A sparse spectral method for Volterra integral equations using orthogonal polynomials on the triangle

Abstract

We introduce and analyse a sparse spectral method for the solution of Volterra integral equations using bivariate orthogonal polynomials on a triangle domain. The sparsity of the Volterra operator on a weighted Jacobi basis is used to achieve high efficiency and exponential convergence. The discussion is followed by a demonstration of the method on example Volterra integral equations of the first and second kind with known analytic solutions as well as an application-oriented numerical experiment. We prove convergence for both first and second kind problems, where the former builds on connections with Toeplitz operators.

A fast sparse spectral method for nonlinear integro-differential Volterra equations with general kernels

“…It was shown in [23] that the Volterra integral operator is sparse with banded structure on appropriate Jacobi polynomial spaces. Based on this, a sparse spectral method with exponential convergence for linear Volterra equations with general kernels was motivated and analyzed.…”

“…In this section we briefly review these methods to the degree necessary to follow the integro-differential and nonlinear extension in this paper. For the full discussion of the linear case, we refer to [23].…”

“…In this paper we present a spectral method which works for linear, nonlinear, integro-differential and many other types of Volterra equations of first, second and third kind while utilizing polynomial spaces in which the Volterra operators have a sparse banded structure. As such this paper is a direct generalization of results in [23], which derived said banded structure of the Volterra operator in Jacobi polynomial bases and on the basis of [40] developed an efficient Clenshaw algorithm based approach to numerically generate the operator. This method thus combines the unmatched exponential convergence of spectral methods with highly efficient sparse linear algebra, a very promising combination which has been successful in recent years [25,38,39,50,52].…”

“…A highly efficient spectral solver for the case of convolution kernels K(x, y) = K(x − y) for linear VIDEs using low rank approximations was recently described by Hale [24]. Gutleb and Olver described a general kernel sparse spectral method for linear Volterra integral equations in [23], which forms the background of the present paper.…”

“…The structure of this paper is as follows: In sections 1.1 and 1.2 we introduce the necessary framework of uni-and multivariate orthogonal polynomial approximation of functions. Section 1.3 briefly recounts relevant results from [23], detailing the banded structure of the Volterra operator on appropriately chosen triangle domain Jacobi polynomial bases. Section 2 details the extension of the linear Volterra integral equation method to a general linear integro-differential case by augmenting the system with appropriate evaluation operators.…”