A hybrid analytical-numerical method for solving advection-dispersion problems on a half-line
•Solution of the advection-dispersion equation on the half-line via the Unified Transform (Fokas Method).•Combines complex analysis with numerics.•Compared to traditional approaches, the unified transform avoids the solution of ordinary differential equations.•Results are successfully compared to ot...
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| Published in: | International journal of heat and mass transfer Vol. 139; pp. 482 - 491 |
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| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
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Oxford
Elsevier Ltd
01.08.2019
Elsevier BV |
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| ISSN: | 0017-9310, 1879-2189 |
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| Abstract | •Solution of the advection-dispersion equation on the half-line via the Unified Transform (Fokas Method).•Combines complex analysis with numerics.•Compared to traditional approaches, the unified transform avoids the solution of ordinary differential equations.•Results are successfully compared to other existing solutions.•Illustrates the advantage of the numerical implementation of the Fokas Method.
This paper employs the unified transform, also known as the Fokas method, to solve the advection-dispersion equation on the half-line. This method combines complex analysis with numerics. Compared to classical approaches used to solve linear partial differential equations (PDEs), the unified transform avoids the solution of ordinary differential equations and, more importantly, constructs an integral representation of the solution in the complex plane which is uniformly convergent at the boundaries. As a consequence, such solutions are well suited for numerical computations. Indeed, the numerical evaluation of the solution requires only the computation of a single contour integral involving an integrand which decays exponentially fast for large values of the integration variable. A novel contribution of this paper, with respect to the solution of linear evolution PDEs in general, and the implementation of the unified transform in particular, is the following: using the advection-dispersion equation as a generic example, it is shown that if the transforms of the given data can be computed analytically, then the unified transform yields a fast and accurate method that converges exponentially with the number of evaluations N yet only has complexity O(N). Furthermore, if the transforms are computed numerically using M evaluations, the unified transform gives rise to a method with complexity O(NM). Results are successfully compared to other existing solutions. |
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| AbstractList | •Solution of the advection-dispersion equation on the half-line via the Unified Transform (Fokas Method).•Combines complex analysis with numerics.•Compared to traditional approaches, the unified transform avoids the solution of ordinary differential equations.•Results are successfully compared to other existing solutions.•Illustrates the advantage of the numerical implementation of the Fokas Method.
This paper employs the unified transform, also known as the Fokas method, to solve the advection-dispersion equation on the half-line. This method combines complex analysis with numerics. Compared to classical approaches used to solve linear partial differential equations (PDEs), the unified transform avoids the solution of ordinary differential equations and, more importantly, constructs an integral representation of the solution in the complex plane which is uniformly convergent at the boundaries. As a consequence, such solutions are well suited for numerical computations. Indeed, the numerical evaluation of the solution requires only the computation of a single contour integral involving an integrand which decays exponentially fast for large values of the integration variable. A novel contribution of this paper, with respect to the solution of linear evolution PDEs in general, and the implementation of the unified transform in particular, is the following: using the advection-dispersion equation as a generic example, it is shown that if the transforms of the given data can be computed analytically, then the unified transform yields a fast and accurate method that converges exponentially with the number of evaluations N yet only has complexity O(N). Furthermore, if the transforms are computed numerically using M evaluations, the unified transform gives rise to a method with complexity O(NM). Results are successfully compared to other existing solutions. This paper employs the unified transform, also known as the Fokas method, to solve the advection-dispersion equation on the half-line. This method combines complex analysis with numerics. Compared to classical approaches used to solve linear partial differential equations (PDEs), the unified transform avoids the solution of ordinary differential equations and, more importantly, constructs an integral representation of the solution in the complex plane which is uniformly convergent at the boundaries. As a consequence, such solutions are well suited for numerical computations. Indeed, the numerical evaluation of the solution requires only the computation of a single contour integral involving an integrand which decays exponentially fast for large values of the integration variable. A novel contribution of this paper, with respect to the solution of linear evolution PDEs in general, and the implementation of the unified transform in particular, is the following: using the advection-dispersion equation as a generic example, it is shown that if the transforms of the given data can be computed analytically, then the unified transform yields a fast and accurate method that converges exponentially with the number of evaluations N yet only has complexity O(N). Furthermore, if the transforms are computed numerically using M evaluations, the unified transform gives rise to a method with complexity O(NM). Results are successfully compared to other existing solutions. |
| Author | Fokas, A.S. de Barros, F.P.J. Colbrook, M.J. |
| Author_xml | – sequence: 1 givenname: F.P.J. surname: de Barros fullname: de Barros, F.P.J. email: fbarros@usc.edu organization: Sonny Astani Dept. of Civil and Environmental Engineering, University of Southern California, Los Angeles, USA – sequence: 2 givenname: M.J. surname: Colbrook fullname: Colbrook, M.J. email: m.colbrook@damtp.cam.ac.uk organization: Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK – sequence: 3 givenname: A.S. surname: Fokas fullname: Fokas, A.S. organization: Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK |
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| Snippet | •Solution of the advection-dispersion equation on the half-line via the Unified Transform (Fokas Method).•Combines complex analysis with numerics.•Compared to... This paper employs the unified transform, also known as the Fokas method, to solve the advection-dispersion equation on the half-line. This method combines... |
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| SubjectTerms | Advection Advection-dispersion equation Analytical solution Complexity Computation Convergence Decay rate Dispersion Environmental flows Fokas method Integrals Numerical methods Ordinary differential equations Partial differential equations Transformations (mathematics) Unified transform |
| Title | A hybrid analytical-numerical method for solving advection-dispersion problems on a half-line |
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