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Institut für Wasserbau - IWS

Selected Topics and International Network Lectures

15:00 Uhr
Philip Binning, Associate Professor in Subsurface Hydrology and Numerical Modelling, Institute of Environment & Resources, Technical University of Denmark

A selective lumping (slumping) scheme for control of oscillations in method

Lumping is often used to control oscillations in weighted-residuals numerical methods. Standard lumping procedures add numerical diffusion indiscriminately, resulting in excessively diffused solutions. Here it is shown that the mass matrix can be selectively lumped (slumped), with an optimal 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 monotonicity problem is recast in the form of a maximum principle. Secondly, for a 2 x 2 element matrix, the amount of diffusion is calculated for an arbitrary right-hand side so that the solution obeys a maximum principle. Thirdly, the result is generalised for larger matrices. And finally, the result is recast to meet the monotonicity requirement. The result is an equation giving the amount of diffusion to be added in terms of a given right-hand-side vector. Intuitively, this diffusion is related to the local "curvature" of the right-hand side.

Selective lumping is shown to be effective for both an Eulerian-Lagrangian localized adjoint method (ELLAM) solution of the transport equation and a finite element solution of the heat equation. In both cases, solutions are monotonic and contain less numerical diffusion than in standard lumping schemes. The slumping concept is general and can be applied to any numerical approximation based on the method of weighted residuals. The particular derivation presented here is limited to symmetric tridiagonal Toeplitz matrices.