On the stability of a modified Nyström method for Mellin convolution equations in weighted spaces

This paper deals with the numerical solution of second kind integral equations with fixed singularities of Mellin convolution type. The main difficulty in solving such equations is the proof of the stability of the chosen numerical method, being the noncompactness of the Mellin integral operator the chief theoretical barrier. Here, we propose a Nyström method suitably modified in order to achieve the theoretical stability under proper assumptions on the Mellin kernel. We also provide an error estimate in weighted uniform norm and prove the well-conditioning of the involved linear systems. Some numerical tests which confirm the efficiency of the method are shown.     

Singular integral equations — The convergence of the Nyström interpolant of the Gauss-Chebyshev method

Nyström's interpolation formula is applied to the numerical solution of singular integral equations. For the Gauss-Chebyshev method, it is shown that this approximation converges uniformly, provided that the kernel and the input functions possess a continuous derivative. Moreover, the error of the Nyström interpolant is bounded from above by the Gaussian quadrature errors and thus convergence is fast, especially for smooth functions. ForC input functions, a sharp upper bound for the error is obtained. Finally numerical examples are considered. It is found that the actual computational error agrees well with the theoretical derived bounds.This research has been partially supported by a grant from the Rutgers Research Council.     

Spline-Gauss rules and the Nyström method for solving integral equations in quantum scattering

This paper is concerned with an n-point Gauss quadrature

where the points {t_n,j}ⁿ_j=1 and weights {w_n,j}ⁿ_j=1 are chosen to be exact when g is defined on a 2n-dimensional space of polynomial splines. The spline-Gauss quadrature is used in a Nyström method for solving integral equations of the second kind. A practical application is provided by solving integral equations that arise in quantum scattering theory.     

Order conditions for General Linear Nyström methods

The purpose of this paper is to analyze the algebraic theory of order for the family of general linear Nyström (GLN) methods introduced in D’Ambrosio et al. (Numer. Algorithm 61(2), 331–349, 2012) with the aim to provide a general framework for the representation and analysis of numerical methods solving initial value problems based on second order ordinary differential equations (ODEs). Our investigation is carried out by suitably extending the theory of B-series for second order ODEs to the case of GLN methods, which leads to a general set of order conditions. This allows to recover the order conditions of numerical methods already known in the literature, but also to assess a general approach to study the order conditions of new methods, simply regarding them as GLN methods: the obtained results are indeed applied to both known and new methods for second order ODEs.     

A note on the Voronovskaja theorem for Mellin–Fejer convolution operators

Here, using Mellin derivatives and a different notion of moment, we state a Voronovskaja approximation formula for a class of Mellin–Fejer type convolution operators. This new approach gives direct and simple applications to various important specific examples.     

High-order accurate methods for Nyström discretization of integral equations on smooth curves in the plane

Boundary integral equations and Nyström discretization provide a powerful tool for the solution of Laplace and Helmholtz boundary value problems. However, often a weakly-singular kernel arises, in which case specialized quadratures that modify the matrix entries near the diagonal are needed to reach a high accuracy. We describe the construction of four different quadratures which handle logarithmically-singular kernels. Only smooth boundaries are considered, but some of the techniques extend straightforwardly to the case of corners. Three are modifications of the global periodic trapezoid rule, due to Kapur–Rokhlin, to Alpert, and to Kress. The fourth is a modification to a quadrature based on Gauss–Legendre panels due to Kolm–Rokhlin; this formulation allows adaptivity. We compare in numerical experiments the convergence of the four schemes in various settings, including low- and high-frequency planar Helmholtz problems, and 3D axisymmetric Laplace problems. We also find striking differences in performance in an iterative setting. We summarize the relative advantages of the schemes.     

On the Nyström discretization of integral equations on planar curves with corners

The Nyström method can produce ill-conditioned systems of linear equations when applied to integral equations on domains with corners. This defect can already be seen in the simple case of the integral equations arising from the Neumann problem for Laplace?s equation. We explain the origin of this instability and show that a straightforward modification to the Nyström scheme, which renders it mathematically equivalent to Galerkin discretization, corrects the difficulty without incurring the computational penalty associated with Galerkin methods. We also present the results of numerical experiments showing that highly-accurate solutions of integral equations on domains with corners can be obtained, irrespective of whether their solutions exhibit bounded or unbounded singularities, assuming that proper discretizations are used.     

A Nyström Method for the Eigenvalue Problem of a Class of Noncompact Operators

We consider the eigenvalue problem of certain kind of noncompact linear operators given as the sum of a multiplication and a kernel operator. Under the assumption that there is a unique (up to normalization) positive eigenfunction f _p , we propose a combination of the finite section and Nyström methods for approximation of f _p and the corresponding eigenvalue. It is proved that the proposed method is convergent. Some examples of the problem are solved numerically using the proposed method.     

A stable numerical method for space fractional Landau–Lifshitz equations

In this paper, we proposed a simple and unconditional stable time-split Gauss–Seidel projection (GSP) method for the space fractional Landau–Lifshitz (FLL) equations. Numerical results are presented to demonstrate the effectiveness and stability of this method.     

Stable Runge–Kutta–Nyström methods for dissipative stiff problems

The definition of stability for Runge–Kutta–Nyström methods applied to stiff second-order in time problems has been recently revised, proving that it is necessary to add a new condition on the coefficients in order to guarantee the stability. In this paper, we study the case of second-order in time problems in the nonconservative case. For this, we construct an $RThe definition of stability for Runge–Kutta–Nystr?m methods applied to stiff second-order in time problems has been recently revised, proving that it is necessary to add a new condition on the coefficients in order to guarantee the stability. In this paper, we study the case of second-order in time problems in the nonconservative case. For this, we construct an -stable Runge–Kutta–Nystr?m method with two stages satisfying this condition of stability and we show numerically the advantages of this new method.This research was supported by MTM 2004-08012 and JCYL VA103/04.     

On using explicit Runge–Kutta–Nyström methods for the treatment of retarded differential equations with periodic solutions

In this work we dial with the treatment of second order retarded differential equations with periodic solutions by explicit Runge–Kutta–Nyström methods. In the past such methods have not been studied for this class of problems. We refer to the underline theory and study the behavior of various methods proposed in the literature when coupled with Hermite interpolants. Among them we consider methods having the characteristic of phase–lag order. Then we consider continuous extensions of the methods to treat the retarded part of the problem. Finally we construct scaled extensions and high order interpolants for RKN pairs which have better characteristics compared to analogous methods proposed in the literature. In all cases numerical tests and comparisons are done over various test problems.     

Convergence analysis of a high-order Nyström integral-equation method for surface scattering problems

In this paper we present a convergence analysis for the Nyström method proposed in [J Comput Phys 169 (1):80–110, 2001] for the solution of the combined boundary integral equation formulations of sound-soft acoustic scattering problems in three-dimensional space. This fast and efficient scheme combines FFT techniques and a polar change of variables that cancels out the kernel singularity. We establish the stability of the algorithms in the $L^2$ norm and we derive convergence estimates in both the $L^2$ and $L^\infty $ norms. In particular, our analysis establishes theoretically the previously observed super-algebraic convergence of the method in cases in which the right-hand side is smooth.     

A criterion for the convergence of the Mellin–Barnes integral for solutions to simultaneous algebraic equations

We obtain a criterion for the convergence of the Mellin–Barnes integral representing the solution to a general system of algebraic equations. This yields a criterion for a nonnegative matrix to have positive principal minors. The proof rests on the Nilsson–Passare–Tsikh Theorem about the convergence domain of the general Mellin–Barnes integral, as well as some theorem of a linear algebra on a subdivision of the real space into polyhedral cones.     

High order explicit Runge–Kutta Nyström pairs

Explicit Runge–Kutta Nyström pairs provide an efficient way to find numerical solutions to second-order initial value problems when the derivative is cheap to evaluate. We present new optimal pairs of orders ten and twelve from existing families of pairs that are intended for accurate integrations in double precision arithmetic. We also present a summary of numerical comparisons between the new pairs on a set of eight problems which includes realistic models of the Solar System. Our searching for new order twelve pairs shows that there is often not quantitative agreement between the size of the principal error coefficients and the efficiency of the pairs for the tolerances we are interested in. Our numerical comparisons, as well as establishing the efficiency of the new pairs, show that the order ten pairs are more efficient than the order twelve pairs on some problems, even at limiting precision in double precision.     

Local Hölder continuity for some doubly nonlinear parabolic equations in measure spaces

We establish the local Hölder continuity for the nonnegative weak solutions of certain doubly nonlinear parabolic equations possessing a singularity in the time derivative part and a degeneracy in the principal part. The proof involves the method of intrinsic scaling and the setting is a measure space equipped with a doubling non-trivial Borel measure supporting a Poincaré inequality.     

A fast Nyström-Broyden solver by Chebyshev compression

When a Nyström-Broyden method is used to solve numerically Urysohn integral equations, the main problem is to evaluate the action of the integral operator at a low cost. Here we suitably approximate the relevant discrete integral operator of dimensionn by itsm-degree truncated Chebyshev series expansion (withm?n), reducing the complexity from the basic ${\mathcal{O}}(n^2 )$ to ${\mathcal{O}}(mn)$ . A technique to evaluate cheaply suchm is presented. Several linear and nonlinear examples are considered.