Abstract
We consider the discretization in time of inhomogeneous parabolic equations, using the technique of Laplace inversion along a contour located in the complex left half-plane which, after transformation to a finite interval, is then evaluated to high accuracy by a contour quadrature rule. This reduces the problem to a finite set of elliptic equations with complex coefficients, which may be solved in parallel. A serious problem is how to treat the source term f(t), because at each quadrature node along the contour we need its Laplace forward transform, which unfortunately is often unavailable. In this paper, we propose a new contour quadrature which does not require direct use of the Laplace forward transform of f(t). Compared to the existing contour quadratures, error analysis shows that the new quadrature possesses competitive asymptotic order of accuracy and numerical results show that when regularity of the initial term and/or differentiability of f(t) is not satisfied, the new quadrature is more accurate.
Funding statement: This work was supported by the NSF of China (11301362, 11371157, 61573010), the project of Sichuan University of Science and Engineering (2015LX01), the China Postdoctoral Foundation(2015M580777, 2016T- 90841), the project of Technology and Education of Sichuan Province (2014 JQ0035, 15A0220).
Acknowledgement
The author is very grateful to the anonymous referees for the careful reading of a preliminary version of the manuscript and their valuable suggestions and comments, which greatly improved the quality of this paper.
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