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CivilComp Proceedings
ISSN 17593433 CCP: 100
PROCEEDINGS OF THE EIGHTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY Edited by: B.H.V. Topping
Paper 81
QuadratureFree Characteristic Methods for ConvectionDiffusion Problems M. Tabata
Department of Mathematics, Waseda University, Tokyo, Japan M. Tabata, "QuadratureFree Characteristic Methods for ConvectionDiffusion Problems", in B.H.V. Topping, (Editor), "Proceedings of the Eighth International Conference on Engineering Computational Technology", CivilComp Press, Stirlingshire, UK, Paper 81, 2012. doi:10.4203/ccp.100.81
Keywords: convectiondiffusion, characteristics, quadraturefree, finite element method, finite difference method, lumped mass, Péclet number, stability, convergence.
Summary
Convectiondiffusion problems appear and are solved in various fields of sciences and technologies.
The convectiondiffusion 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 wellknown 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 highPéclet numbers. Galerkincharacteristics method also has the advantage of the finite element method, the geometrical flexibility and the extension to higherorder 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 nonsmoothness 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 Galerkincharacteristics finite element scheme of lumped mass type [2] is considered. The other way is to use a finite difference method [3] derived from a Galerkincharacteristics finite element scheme [4]. Both schemes are free from numerical quadrature. For these schemes the stability and convergence are discussed. References
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