Related papers: A Levin method for logarithmically singular oscill…
In this paper, new Levin methods are presented for calculating oscillatory integrals with algebraic and/or logarithmic singularities. To avoid singularity, the technique of singularity separation is applied and then the singular ODE…
A practical and simple stable method for calculating Fourier integrals is proposed, effective both at low and at high frequencies. An approach based on the fruitful idea of Levin, to use of the collocation method to approximate the slowly…
The Levin method is a well-known technique for evaluating oscillatory integrals, which operates by solving a certain ordinary differential equation in order to construct an antiderivative of the integrand. It was long believed that this…
We develop two classes of composite moment-free numerical quadratures for computing highly oscillatory integrals having integrable singularities and stationary points. The first class of the quadrature rules has a polynomial order of…
In this work we propose and analyse a numerical method for computing a family of highly oscillatory integrals with logarithmic singularities. For these quadrature rules we derive error estimates in terms of $N$, the number of nodes, $k$ the…
We present a methodology for numerically integrating ordinary differential equations containing rapidly oscillatory terms. This challenge is distinct from that for differential equations which have rapidly oscillatory solutions: here the…
The efficient approximation of highly oscillatory integrals plays an important role in a wide range of applications. Whilst traditional quadrature becomes prohibitively expensive in the high-frequency regime, Levin methods provide a way to…
We present an adaptive delaminating Levin method for evaluating bivariate oscillatory integrals over rectangular domains. Whereas previous analyses of Levin methods impose non-resonance conditions that exclude stationary and resonance…
In this paper, we develop an oscillation free local discontinuous Galerkin (OFLDG) method for solving nonlinear degenerate parabolic equations. Following the idea of our recent work [J. Lu, Y. Liu, and C.-W. Shu, SIAM J. Numer. Anal.…
We propose a third-order numerical integrator based on the Neumann series and the Filon quadrature, designed mainly for highly oscillatory partial differential equations. The method can be applied to equations that exhibit small or moderate…
In this work, we present a method of generating a class of nonlinear ordinary differential equations (ODEs), representing the dynamics of appropriate nonlinear oscillators, that have the characteristics of either amplitude independent…
The solutions of scalar ordinary differential equations become more complex as their coefficients increase in magnitude. As a consequence, when a standard solver is applied to such an equation, its running time grows with the magnitudes of…
We propose an efficient approach for time integration of Klein-Gordon equations with highly oscillatory in time input terms. The new methods are highly accurate in the entire range, from slowly varying up to highly oscillatory regimes. Our…
This paper investigates a class of non-autonomous highly oscillatory ordinary differential equations characterized by a linear component inversely proportional to a small parameter $\varepsilon$, with purely imaginary eigenvalues, and an…
A very simple and efficient local variational iteration method for solving problems of nonlinear science is proposed in this paper. The analytical iteration formula of this method is derived first using a general form of first order…
We introduce an efficient numerical method for second order linear ODEs whose solution may vary between highly oscillatory and slowly changing over the solution interval. In oscillatory regions the solution is generated via a nonoscillatory…
We prove a sharp asymptotic formula for certain oscillatory integrals that may be approached using the stationary phase method. The estimates are uniform in terms of auxiliary parameters, which is crucial for application in analytic number…
It is well known that second order homogeneous linear ordinary differential equations with slowly varying coefficients admit slowly varying phase functions. This observation underlies the Liouville-Green method and many other techniques for…
We propose a new method of estimating oscillatory integrals, which we call a stationary set method. We use it to obtain the sharp convergence exponents of Tarry's problems in dimension two for every degree $k\ge 2$. As a consequence, we…
Inferring the parameters of ordinary differential equations (ODEs) from noisy observations is an important problem in many scientific fields. Currently, most parameter estimation methods that bypass numerical integration tend to rely on…