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The Eulerian-Lagrangian Method with Accurate Numerical Integration

Published online by Cambridge University Press:  03 June 2015

Kun Li
Kun Li*
Affiliation:
LMAM & School of Mathematical Sciences, Peking University, Beijing 100080, China
*
*Corresponding author. URL: http://dsec.pku.edu.cn/∼kli/indexch.htm Email:kli@math.pku.edu.cn
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Abstract

This paper is devoted to the study of the Eulerian-Lagrangian method (ELM) for convection-diffusion equations on unstructured grids with or without accurate numerical integration. We first propose an efficient and accurate algorithm to calculate the integrals in the Eulerian-Lagrangian method. Our approach is based on an algorithm for finding the intersection of two non-matching grids. It has optimal algorithmic complexity and runs fast enough to make time-dependent velocity fields feasible. The evaluation of the integrals leads to increased precision and the unconditional stability. We demonstrate by numerical examples that the ELM with our proposed algorithm for accurate numerical integration has the following two features: first it is much more accurate and more stable than the ones with traditional numerical integration techniques and secondly the overall cost of the proposed method is comparable with the traditional ones.

Keywords

65D1865D3065M2565N30Eulerian-Lagrangian methodintersection of non-matching grids exact integration
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 4 , Issue 2 , April 2012 , pp. 156 - 174
Copyright
Copyright © Global-Science Press 2012

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References

[1] De Berg, M., Cheong, O., Van Kreveld, M. and Overmars, M., Computational Geometry: Algorithms and Applications, Springer Verlag, 2008.Google Scholar
[2] Douglas, J. Jr and Russell, T. F., Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures, SIAM J. Numer. Anal., 5 (1982), pp. 871–885.Google Scholar
[3] Farrell, P. E. and Maddison, J. R., Conservative interpolation between volume meshes by local Galerkin projection, Comput. Methods Appl. Mech. Eng., accepted, 2010.Google Scholar
[4] Gander, M. J. and Japhet, C., An algorithm for non-matching grid projections with linear complexity, Lect. Notes Comp. Sci. XVIII, (2009), pp. 185–192.Google Scholar
[5] Jia, J., Hu, X., Xu, J. and Zhang, C., Effects of integrations and adaptivity for the Eulerian-Lagrangian method, J. Comput. Math., accepted.Google Scholar
[6] Morton, K. W. and Kellogg, R. B., Numerical Solution of Convection-Diffusion Problems, Chapman & Hall London, 1996.Google Scholar
[7] Morton, K. W., Priestley, A. and Suüli, E., Stability of the Lagrange-Galerkin method with nonexact integration, RAIRO Modél, Math. Anal. Numér., 4 (1988), pp. 625–653.Google Scholar
[8] Morton, K. W. and Suüli, E., Evolution-Galerkin methods and their supraconvergence, Numer. Math., 3 (1995), pp. 331–355.Google Scholar
[9] Pironneau, O., On the transport-diffusion algorithm and its applications to the Navier-Stokes equations, Numer. Math., 3 (1982), pp. 309–332.Google Scholar
[10] Priestley, A., Exact projections and the Lagrange-Galerkin method: a realistic alternative to quadrature, J. Comput. Phys., 2 (1994), pp. 316–333.CrossRefGoogle Scholar
[11] Russell, T. F. and Celia, M. A., An overview of research on Eulerian-Lagrangian localized adjoint methods (ELLAM), Adv. Water Resour., 8-12 (2002), pp. 1215–1231.Google Scholar
[12] Xiu, D. and Karniadakis, G. E., An algorithm for non-matching grid projections with linear complexity, J. Sci. Comput., 1 (2002), pp. 585–597.Google Scholar