A selective lumping slumping scheme for control of oscillations in method

13. Oktober 2004, 16:00 Uhr

Zeit: 13.10.04, 16:00 – 17:30 Uhr
Referent*in: Philip Binning, Associate Professor in Subsurface Hydrology and Numerical Modelling, Institute of Environment & Resources,Technical University of Denmark
Veranstaltungsort: Pfaffenwaldring 61, Raum U1.003 (MML), Universität Stuttgart
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Lumping is often used to control oscillations in weighted-residualsnumerical methods. Standard lumping procedures add numerical diffusionindiscriminately, resulting in excessively diffused solutions. Here it isshown that the mass matrix can be selectively lumped (slumped), with anoptimal amount of diffusion added to each element matrix of the mass matrix.The amount of diffusion added is calculated from the right-hand-side vector.

The optimal amount of diffusion is found in 4 steps. First the monotonicityproblem is recast in the form of a maximum principle. Secondly, for a 2 x 2element matrix, the amount of diffusion is calculated for an arbitraryright-hand side so that the solution obeys a maximum principle. Thirdly, theresult is generalised for larger matrices. And finally, the result is recastto meet the monotonicity requirement. The result is an equation giving theamount of diffusion to be added in terms of a given right-hand-side vector.Intuitively, this diffusion is related tothe local "curvature" of the right-hand side.

Selective lumping is shown to be effective for both an Eulerian-Lagrangianlocalized adjoint method (ELLAM) solution of the transport equation and afinite element solution of the heat equation. In both cases, solutions aremonotonic and contain less numerical diffusion than in standard lumpingschemes. The slumping concept is general and can be applied to any numericalapproximation based on the method of weighted residuals. The particularderivation presented here is limited to symmetric tridiagonal Toeplitzmatrices.
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