Compact Integration Rules as a quadrature method with some applications

In many instances of computational science and engineering the value of a definite integral of a known function f(x) is required in an interval. Nowadays there are plenty of methods that provide this quantity with a given accuracy. In one way or another, all of them assume an interpolating function, usually polynomial, that represents the original function either locally or globally. This paper presents a new way of calculating integral(x2)(x1) f(x) dx by means of compact integration, in a similar way to the compact differentiation employed in computational physics and mathematics. Compact integration is a linear combination of definite integrals associated to an interval and its adjacent ones, written in terms of nodal values of f(x). The coefficients that multiply both the integrals and f(x) at the nodes are obtained by matching terms in a Taylor series expansion. In this implicit method a system of algebraic equations is solved, where the vector of unknowns contains the integrals in each interval of a uniform discrete domain. As a result the definite integral over the whole domain is the sum of all these integrals. In this paper the mathematical tool is analyzed by deriving the appropriate coefficients for a given accuracy, and is exploited in various numerical examples and applications. The great accuracy of the method is highlighted. (C) 2019 Elsevier Ltd. All rights reserved.