***********************
**Porous Mass Lumping**
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::

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

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**Description / Usage**
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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.

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**Examples**
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::

   Porous Mass Lumping = true

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**Technical Discussion**
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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.



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**References**
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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).