\psi }_k\left(x_j\right)=f\left(x_j\right)\left(x_j\in \left[-1,1\right],j=1,\dots ,n\right).$$

(6)

In his original work, Levin used a linear set of nodes $x_j$ and monomials for ${\psi }_k\left(x\right)$. Later, Evans and Webster [12] adopted Chebyshev nodes and Chebyshev polynomials as basis functions. However, since no initial conditions are provided, this collocation approach often results in an ill-conditioned system of linear equations [12,29,31].

3. Spectral Galerkin–Levin Approach

This section combines spectral methods—specifically the Galerkin and Levin approaches—to develop a spectral Galerkin–Levin quadrature for highly oscillatory integrals. Furthermore, it demonstrates that applying Galerkin orthogonalization enables the use of spectral methods with arbitrary polynomial bases, whether monomials or orthogonal polynomials.

3.1. Orthogonal Polynomial-Based Galerkin–Levin Approach

Consider approximating the sought function $\varphi \left(x\right)$ with the Legendre polynomial, $P_k\left(x\right)$, through the finite sum of Fourier–Legendre series

$$\varphi \left(x\right)=\sum _{k=0}^n{\varphi }_k{P}_k\left(x\right),$$

(7)

which converges uniformly on any compact domain within an ellipse with foci at $x=\pm 1$ as $n\to \infty$ ([32], par. 18.18). The coefficients ${\varphi }_k$ are defined by

$${\varphi }_k={\displaystyle \frac{2k+1}{2}{\int }_{-1}^1\varphi \left(x\right)P_k\left(x\right)dx.}$$

(8)

Due to Equation (8), the Legendre polynomials satisfy the orthogonality relation

$${\int }_{-1}^1{P}_k\left(x\right)P_j\left(x\right)dx={\displaystyle \frac{2}{2k+1}}{\delta }_{kj},$$

(9)

where ${\delta }_{kj}$ is the Kronecker delta. This orthogonality relation (9) is fundamental for applying the Galerkin method to approximate the solution to the Levin Equation (4).

The functions $f\left(x\right)$ and $g\left(x\right)$ can also be expressed in terms of Legendre polynomials as

$$f\left(x\right)=\sum _{k=0}^n{f}_k{P}_k\left(x\right),$$

(10)

$$g\left(x\right)=\sum _{k=0}^n{g}_k{P}_k\left(x\right).$$

(11)

According to Olver et al. [32], the following differentiation rule holds for Legendre polynomials:

$${\displaystyle \frac{d}{dx}}{P}_{k+1}\left(x\right)=\sum _{j=0}^{k/2}(2(k-2j)+1)P_{k-2j}\left(x\right)\left(k=0,1,\dots \right).$$

(12)

Using this, one can define the differentiation operator (matrix) $D_{jk}$ as

$$D_{jk}=\left\{\begin{array}{cc}2j+1,\hfill & k>j\wedge \left((k+j)mod2\right)=1\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(13)

Then, the derivatives of functions (8) and (11) can be written in the following compact form

$${\varphi }^{\prime }\left(x\right)=\sum _{j=0}^n\sum _{k=0}^n{D}_{jk}{\varphi }_k{P}_j\left(x\right),g^{\prime }\left(x\right)=\sum _{j=0}^n\sum _{k=0}^n{D}_{jk}{g}_k{P}_j\left(x\right).$$

(14)

Substituting Equations (7), (10), (11) and (14) into the Levin Equation (4) yields

$$\sum _{j=0}^n\sum _{k=0}^n\left(D_{jk}{\varphi }_k{P}_j\left(x\right)+\mathrm{i}\omega \sum _{q=0}^n{D}_{qk}{g}_k{\varphi }_j{P}_q\left(x\right)P_j\left(x\right)\right)=\sum _{j=0}^n{f}_j{P}_j\left(x\right).$$

(15)

According to Dougall [33], the product of two Legendre polynomials can be expressed as

$$P_p\left(x\right)P_q\left(x\right)=\sum _{s=0}^{(p+q)/2}{A}_{2s}(p,q)P_{p+q-2s}\left(x\right),$$

(16)

where

$$A_{2s}(p,q)={\displaystyle \frac{2p+2q-4s+1}{2p+2q-2s+1}}{\displaystyle \frac{{\lambda }_s{\lambda }_{p-s}{\lambda }_{q-s}}{{\lambda }_{p+q-s}}}$$

(17)

and

$${\lambda }_s={\displaystyle \frac{\left(2s\right)!}{2^{s}s!s!}}.$$

(18)

Using this result, Equation (15) can be rewritten as

$$\begin{array}{c}{\displaystyle \sum _{j=0}^n\sum _{k=0}^n\left(D_{jk}{\varphi }_k{P}_j\left(x\right)+\mathrm{i}\omega \sum _{q=0}^n\sum _{s=0}^{(j+q)/2}{D}_{qk}{A}_{2s}(j,q)g_k{\varphi }_j{P}_{j+q-2s}\left(x\right)\right)=}\hfill \\ {\displaystyle =\sum _{j=0}^n{f}_j{P}_j\left(x\right).}\hfill \end{array}$$

(19)

Applying the Galerkin orthogonality conditions

$${\int }_{-1}^1\left(\mathcal{L}\left(\varphi \left(x\right)\right)P_k\left(x\right)\right)dx=0\left(k=0,1,\dots ,n\right)$$

(20)

to Equation (19)—that is, multiplying by $P_k\left(x\right)$ and integrating over $(-1,1)$—leads to the following well-conditioned system for determining the coefficients ${\varphi }_k$:

$${\varphi }_k=\sum _{m=0}^n{Z}_{km}^{-1}{f}_m(k=0,1,\dots ,n),$$

(21)

where $Z_{km}^{-1}$ are the coefficients of the matrix ${\mathbf{Z}}^{-1}$, which is the inverse of the matrix $\mathbf{Z}$ with coefficients

$$Z_{mk}=D_{mk}+\mathrm{i}\omega \sum _{p=0}^n{B}_{mkp}{g}_p\left(m,k=0,1,\dots ,n\right),$$

(22)

and

$$B_{mkp}=\sum _{q=0}^n{A}_{k+q-m}(k,q)D_{qp}\left(m,k,p=0,1,\dots ,n\right).$$

(23)

According to Olver et al. [32], the Legendre polynomials satisfy the endpoint conditions

$$P_k\left(1\right)=1,P_k(-1)={(-1)}^k.$$

(24)

Using these properties along with Equations (3), (7), (10), (11), and (21), the sought integral $\mathcal{I}\left(\omega \right)$ from Equation (1) can be approximated as

$$\mathcal{I}\left(\omega \right)\approx {\mathrm{e}}^{\mathrm{i}\omega g\left(1\right)}\sum _{k=0}^n{\varphi }_k-{\mathrm{e}}^{\mathrm{i}\omega g(-1)}\sum _{k=0}^n{(-1)}^k{\varphi }_k.$$

(25)

In numerical analysis, it is often more convenient to work directly with the values of the functions $f\left(x\right)$ and $g\left(x\right)$ at sampling points rather than with their Legendre coefficients $f_k$ and $g_k$, as used in Equation (25). Therefore, to provide a more classical form for the resulting quadrature and enhance its practical implementation, we consider the following numerical approach to evaluate the Legendre coefficients $f_k$ and $g_k$.

Consider a set of sampling points $x_k\in \left[-1,1\right]$ for $k=0,1,\dots ,n$. The functions $f\left(x\right)$ and $g\left(x\right)$ can be approximated by their values $f\left(x_k\right)$ and $g\left(x_k\right)$ at these points using Lagrange interpolation polynomials $L_k\left(x\right)$:

$$f\left(x\right)\approx \sum _{k=0}^{n}f\left(x_k\right)L_k\left(x\right),g\left(x\right)\approx \sum _{k=0}^{n}g\left(x_k\right)L_k\left(x\right),$$

(26)

where

$$L_k\left(x\right)=\prod _{\begin{array}{c}j=0\\ j\ne k\end{array}}^n{\displaystyle \frac{\left(x-x_j\right)}{\left(x_k-x_j\right)}}.$$

(27)

According to Equation (8), the coefficients $f_k$ and $g_k$ can then be approximated as

$$f_k=\sum _{j=0}^n{T}_{kj}f\left(x_j\right),g_k=\sum _{j=0}^n{T}_{kj}g\left(x_j\right),$$

(28)

where

$$T_{kj}={\displaystyle \frac{2k+1}{2}}{\int }_{-1}^1{L}_j\left(x\right)P_k\left(x\right)dx.$$

(29)

Since the integrand in Equation (29) is a polynomial with a degree of at most $2n$, the coefficients $T_{kj}$ can be computed exactly using an $(n+1)$-point Gaussian quadrature, which is exact for polynomials with a degree up to $2n+1$.

The sampling points $x_k$ can, in principle, be chosen arbitrarily. However, to achieve optimal approximation and avoid Runge’s phenomenon, it is advantageous to use Chebyshev–Gauss–Lobatto (CGL) points [34]

$$x_k=-cos\left({\displaystyle \frac{k\pi }{n}}\right)\left(k=0,1,\dots ,n\right).$$

(30)

In the present method, CGL points are used exclusively for approximating the amplitude and phase functions $f\left(x\right)$ and $g\left(x\right)$, both of which are assumed to be smooth and non-oscillatory. The primary purpose of employing these nodes is to construct accurate polynomial interpolants of these smooth functions using global basis functions, thereby ensuring spectral accuracy and mitigating Runge’s phenomenon. The latter typically arises when interpolating smooth functions using high-degree polynomials at equidistant points, where increasing the number of nodes leads to large oscillations near the interval boundaries. In contrast, the clustering of CGL points near the endpoints suppresses such behavior and promotes numerical stability.

It is important to emphasize that these sampling points are not intended to resolve the high-frequency content of the full oscillatory integrand $f\left(x\right)exp\left(\mathrm{i}\omega g\right(x\left)\right)$. Instead, the oscillatory behavior is handled analytically through the proposed Galerkin–Levin formulation, which decouples the resolution of the smooth core functions from the treatment of the oscillatory exponential term.

3.2. Convergence Properties of the Spectral Galerkin–Levin Approach

The convergence of the proposed spectral Galerkin–Levin quadrature is intrinsically tied to the smoothness and analytic properties of the solution $\varphi \left(x\right)$ to the associated Levin differential Equation (4). This subsection presents a theoretical analysis of the convergence rate of the approximation (see Equation (7))

$${\varphi }_n\left(x\right)=\sum _{k=0}^n{\varphi }_k{P}_k\left(x\right),$$

where $P_k\left(x\right)$ are Legendre polynomials and ${\varphi }_k$ are computed via Galerkin projection.

Consider a case where the amplitude function $f\left(x\right)$ and the phase function $g\left(x\right)$, and hence, the solution $\varphi \left(x\right)$ are analytic in a closed Bernstein ellipse ${\mathcal{E}}_{\rho }$ in the complex plane containing the interval $[-1,1]$. This condition is satisfied when $g^{\prime }\left(x\right)\ne 0$ on $[-1,1]$, ensuring that the phase function has no stationary points and the transformation to the Levin Equation (4) does not introduce singularities.

In this setting, the Legendre approximation ${\varphi }_n\left(x\right)$ converges exponentially in the $L^2$-norm:

$$\parallel \varphi -{\varphi }_n{\parallel }_{L^2(-1,1)}\le C{\rho }^{-n},\mathrm{for}\mathrm{some}\rho >1,$$

where $\rho$ is determined by the size of the largest ellipse ${\mathcal{E}}_{\rho }$ in which $\varphi$ is analytic, and C is a constant independent of n [35,36]. The corresponding error in the final integral approximation

$$\mathcal{I}\left(\omega \right)\approx {\varphi }_n\left(1\right){\mathrm{e}}^{\mathrm{i}\omega g\left(1\right)}-{\varphi }_n(-1){\mathrm{e}}^{\mathrm{i}\omega g(-1)}$$

inherits this exponential convergence.

Furthermore, since Galerkin projection is an orthogonal projection in the Hilbert space $L^2\left([-1,1]\right)$, the Galerkin approximation achieves the best possible accuracy within the chosen polynomial subspace. Thus, the obtained convergence rate is not only guaranteed but also optimal for polynomial spaces of degree n.

When the phase function $g\left(x\right)$ contains stationary points (i.e., $g^{\prime }\left(x\right)=0$) at some $x=x_0\in [-1,1])$, the solution $\varphi \left(x\right)$ may exhibit reduced smoothness or local singular behavior. This affects the decay rate of Legendre coefficients ${\varphi }_k$, and therefore the convergence of ${\varphi }_n\left(x\right)$.

In such cases, the convergence of the Galerkin projection is algebraic [35,36]. If $\varphi \left(x\right)\in H^s\left([-1,1]\right)$, for some $s>0$, the approximation satisfies the bound:

$$\parallel \varphi -{\varphi }_n{\parallel }_{L^2(-1,1)}\le C{n}^{-s},$$

where s reflects the degree of smoothness, and C is a constant [35,36]. In the worst case, convergence may be significantly slower, and more basis functions are required to achieve high precision.

The numerical results presented in the Numerical Examples section corroborate these theoretical findings.

3.3. Discussion on the Selection of Polynomial Basis Functions and Approximation Points

Some authors [12,29,34] highlight the advantages of Chebyshev polynomials over monomials or other classical orthogonal polynomials (such as Jacobi polynomials) for approximating solutions to certain equations, including the Levin Equation (4). However, each approach has its own peculiarities, which may or may not be influenced by the choice of polynomial basis functions.

Consider three different polynomial bases:

• Monomials

  $$M_k\left(x\right)=x^k\left(k=0,1,\dots ,n\right),$$

  (31)
• Legendre polynomials

  $$P_0\left(x\right)=1,P_1\left(x\right)=x,P_{k+1}\left(x\right)={\displaystyle \frac{2k+1}{k+1}}x{P}_k\left(x\right)-{\displaystyle \frac{k}{k+1}}{P}_{k-1}\left(x\right)$$

  (32)
• Chebyshev polynomials

  $$T_0\left(x\right)=1,T_1\left(x\right)=x,T_{k+1}\left(x\right)=2x{T}_k\left(x\right)-T_{k-1}\left(x\right).$$

  (33)

In Equations (32) and (33), the index $k=1,2,\dots$.

When considering a finite set of $n+1$ polynomial basis functions $(k=0,\dots ,n)$, it can be shown that

$$P_k\left(x\right)=\sum _{j=0}^n{\alpha }_{kj}{M}_j\left(x\right),T_k\left(x\right)=\sum _{j=0}^n{\beta }_{kj}{M}_j\left(x\right),$$

(34)

where, according to Equations (32) and (33), the nonzero coefficients satisfy the recurrence relations

$${\alpha }_{00}=1,{\alpha }_{11}=1,{\alpha }_{(k+1)j}={\displaystyle \frac{2k+1}{k+1}}{\alpha }_{k(j-1)}-{\displaystyle \frac{k}{k+1}}{\alpha }_{(k-1)j},$$

(35)

$${\beta }_{00}=1,{\beta }_{11}=1,{\beta }_{(k+1)j}=2{\beta }_{k(j-1)}-{\beta }_{(k-1)j},$$

(36)

with $k=1,2,...$.

Moreover, the inverse coefficient matrices ${\alpha }_{kj}^{-1}$ and ${\beta }_{kj}^{-1}$ exist such that

$$M_k\left(x\right)=\sum _{j=0}^n{\alpha }_{kj}^{-1}{P}_j\left(x\right)=\sum _{j=0}^n{\beta }_{kj}^{-1}{T}_j\left(x\right).$$

(37)

Therefore, the Galerkin Equation (20) can be rewritten for monomial and Chebyshev basis functions as follows:

$${\int }_{-1}^1\left(\mathcal{L}\left(\varphi \left(x\right)\right)P_k\left(x\right)\right)dx=\sum _{j=0}^n{\alpha }_{kj}{\int }_{-1}^1\left(\mathcal{L}\left(\varphi \left(x\right)\right)M_j\left(x\right)\right)dx=0,$$

(38)

$${\int }_{-1}^1\left(\mathcal{L}\left(\varphi \left(x\right)\right)P_k\left(x\right)\right)dx=\sum _{j=0}^n\sum _{m=0}^n{\alpha }_{kj}{\beta }_{jm}^{-1}{\int }_{-1}^1\left(\mathcal{L}\left(\varphi \left(x\right)\right)T_m\left(x\right)\right)dx=0.$$

(39)

Therefore, regardless, the monomial, Legendre, or Chebyshev polynomial basis is selected, according to Equations (34), (37)–(39). The resulting Galerkin–Levin system of equations is just a linear mapping of the system with the Legendre polynomial basis and vice versa. Thus, the main advantage of the proposed approach, as well as other Galerkin-type or spectral methods, can be achieved with the appropriate selection of approximation points $x_k$ in Equations (28) and (29).

Numerical tests for selected highly oscillatory integrals, as well as those of Evans and Webster [12], enable Chebyshev nodes to allow us to obtain as much as two times greater accuracy compared to the uniform distribution of sampling points for low numbers of these points. Increasing the number of sampling points with a simultaneous increase in the number of resulting equations significantly decreases the error of integration, and the deviation between different sampling becomes insufficient. Nevertheless, Chebyshev–Gauss–Lobatto (30) points allow us to exclude Runge’s phenomenon; therefore, this sampling points set is chosen for numerical integration with the proposed spectral Galerkin–Levin approach.

4. Spectral Galerkin–Levin Quadrature for HOIs of Matrix Functions

Consider the integral

$$\mathcal{I}\left(\omega \right)={\int }_{-1}^1\mathbf{F}\left(x\right)exp\left(\mathrm{i}\omega \mathbf{G}\left(x\right)\right)dx,$$

(40)

where $\mathbf{F}\left(x\right)$ and $\mathbf{G}\left(x\right)$ are square matrices, and $\mathbf{G}\left(x\right)$ is positive definite. Following the Levin approach (2), we assume there exists a matrix function $\mathbf{\Phi }\left(x\right)$ such that

$${\displaystyle \frac{d}{dx}}\left(\mathbf{\Phi }\left(x\right)exp\left(\mathrm{i}\omega \mathbf{G}\right(x\left)\right)\right)=\mathbf{F}\left(x\right)exp\left(\mathrm{i}\omega \mathbf{G}\left(x\right)\right).$$

(41)

Then, the integral (40) evaluates to

$$\mathcal{I}\left(\omega \right)=\mathbf{\Phi }\left(1\right)exp\left(\mathrm{i}\omega \mathbf{G}\right(1\left)\right)-\mathbf{\Phi }(-1)exp\left(\mathrm{i}\omega \mathbf{G}\right(-1\left)\right).$$

(42)

By applying the product rule to the derivative in Equation (41), one obtains

$${\displaystyle \frac{d\mathbf{\Phi }\left(x\right)}{dx}}exp\left(\mathrm{i}\omega \mathbf{G}\left(x\right)\right)+\mathrm{i}\omega \mathbf{\Phi }\left(x\right){\displaystyle \frac{d\mathbf{G}\left(x\right)}{dx}}exp\left(\mathrm{i}\omega \mathbf{G}\left(x\right)\right)=\mathbf{F}\left(x\right)exp\left(\mathrm{i}\omega \mathbf{G}\left(x\right)\right)$$

(43)

Since $exp\left(\mathrm{i}\omega \mathbf{G}\right(x\left)\right)\ne \mathbf{0}$, the latter equation can be rewritten as

$${\displaystyle \frac{d\mathbf{\Phi }\left(x\right)}{dx}}+\mathrm{i}\omega \mathbf{\Phi }\left(x\right){\displaystyle \frac{d\mathbf{G}\left(x\right)}{dx}}=\mathbf{F}\left(x\right).$$

(44)

Transposing the matrices, we have

$${\displaystyle \frac{d{\mathbf{\Phi }}^{\mathrm{T}}\left(x\right)}{dx}}+\mathrm{i}\omega {\displaystyle \frac{d{\mathbf{G}}^{\mathrm{T}}\left(x\right)}{dx}}{\mathbf{\Phi }}^{\mathrm{T}}\left(x\right)={\mathbf{F}}^{\mathrm{T}}\left(x\right).$$

(45)

Assuming the expansions

$$\mathbf{\Phi }\left(x\right)=\sum _{k=0}^n{\mathbf{\Phi }}_k{P}_k\left(x\right),\mathbf{G}\left(x\right)=\sum _{k=0}^n{\mathbf{G}}_k{P}_k\left(x\right),\mathbf{F}\left(x\right)=\sum _{k=0}^n{\mathbf{F}}_k{P}_k\left(x\right)$$

(46)

and following the spectral Galerkin–Levin approach presented above, the coefficients ${\mathbf{\Phi }}_k$ are determined by solving the system

$$\sum _{k=0}^n\left(D_{mk}+\mathrm{i}\omega \sum _{p=0}^n{B}_{mkp}{\mathbf{G}}_p^{\mathrm{T}}\right){\mathbf{\Phi }}_k^{\mathrm{T}}={\mathbf{F}}_m^{\mathrm{T}},$$

(47)

where the matrices $D_{mk}$ and $B_{mkp}$ are defined by Equations (13) and (23), respectively.

This approach avoids the need to compute matrix eigenvectors and eigenvalues at each collocation point, which are usually required to evaluate the matrix exponential in Equation (40). Instead, the matrix exponential is evaluated only at the endpoints. Accordingly, using Equations (24), (42), and (46), the integral is approximated as

$$\mathcal{I}\left(\omega \right)\approx \left(\sum _{k=0}^n{\mathbf{\Phi }}_k\right){\mathrm{e}}^{\mathrm{i}\omega \mathbf{G}\left(1\right)}-\left(\sum _{k=0}^n{(-1)}^k{\mathbf{\Phi }}_k\right){\mathrm{e}}^{\mathrm{i}\omega \mathbf{G}(-1)}.$$

(48)

Equation (48) can be effectively used to evaluate the 3D time-harmonic Green’s function. According to Pasternak et al. [37], the Radon transform technique allows the expression of the 3D time-harmonic Green’s function (fundamental solution) for anisotropic multifield or quasicrystalline media in the following matrix form:

$$\tilde{\mathbf{U}}(\mathbf{x}-{\mathbf{x}}_0)={\tilde{\mathbf{U}}}^R(\mathbf{x}-{\mathbf{x}}_0)+{\tilde{\mathbf{U}}}^s(\mathbf{x}-{\mathbf{x}}_0),$$

(49)

where

$${\tilde{\mathbf{U}}}^R\left(\mathbf{x}\right)={\displaystyle \frac{\mathrm{i}\omega }{16{\pi }^2}}{\int \int }_{\left|\xi \right|=1}{\mathbf{\Gamma }}^{-1}\sqrt{\mathbf{M}{\mathbf{\Gamma }}^{-1}}exp\left(\mathrm{i}\omega \sqrt{\mathbf{M}{\mathbf{\Gamma }}^{-1}}\left|\xi ·\mathbf{x}\right|\right)dS\left(\xi \right),$$

(50)

$${\tilde{\mathbf{U}}}^S\left(\mathbf{x}\right)={\displaystyle \frac{1}{8{\pi }^2\left|\mathbf{x}\right|}}{\oint }_{\begin{array}{c}\left|\xi \right|=1,\\ \xi ·\mathbf{x}=0\end{array}}{\mathbf{\Gamma }}^{-1}\left(\xi \right)dl\left(\xi \right).$$

(51)

Here, ${\mathbf{\Gamma }}_{IJ}={\sum }_{k=1}^3{\sum }_{m=1}^3{\tilde{C}}_{IkJm}{\xi }_k{\xi }_m$, where ${\tilde{C}}_{IjKm}$ is the extended tensor of material constants, and $M_{IJ}$ is the diagonal matrix of inertial properties [37]. The range of the capital indices depends on the type of material considered (e.g., crystalline, quasicrystalline, ferroelectric, etc.).

In the evaluation of the time-harmonic Green’s function (49), its component (50) is a highly oscillatory integral over the surface of the unit sphere, which can be reduced to the following double integral [37]:

$${\tilde{\mathbf{U}}}^R\left(\mathbf{x}\right)={\displaystyle \frac{\mathrm{i}\omega }{8{\pi }^2}}{\int }_{-\pi }^{\pi }{\int }_0^1{\mathbf{\Gamma }}^{-1}\sqrt{\mathbf{M}{\mathbf{\Gamma }}^{-1}}exp\left(\mathrm{i}\omega \sqrt{\mathbf{M}{\mathbf{\Gamma }}^{-1}}\left|\mathbf{x}\right|b\right)dbd\phi ,$$

(52)

where $\xi =\sqrt{1-b^2}\lambda \left(\phi \right)+b\mathbf{x}/\left|\mathbf{x}\right|$ and $\lambda \left(\phi \right)\perp \mathbf{x}$.

The inner integral in Equation (52) can be efficiently evaluated using the proposed spectral Galerkin–Levin matrix quadrature (48). However, Levin’s method is not suitable for the outer integral because the integrand is periodic over the integration domain, resulting in multiple stationary points of $g\left(x\right)$ in the general case. Consequently, the sought function $\mathbf{\Phi }\left(x\right)$ exhibits highly oscillatory behavior similar to that of the integrand, making the direct application of Levin’s method ineffective. Therefore, the outer integral requires alternative quadrature methods or a high number of integration nodes.

Nonetheless, applying the proposed approach to the inner integral substantially reduces the overall number of nodes required for numerical integration, which is a significant computational advantage.

5. Extension to Higher-Dimensional Oscillatory Integrals via Nested Quadrature

Although the proposed spectral Galerkin–Levin quadrature is formulated for one-dimensional integrals, it can be systematically extended to multidimensional highly oscillatory integrals (HOIs) defined over square or product-type domains. The key idea is to interpret the multidimensional integral as a sequence of nested one-dimensional integrals, allowing the Galerkin–Levin approach to be applied iteratively in each coordinate direction.

Consider the two-dimensional oscillatory integral over a square domain:

$${\mathcal{I}}^{\left(2D\right)}\left(\omega \right)={\int }_{-1}^1{\int }_{-1}^{1}f(x,y){\mathrm{e}}^{\mathrm{i}\omega g(x,y)}dxdy,$$

(53)

where $f(x,y)$ and $g(x,y)$ are smooth amplitude and phase functions, and $\omega \gg 1$.

We begin by expressing the integral in a nested form:

$${\mathcal{I}}^{\left(2D\right)}\left(\omega \right)={\int }_{-1}^1{I}_{\mathrm{inner}}\left(y\right)dy,\mathrm{with}{I}_{\mathrm{inner}}\left(y\right)={\int }_{-1}^{1}f(x,y){\mathrm{e}}^{\mathrm{i}\omega g(x,y)}dx.$$

(54)

To evaluate $I_{\mathrm{inner}}\left(y\right)$, we apply the Galerkin–Levin technique in the x-direction. We construct a function $\varphi (y,x)$ satisfying:

$${\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }x}}\left(\varphi (y,x){\mathrm{e}}^{\mathrm{i}\omega g(x,y)}\right)=f(x,y){\mathrm{e}}^{\mathrm{i}\omega g(x,y)}.$$

(55)

According to the Levin identity, the inner integral reduces to

$$I_{\mathrm{inner}}\left(y\right)=\varphi (y,1){\mathrm{e}}^{\mathrm{i}\omega g(1,y)}-\varphi (y,-1){\mathrm{e}}^{\mathrm{i}\omega g(-1,y)}.$$

(56)

Substituting Equation (56) into Equation (54), the two-dimensional integral becomes

$$\begin{array}{}\hfill \mathrm{(57)}& \hfill {\mathcal{I}}^{\left(2D\right)}\left(\omega \right)& ={\int }_{-1}^1\varphi (y,1){\mathrm{e}}^{\mathrm{i}\omega g(1,y)}dy-{\int }_{-1}^1\varphi (y,-1){\mathrm{e}}^{\mathrm{i}\omega g(-1,y)}dy\hfill \hfill \mathrm{(58)}& \hfill & ={\mathcal{I}}_{+}\left(\omega \right)-{\mathcal{I}}_{-}\left(\omega \right),\hfill \end{array}$$

where ${\mathcal{I}}_{\pm }\left(\omega \right)$ denotes 1D oscillatory integrals in y with modified amplitudes $\varphi (y,\pm 1)$ and phase functions $g(\pm 1,y)$.

The above transformation demonstrates that the original two-dimensional HOI (53) is reduced to th difference between two one-dimensional HOIs, where the highly oscillatory behavior in the x-direction is handled by the Galerkin–Levin procedure. Importantly, the remaining y-dependent integrals ${\mathcal{I}}_{\pm }\left(\omega \right)$ retain a highly oscillatory structure and are likewise efficiently evaluated using the same spectral Galerkin–Levin technique. This recursive application of the method enables the systematic and accurate treatment of multidimensional oscillatory integrals over square or tensor-product domains.

In the multidimensional nested approach, the functions $\varphi (y,1)$ and $\varphi (y,-1)$, which appear as boundary values in the inner integral evaluation, are computed at a discrete set of points $y_j\in [-1,1]$. These nodal values are then approximated by expanding $\varphi (y,\pm 1)$ in terms of Legendre polynomials as

$$\varphi (y,\pm 1)\approx \sum _{m=0}^n{\varphi }_m^{\pm }{P}_m\left(y\right),$$

where the coefficients ${\varphi }_m^{\pm }$ are determined via Equation (28). This approach preserves the spectral accuracy in each coordinate direction by systematically approximating the boundary functions of the inner integrals, thereby enabling efficient and accurate computation of the overall multidimensional highly oscillatory integral.

The nested framework naturally extends the spectral Galerkin–Levin approach to multidimensional highly oscillatory integrals, enabling accurate and efficient numerical evaluation over tensor-product domains. Such multidimensional integrals commonly arise in various applied fields, including computational electromagnetics, wave propagation, and quantum mechanics, where handling oscillations in multiple variables is critical. By recursively applying the one-dimensional Galerkin–Levin quadrature in each coordinate direction, the method achieves spectral accuracy and computational efficiency in higher dimensions.

6. Numerical Examples

In this section, we present several numerical examples to demonstrate the accuracy, stability, and efficiency of the proposed spectral Galerkin – Levin method for evaluating highly oscillatory integrals. All computations were performed using the freely available SciLab software package. A custom SciLab source code was developed specifically to implement the proposed approach. The code closely follows the algorithmic structure described in the paper and was used to produce all of the numerical results presented below.

6.1. Example 1: Comparison with Levin’s Approach

Following Levin [11], consider the highly oscillatory integral

$$I_1={\int }_0^{1}sinx{\mathrm{e}}^{\mathrm{i}W(x+c{x}^2)}dx$$

(59)

where the parameters are set to $c=1$ and $W=500$. This integral features a quadratic phase function that induces rapid oscillations for large W, making it a suitable test case for evaluating the performance of numerical quadrature methods designed for highly oscillatory integrals.

Table 1. A comparison of the proposed method and Levin’s approach for evaluating Equation (59).

Approach	$\mathit{n}=5$	$\mathit{n}=10$
Levin [11], ${10}^4\Re I_1$	$4.60098$	$4.59863$
Proposed, ${10}^4\Re I_1$	$4.58840$	$4.59864$
Romberg integration [11], ${10}^4\Re I_1$			$4.59859$

presents a detailed comparison between the numerical results obtained by the proposed spectral Galerkin–Levin method and those originally reported by Levin [11]. The table lists the approximated values of $I_1$ for varying numbers of basis functions n, which control the approximation accuracy in both methods.

The first column of Table 1 specifies the method used, while the second and third columns show the computed values of the real part of $I_1$ for different values of n. The final column provides the reference value of the integral obtained using Romberg integration.

From Table 1, it is evident that both methods achieve high accuracy, with the proposed method closely matching Levin’s results. Notably, the spectral Galerkin–Levin approach attains this accuracy with relatively small n, demonstrating fast convergence and numerical stability even for a highly oscillatory integrand with large W.

6.2. Example 2: Numerical Evaluation of Oscillatory Integral with Known Exact Value

Consider the highly oscillatory integral introduced by Molabahrami [31]

$$I_2\left(\omega \right)={\int }_{-1}^{1}xcos(x^3+x)exp\left(\mathrm{i}\omega \pi x^2\right)dx,$$

(60)

whose exact value is known to be zero for all $\omega$, i.e., $I_2\left(\omega \right)=0.$

Table 2. Numerical evaluation of integral (60) using proposed spectral Galerkin–Levin method.

	$\mathit{n}=4$	$\mathit{n}=8$	$\mathit{n}=16$
$\omega =100$	$\begin{array}{c}1.694·{10}^{-20}\hfill \\ -4.337·{10}^{-19}\mathrm{i}\hfill \end{array}$	$\begin{array}{c}-7.454·{10}^{-20}\hfill \\ +8.674·{10}^{-19}\mathrm{i}\hfill \end{array}$	$\begin{array}{c}-7.438·{10}^{-17}\hfill \\ -1.930·{10}^{-17}\mathrm{i}\hfill \end{array}$
$\omega =200$	$\begin{array}{c}4.235·{10}^{-21}\hfill \\ -2.168·{10}^{-19}\mathrm{i}\hfill \end{array}$	$\begin{array}{c}-1.863·{10}^{-20}\hfill \\ +4.337·{10}^{-19}\mathrm{i}\hfill \end{array}$	$\begin{array}{c}-7.507·{10}^{-17}\hfill \\ -9.595·{10}^{-18}\mathrm{i}\hfill \end{array}$
$\omega =400$	$\begin{array}{c}1.059·{10}^{-21}\hfill \\ -1.084·{10}^{-19}\mathrm{i}\hfill \end{array}$	$\begin{array}{c}-4.659·{10}^{-21}\hfill \\ +2.168·{10}^{-19}\mathrm{i}\hfill \end{array}$	$\begin{array}{c}-7.524·{10}^{-17}\hfill \\ -4.852·{10}^{-18}\mathrm{i}\hfill \end{array}$
$\omega =600$	$\begin{array}{c}5.020·{10}^{-17}\hfill \\ +5.421·{10}^{-20}\mathrm{i}\hfill \end{array}$	$\begin{array}{c}5.019·{10}^{-17}\hfill \\ +3.253·{10}^{-19}\mathrm{i}\hfill \end{array}$	$\begin{array}{c}-2.510·{10}^{-16}\hfill \\ -3.754·{10}^{-18}\mathrm{i}\hfill \end{array}$

displays the results of applying the proposed spectral Galerkin–Levin method to evaluate $I_2\left(\omega \right)$ for various truncation orders n and oscillation frequencies $\omega$. Each cell shows the real part of the integral value on the top and the imaginary part on the bottom. The computations were performed using IEEE double-precision floating-point arithmetic (64-bit), which ensures numerical precision up to approximately 16 significant decimal digits.

From Table 2, it is evident that the proposed method achieves excellent accuracy even for very high values of $\omega$ and relatively small numbers of basis functions n. All computed integral values lie within the machine epsilon $\left(\approx 2.22·{10}^{-16}\right)$ of the exact zero value, confirming the numerical stability and high precision of the approach in handling highly oscillatory integrals.

6.3. Example 3: Numerical Evaluation of Oscillatory Integral with Known Exact Value

Consider the following integral [31]:

$$I_3\left(\omega \right)={\int }_0^{1}cos(sin\left(x\right))cos\left(x\right)exp(\mathrm{i}\omega sin\left(x\right))dx,$$

(61)

which has the exact analytical value given by

$$I_3\left(\omega \right)={\displaystyle \frac{\mathrm{i}\left(\omega +{\mathrm{e}}^{\mathrm{i}\omega sin\left(1\right)}\left(-\omega cos(sin(1\left)\right)+\mathrm{i}sin(sin(1\left)\right)\right)\right)}{{\omega }^2-1}}.$$

(62)

Table 3. Numerical evaluation of the integral (61) with the proposed approach.

	$\mathit{n}=4$	$\mathit{n}=8$	$\mathit{n}=16$	Exact, Equation (62)
$\omega =100$	$\begin{array}{c}\underline{4.2}321203·{10}^{-3}\hfill \\ +\underline{1.51}41538·{10}^{-2}\mathrm{i}\hfill \end{array}$	$\begin{array}{c}\underline{4.22739}71·{10}^{-3}\hfill \\ +\underline{1.51534}48·{10}^{-2}\mathrm{i}\hfill \end{array}$	$\begin{array}{c}\underline{4.2273968}·{10}^{-3}\hfill \\ +\underline{1.5153477}·{10}^{-2}\mathrm{i}\hfill \end{array}$	$\begin{array}{c}4.227396852·{10}^{-3}\hfill \\ +1.515347744·{10}^{-2}\mathrm{i}\hfill \end{array}$
$\omega =200$	$\begin{array}{c}-\underline{3.2}692586·{10}^{-3}\hfill \\ +\underline{4.2}7367·{10}^{-3}\mathrm{i}\hfill \end{array}$	$\begin{array}{c}-\underline{3.2563}837·{10}^{-3}\hfill \\ +\underline{4.29448}52·{10}^{-3}\mathrm{i}\hfill \end{array}$	$\begin{array}{c}-\underline{3.256391}7·{10}^{-3}\hfill \\ +\underline{4.294482}4·{10}^{-3}\mathrm{i}\hfill \end{array}$	$\begin{array}{c}-3.256391695·{10}^{-3}\hfill \\ +4.294482267·{10}^{-3}\mathrm{i}\hfill \end{array}$
$\omega =400$	$\begin{array}{c}-\underline{7.0}552709·{10}^{-4}\hfill \\ +\underline{4.0}074664·{10}^{-3}\mathrm{i}\hfill \end{array}$	$\begin{array}{c}-\underline{7.0231}465·{10}^{-4}\hfill \\ +\underline{4.0106}665·{10}^{-3}\mathrm{i}\hfill \end{array}$	$\begin{array}{c}-\underline{7.023177}6·{10}^{-4}\hfill \\ +\underline{4.0106728}·{10}^{-3}\mathrm{i}\hfill \end{array}$	$\begin{array}{c}-7.023177229·{10}^{-4}\hfill \\ +4.010672839·{10}^{-3}\mathrm{i}\hfill \end{array}$
$\omega =600$	$\begin{array}{c}\underline{8.8}360944·{10}^{-4}\hfill \\ +\underline{2.34}01971·{10}^{-3}\mathrm{i}\hfill \end{array}$	$\begin{array}{c}\underline{8.80}69819·{10}^{-4}\hfill \\ +\underline{2.34330}21·{10}^{-3}\mathrm{i}\hfill \end{array}$	$\begin{array}{c}\underline{8.807008}2·{10}^{-4}\hfill \\ +\underline{2.3433060}·{10}^{-3}\mathrm{i}\hfill \end{array}$	$\begin{array}{c}8.807008121·{10}^{-4}\hfill \\ +2.343306023·{10}^{-3}\mathrm{i}\hfill \end{array}$

presents a detailed comparison between the numerical results obtained using the proposed spectral Galerkin–Levin method and the exact values calculated from the above expression for various oscillation frequencies $\omega$ and truncation orders n.

These results demonstrate very rapid convergence of the numerical approximation as the number of sampling points n increases. Remarkably, in this particular example, the accuracy of the method shows negligible sensitivity to the frequency parameter $\omega$, maintaining high precision even for large $\omega$. This highlights the robustness of the proposed approach in handling highly oscillatory integrals with nonlinear oscillatory phases.

To assess the accuracy and convergence of the proposed method, we define the relative error of the numerical computation as

$${\delta }_3\left(\omega \right)={\displaystyle \frac{\left|\left|I_3^{\mathrm{num}}\left(\omega \right)\right|-\left|I_3\left(\omega \right)\right|\right|}{\left|I_3\left(\omega \right)\right|}},$$

where $I_3^{\mathrm{num}}\left(\omega \right)$ is the numerical result obtained using the spectral Galerkin–Levin approach, and $I_3\left(\omega \right)$ is the exact value given by Equation (62).

Figure 1 displays the computed relative errors ${\delta }_3\left(\omega \right)$ for various values of $\omega$ and truncation orders n corresponding to the results in Table 3. The figure clearly demonstrates the exponential convergence of the proposed method: the relative error decreases rapidly with increasing n. This behavior confirms the high accuracy and numerical stability of the spectral Galerkin–Levin quadrature, even for integrals with nonlinear phase functions.

6.4. Example 4: An Integral with a Stationary Point

As the fourth example, consider the integral [29]

$$I_4\left(\omega \right)={\int }_{-1}^{1}cos\left(x\right)exp\left(\mathrm{i}\omega {(x-1)}^2\right)dx,$$

(63)

which features a stationary point at $x=1$. The exact analytical value of this integral is given by

$$I_4\left(\omega \right)={\displaystyle \frac{{\mathrm{e}}^{\mathrm{i}\pi /4}}{4}}\sqrt{{\displaystyle \frac{\pi }{\omega }}}\left(\mathrm{erf}\left(A\left(4\omega -1\right)\right){\mathrm{e}}^B+{\mathrm{e}}^C\mathrm{erf}\left(A\left(4\omega +1\right)\right)-\mathrm{erf}\left(A\right)\left({\mathrm{e}}^C-{\mathrm{e}}^B\right)\right),$$

(64)

where the constants are defined as

$$A={\displaystyle \frac{\left(1-\mathrm{i}\right)\sqrt{2}}{4\sqrt{\omega }}},B={\displaystyle \frac{\mathrm{i}\left(4\omega -1\right)}{4\omega }},C=-{\displaystyle \frac{\mathrm{i}\left(4\omega +1\right)}{4\omega }}.$$

Table 4. Numerical evaluation of the integral (63) with the proposed approach.

	$\mathit{n}=16$	$\mathit{n}=32$	$\mathit{n}=64$	Exact, Equation (64)
$\omega =100$	$\begin{array}{c}\underline{0.0328}942\hfill \\ +\underline{0.038}9471\mathrm{i}\hfill \end{array}$	$\begin{array}{c}\underline{0.0328043}\hfill \\ +\underline{0.0386964}\mathrm{i}\hfill \end{array}$	$\begin{array}{c}\underline{0.0328043}\hfill \\ +\underline{0.0386965}\mathrm{i}\hfill \end{array}$	$\begin{array}{c}0.03280425888\hfill \\ +0.03869647188\mathrm{i}\hfill \end{array}$
$\omega =200$	$\begin{array}{c}\underline{0.02}39407\hfill \\ +\underline{0.0263}233\mathrm{i}\hfill \end{array}$	$\begin{array}{c}\underline{0.024577}2\hfill \\ +\underline{0.02631}58\mathrm{i}\hfill \end{array}$	$\begin{array}{c}\underline{0.0245777}\hfill \\ +\underline{0.0263164}\mathrm{i}\hfill \end{array}$	$\begin{array}{c}0.02457772755\hfill \\ +0.02631635525\mathrm{i}\hfill \end{array}$
$\omega =400$	$\begin{array}{c}\underline{0.016}7674\hfill \\ +\underline{0.01}69926\mathrm{i}\hfill \end{array}$	$\begin{array}{c}\underline{0.016669}3\hfill \\ +\underline{0.01817}84\mathrm{i}\hfill \end{array}$	$\begin{array}{c}\underline{0.0166699}\hfill \\ +\underline{0.0181729}\mathrm{i}\hfill \end{array}$	$\begin{array}{c}0.01666992395\hfill \\ +0.01817287590\mathrm{i}\hfill \end{array}$
$\omega =600$	$\begin{array}{c}\underline{0.01}50312\hfill \\ +\underline{0.01}33997\mathrm{i}\hfill \end{array}$	$\begin{array}{c}\underline{0.0137}769\hfill \\ +\underline{0.0142}887\mathrm{i}\hfill \end{array}$	$\begin{array}{c}\underline{0.0137888}\hfill \\ +\underline{0.0142965}\mathrm{i}\hfill \end{array}$	$\begin{array}{c}0.01378881523\hfill \\ +0.01429652647\mathrm{i}\hfill \end{array}$

presents the results of evaluating $I_4\left(\omega \right)$ using the proposed spectral Galerkin–Levin quadrature for various values of $\omega$ and truncation order n. Both the real and imaginary parts of the computed values are compared with the analytical solution given above.

Although the presence of a stationary point generally degrades the convergence rate of Levin-type methods, the proposed approach remains effective. As shown in Table 4, even for high oscillation frequencies (e.g., $\omega =600$), accurate results are obtained with as few as 64 quadrature nodes.

This example illustrates that, despite slower convergence in the presence of stationary points, the proposed quadrature is still applicable and robust. However, in such cases, special attention should be paid to the convergence behavior, and a higher number of interpolation points may be required.

7. Conclusions

The presented spectral Galerkin–Levin approach defines a robust and efficient quadrature rule for the evaluation of highly oscillatory integrals of the form (1), relying solely on the nodal values of the core functions $f\left(x\right)$ and $g\left(x\right)$. Unlike the classical Levin collocation methods, this approach employs Galerkin orthogonalization combined with Legendre polynomial interpolation, resulting in a well-posed and numerically stable system of linear algebraic equations. The flexibility of the method allows the use of various interpolation nodes without loss of stability or accuracy. Furthermore, it is rigorously demonstrated that the Galerkin orthogonalization and the resulting solution are invariant with respect to the choice of polynomial basis—be it monomials or orthogonal polynomials—thus highlighting the generality and robustness of the approach.

The proposed method extends naturally to matrix-valued HOIs, significantly broadening its applicability to complex problems in mathematical physics, such as those involving anisotropic media and multifield models. This extension is achieved without incurring additional computational complexity related to matrix exponential evaluations, which are confined to boundary points.

Numerical experiments confirm that the method exhibits high convergence rates and excellent stability for HOIs both with and without stationary points in the oscillatory phase function. While the presence of stationary points naturally demands a higher density of sampling nodes to maintain accuracy, the approach remains computationally efficient and significantly reduces the overall complexity compared to traditional quadrature techniques.

This work provides a simple, flexible, and reliable numerical tool for tackling a broad class of highly oscillatory integrals, with potential for impactful applications in scientific computing and applied mathematics.