Efficient Numerical Methods For Highly Oscillatory Problesms

Highly oscillatory problems, such as highly oscillatory integrals and numerical solutions of highly oscillatory differential equations and integral equations, occur in a wide range of applications ranging from quantum chemistry, image analysis, fluid mechanics, etc, and are widely perceived as difficult problems. This thesis is devoted to explore some efficient methods for the highly oscillatory problems and related function approximation problems. For the parts of highly oscillatory problems, we are mainly concerned with the computation of highly oscillatory Fourier and Bessel integrals, and the Cauchy principal value integrals of oscillatory functions and numerical solution of a Volterra integral equation of the first kind with a highly oscillatory Bessel kernel. For the parts of Legendre approximations, our main concerns are the decay rate of the Legendre coefficients and the fast and stable algorithm for the implementation of Legendre interpolation formula.In Chapter 1, we show the background of the highly oscillatory problems. In Chapter 2 we review several efficient methods for highly oscillatory integrals such as asymptotic methods, Filon-type methods, Levin-type methods, generalized quadrature rules and numerical steepest descent methods.In Chapter 3, we discuss the Levin iterative method and prove the identity of the Levin iterative method and the asymptotic method for the Fourier type integrals when the integrals are free of stationary points. We also extend the Levin iterative method to the Bessel type integrals.In Chapter 4, we devote our attention to the computation of the finite Hilbert transform of the following form and three new algorithms are presented. When the function f(x) is analytic in a sufficiently large complex region containing the interval [-1,1], we present a numerical steepest descent method which can be achieved efficiently by the classical Gauss-Laguerre quadrature rule. The accuracy of this method can be dramatically improved whenωincreases. When f(x) is analytic in a small neighborhood of the integration interval, we then present a Chebyshev interpolatory type method which can be achievd efficiently by using a three-term recurrence relation and the fast Fourier transform. We demonstrate that this Chebyshev algorithm is uniformly convergent for approximating the finite Hilbert transform, namely the approximation error can be bounded independently of the values ofτ. Based on the above two methods, we extend the Filon-type method to the current integrals. Filon-type method, which does not require the function f(x) to be analytic, can also be computed by using recurrence relation and its accuracy can be improved asωgrows.In Chapter 5, we concentrate on the numerical solution of a Volterra convolution equation of the first kind with a highly oscillatory Bessel kernel. Firstly, for highly oscillatory Bessel type integrals, if the integration interval contains 0, we prove that the asymptotic order of the asymptotic and Filon-type methods can be improved when the order of the Bessel function is an integer. Using this new result, we study the asymptotic expansion and numerical methods for a Volterra convolution equation of the first kind with a highly oscillatory Bessel kernel. By using Laplace transform, the exact solution is transformed into a highly oscillatory Bessel integral, then the asymptotic expansion of the exact solution is obtained. Further, we design a Filon-type method, which is a convergent method for any fixed frequency, to approximate the solution. The accuracy of the Filon-type method can be greatly improved by adding either the derivatives or the number of interpolation nodes. Several examples are presented to confirm our theoretical analysis.In Chapter 6, we focus on the problems of Legendre approximation. The decay rate of the Legendre coefficients is established and hence the error bound of the truncated Legendre expansion is derived immediately. Meanwhile, the computation of the Legendre interpolation formula is studied. It is well known that the computation of the Chebyshev interpolation formula can be achieved in O(n) operations by its barycentric form and this process is stable. However, the computation of the Legendre interpolation formula has not investigated as extensively in the literature as that of Chebyshev cases. In this chapter, explicit barycentric weights, in terms of the nodes and weights of the Gauss Legendre quadrature rule, are presented. Since the nodes and weights of Gauss Legendre quadrature can be evaluated by Glaser-Liu-Rokhlin algorithm in O(n) operations, hence the Legendre interpolation formula can also be evaluated in O(n) operations and this is the same work required to compute the Chebyshev interpolation formula. Similarly, we present the explicit barycentric weights for the Gauss-Legendre-Lobatto interpolation formula in terms of the weights of the Gauss-Legendre-Lobatto rule. Thus, the Gauss-Legendre-Lobatto interpolation formula can also be evaluated in a fast and stable way.