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IMA Journal of Numerical Analysis Advance Access published online on March 30, 2009
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drn044
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© The author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

A semi-Lagrangian–Galerkin projection scheme for convection equations

R. Bermejo{dagger} and J. Carpio{ddagger}

Departamento de Matemática Aplicada, Escuela Técnica Superior de Ingenieros Industriales, Universidad Politécnica de Madrid, C/ José Gutiérrez Abascal 2, 28006 Madrid, Spain

{dagger} Corresponding author. Email: rbermejo{at}etsii.upm.es

{ddagger} Email: jaime.carpio{at}upm.es

Received on 20 July 2006. Revised on 6 May 2008.


  Abstract

We introduce in this paper a semi-Lagrangian–Galerkin projection scheme to discretize backwards in time along the characteristics the convection terms of convection–diffusion equations. The scheme consists of a transport step in which the elements of the fixed mesh are transported backwards along the characteristic curves, thus generating a new mesh composed of curved elements, followed by an approximate L2-projection onto the finite-element space associated with the transported mesh. The new scheme is to some extent related to the so-called Lagrange–Galerkin (or characteristic-Galerkin) methods, but it may be more efficient because the number of trajectories per element to be calculated in the new scheme is smaller than that of the conventional characteristic-Galerkin scheme. It is also proved that, for linear convection problems with the velocity sufficiently smooth, the new scheme is unconditionally stable in the L2-norm and its order of convergence is Formula , where m is the degree of the polynomials of the finite-element space, and the velocity is in L{infty}(0, T;Wq+1, {infty}) with integer q≥ 1.

Key Words: convection; characteristics; Lagrange–Galerkin; Galerkin projection; finite elements; semi-Lagrangian schemes


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