A Comparison of a Family of Eulerian and Semi-Lagrangian Finite Element Methods for the Advection-Diffusion Equations
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Authors
Neta, Beny
Giraldo, Francis X.
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1997
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Wiley
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Abstract
Eulerian and semi-Lagrangian finite element methods are analyzed for stability and accuracy for the one-dimensional advection-diffusion equations. The methods studied are a class of schemes called theta algorithms that yield the explicit (θ = 0), semi-implicit (θ = 1/2), and implicit (θ = 1) methods. The stability analysis shows that the semi-Lagrangian method is unconditionally stable for all values of θ while the Eulerian method is only unconditionally stable for 1/2 ≤ θ ≤ 1. The accuracy analysis shows that the semi-Lagrangian and Eulerian methods are second order accurate in both space and time only for θ = 1/2. This analysis shows that the best methods are the θ = 1/2 which are the semi-implicit methods. In essence this paper compares the semi-implicit Eulerian method with a semi-implicit semi-Lagrangian method, analytically and numerically. The analysis shows that the semi-implicit, semi-Lagrangian method exhibits better amplitude, dispersion and group velocity errors than the semi-implicit Eulerian method thereby achieving better results. Numerical experiments are performed on the two-dimensional advection and advection-diffusion equations having known analytic solutions. The numerical results corroborate the analysis by demonstrating that the semi-Lagrangian method is superior to the Eulerian methods particularly for integrating atmospheric and ocean equations because long time histories are sought for such problems.
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Article
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Applied Mathematics
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This publication is a work of the U.S. Government as defined in Title 17, United States Code, Section 101. Copyright protection is not available for this work in the United States.