- National Science Foundation, United States
Lumping is often used to control oscillations in method of 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 row 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 two by two, or 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. 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 were monotonic and contained less numerical diffusion than in standard lumping schemes. The SLUMPING method is general and can be applied to any numerical approximation based on the method of weighted residuals. The project will also investigate the relationship of SLUMPING schemes to flux limiter schemes used in finite difference approximations and to streamline upwind Petrov Galerkin methods.
|Period||01/01/2003 → 01/01/2006|