**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. |