Abstract
Following a previous result stating their equivalence under constant advection speed, Semi-Lagrangian and Lagrange–Galerkin schemes are compared in this paper in the situation of variable coefficient advection equations. Once known that Semi-Lagrangian schemes can be proved to be equivalent to area-weighted Lagrange–Galerkin schemes via a suitable definition of the basis functions, we will further prove that area-weighted Lagrange–Galerkin schemes represent a “small” (more precisely, an \(O(\Delta t\))) perturbation of exact Lagrange–Galerkin schemes. This equivalence implies a general result of stability for Semi-Lagrangian schemes.
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Notes
In fact, the method proposed in [14] applies this technique element! by element following the displacement of centroids; however, it is conceptually equivalent to apply the same technique to the whole basis function \(\phi _j\) provided it is piecewise polynomial and compactly supported as assumed in [14].
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Acknowledgments
I am indebted with at least two colleagues. The first is Rodolfo Bermejo who first suggested to me the idea of studying the equivalence of SL and LG schemes. The second is Corrado Falcolini for his kind help with Mathematica.
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Ferretti, R. On the relationship between Semi-Lagrangian and Lagrange–Galerkin schemes. Numer. Math. 124, 31–56 (2013). https://doi.org/10.1007/s00211-012-0505-5
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DOI: https://doi.org/10.1007/s00211-012-0505-5