Parabolic and hyperbolic contours for computing the Bromwich integral

Abstract

Abstract. Some of the most effective methods for the numerical inversion of the Laplace transform are based on the approximation of the Bromwich contour integral. The accuracy of these methods often hinges on a good choice of contour, and several such contours have been proposed in the literature. Here we analyze two recently proposed contours, namely a parabola and a hyperbola. Using a representative model problem, we determine estimates for the optimal parameters that define these contours. An application to a fractional diffusion equation is presented.

Diffusive Realization of a Lyapunov Equation Solution, and Parallel Algorithms Implementation

“…Then, computational algorithms for history functions ψ ± and for z ± are detailed in Subsection 3.2 and 3.3. In this paper, we only use the hyperbolic contours −θ ± proposed by Weideman and Trefethen in [10], in the context of inverse Laplace transform computation, namely…”

“…Weideman and Trefethen [10] have developed a contour optimization based on a balance between the truncation error estimate and the discretization error for the numerical integration of the Laplace inversion at points t ℓ,j ∈ I = {h, 2h, ..., 1} excluding point 0. This approach is not efficient in the present case since the ratio between the upper and lower bounds of the set I, i.e.…”

“…López-Fernández et al [6], apply a similar idea to convolution operators and develop error estimates as well as optimized numerical methods. In the related field of numerical inversion of the Laplace transform, Weideman and Trefethen [10] provided an error estimate and its application to contour optimization for computation cost reduction. It is based on the balancing of discretization and truncation errors.…”

“…First, we apply an additional change of variables to the kernel p and its extension so that they become defined in ω which eliminates an important source of error. Then, as in [10], but with a significantly different theoretical approach, we provide, for the first time, an error estimate of the DR and use it in a contour optimization method. Moreover, the method has been extensively tested and some significant results are reported.…”

An improved Talbot method for numerical Laplace transform inversion

“…That is, one needs to consider only quadrature nodes in the upper (or lower) half-plane. When comparing numerical results it is important to keep this point in mind as several other papers, including [Duf93,Wei06,WT07], consider quadrature rules with a total of 2N nodes. Popular contours C are the parabola [But57,GM05] and the hyperbola [GM05, LFP04,SST03], as well as the cotangent contour introduced by Talbot in [Tal79].…”