Keywords: convection-diffusion, characteristics, quadrature-free, finite element method, finite difference method, lumped mass, Péclet number, stability, convergence.

Convection-diffusion problems appear and are solved in various fields of sciences and technologies. The convection-diffusion equation is linear, but to solve it is not always an easy task. When the Péclet number is high, that is, convection dominant cases, it is well-known that the Galerkin finite element scheme, or equivalently, the centered finite difference scheme, produces easily oscillation solutions. Hence, elaborate numerical schemes with new ideas have been developed to perform stable computation. Among them we focus on the method based on characteristics.

The procedure of the characteristic method is natural from the physical point of view since it approximates particle movements. It is also attractive from the computational point of view since it leads to a symmetric system of linear equations. Schemes derived from the characteristic method are recognized to be robust for high-Péclet numbers. Galerkin-characteristics method also has the advantage of the finite element method, the geometrical flexibility and the extension to higher-order schemes. A unique disadvantage of this method is in the computation of composite function terms. Since the terms are not polynomials, some numerical quadrature is usually employed to compute them. It is, however, reported that much attention should be paid to the numerical quadrature, because a rough numerical integration formula may yield oscillating results caused by the non-smoothness of the composite function terms [1].

In this paper we discuss two ways to avoid numerical quadrature, referring to recent results. One way is to use the lumping technique. A Galerkin-characteristics finite element scheme of lumped mass type [2] is considered. The other way is to use a finite difference method [3] derived from a Galerkin-characteristics finite element scheme [4]. Both schemes are free from numerical quadrature. For these schemes the stability and convergence are discussed.

1

M. Tabata, S. Fujima, "Robustness of a characteristic finite element scheme of second order in time increment", in C. Groth, D.W. Zingg, (Editors), "Computational Fluid Dynamics 2004", Springer, 177-182, 2006. doi:10.1007/3-540-31801-1_22

2

O. Pironneau, M. Tabata, "Stability and convergence of a Galerkin-characteristics finite element scheme of lumped mass type", International Journal for Numerical Methods in Fluids, 64, 1240-1253, 2010. doi:10.1002/fld.2459

3

H. Notsu, H. Rui, M. Tabata, "Charactersitics finite difference schemes of second order in time for convection-diffustion problems", submitted to Numerische Mathematik, 2012.

4

H. Rui, M. Tabata, "A second order characteristic finite element scheme for convection diffusion problems", Numerische Mathematik, 92, 161-177, 2002. doi:10.1007/s002110100364

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