# Porous Mass Lumping#

```Porous Mass Lumping = {yes | true | no | false}
```

## Description / Usage#

Mass lumping is a technique for handling stiff problems with propagation of discontinuities. By “Mass” we mean the so-called mass matrix, or the submatrix generated by the time-derivative term in the physical equations. Discretization of this term with the standard Galerkin finite element method produces a symmetric, but nondiagonal matrix, also known as the consistent mass matrix as it adheres to the proper weak form. This required card specifies the mode in which the mass matrix is computed. If mass lumping is turned on, then the matrix is formed on a nodal, collocated basis and the mass matrix becomes diagonal. This technique expedites timeintegration during the propagation of steep fronts.

The mass lumping here applies ONLY to the time-derivative term in the EQ=porous_liq or EQ=porous_gas equations in Goma, and only when the Media Type is either POROUS_UNSATURATED or POROUS_TWO_PHASE. Mass lumping is not enabled for saturated porous flow. Please see technical discussion below for other usage tips. The card options are as follows:

yes | true Compute mass matrix with the lumped approach.

no | false Compute mass matrix with the standard Galerkin approach. This is the default.

## Examples#

```Porous Mass Lumping = true
```

## Technical Discussion#

Mass lumping is almost essential for unsaturated porous flow problems, especially at low permeabilities and in conditions for which the saturation front is sharp. It is recommended that mass lumping always be used for all unsaturated porous flow problems. However, with such use it is also recommended to use ONLY 1st order time integration (see Time step parameter card and choose Backward-Euler, 0.0). For second order time integration on the porous flow equations, mass lumping does not provide any benefit as the increased accuracy in time tends to lead to insufficient accuracy in space, and wiggles form.

Mass lumping is not currently available for saturated deformable porous flow.

## References#

GT-008.2: Porous Media Capabilities/Tutorial for GOMA. User Guidance for Saturated Porous Penetration Problems, August 11, 1999, P. R. Schunk

GT-009.3: GOMA’s Capabilities for Partially Saturated Flow in Porous Media, September 1, 2002, P. R. Schunk

GTM-029.0: SUPG Formulation for the Porous Flow Equations in Goma, H. K. Moffat, August 2001 (DRAFT).

Bradford, S. F. and N. D. Katopodes, “The anti-dissipative, non-monotone behavior of Petrov-Galerkin Upwinding,” International J. for Numerical Methods in Fluids, v. 33, 583-608 (2000).

Gundersen, E. and H. P. Langtangen, “Finite Element Methods for Two-Phase Flow in Heterogeneous Porous Media,” in Numerical Methods and Software Tools in Industrial Mathematics, Morten Daehlen, Aslak Tveito, Eds., Birkhauser, Boston, 1997.

Helmig, R. and R. Huber, “Comparison of Galerkin-type discretization techniques for two-phase flow in heterogeneous porous media,” Advances in Water Resources, 21, 697-711 (1998).

Unger, A. J. A., P. A. Forsyth and E. A. Sudicky, “Variable spatial and temporal weighting schemes for use in multi-phase compositional problems,” Advances in Water Resources, 19, 1 - 27 (1996).