# MassLumpedTimeDerivative

## Description

Lumping of the time derivative can be useful for a couple of reasons. Lumping helps ensure conservation of mass at a node. In a standard node-based Galerkin approximation, fluxes from spatial terms can be thought of as "entering" nodes. If there is no flux to a node, then the mass at that node should stay fixed. However, if the standard Galerkin method is applied to a time derivative term, (\psi_i, \frac{\partial u_h}{\partial t} the corresponding coefficient matrix is tri-diagonal and the mass at a node is affected by fluxes to neighboring nodes. This can lead to violation of local mass conservation and generation of spurious oscillations with unphysical under- and over-shoot phenomena. Lumping fixes this problem by isolating a nodal solution from neighboring nodal solutions in the time derivative term. Mathematically, lumping looks like this. We start with our governing equation u_t = Au where A is a differential operator. We write our finite element solution as

u(t, x) = \sum u_j(t) \phi_j(x)

Substituting into our governing equation, we have:

\sum u_j'\phi_j = Sum u_jA\phi_j

Now we apply our test functions and integrate over the volume:

\sum u_j' (\psi_i\phi_j) = \sum u_j (\psi_i, Au_j)

After applying all of our test functions, we have the matrix system

\widetilde{M}\vec{u'} = \widetilde{K}\vec{u}

where here denotes the vector of coefficients . is the mass matrix and is the stiffness matrix. Note that neither of these matrices are ever explicitly formed in MOOSE but they are still directly relevant. As mentioned previously, a standard Galerkin procedure results in a tri-diagonal . As it turns out, the sum of the matrix row elements in is one. Lumping then consists of summing the matrix row elements (result 1) and placing the sums on the diagonals. The result is the identity matrix and becomes simply :

\vec{u'} = \widetilde{K}\vec{u}

As seen by the above equation, besides helping with local conservation of mass, mass lumping makes explicit time stepping feasible because it removes the need for solving a linear system.

## Example Syntax

The MassLumpedTimeDerivative syntax is simple, only taking type and variable as shown in the kernel block below. This particular test file, from which the kernel block is taken, demonstrates a case where the concentration of would become negative in a non-lumped scheme for sufficiently small time steps.


[Kernels]
[./time_deriv]
type = MassLumpedTimeDerivative
variable = u
[../]
[./diff]
type = FuncCoefDiffusion
variable = u
coef = diff_f
[../]
[]
(moose/test/tests/kernels/mass_lumping/mass_lumping.i)

## Input Parameters

• variableThe name of the variable that this Kernel operates on

C++ Type:NonlinearVariableName

Description:The name of the variable that this Kernel operates on

### Required Parameters

• blockThe list of block ids (SubdomainID) that this object will be applied

C++ Type:std::vector

Description:The list of block ids (SubdomainID) that this object will be applied

### Optional Parameters

• enableTrueSet the enabled status of the MooseObject.

Default:True

C++ Type:bool

Description:Set the enabled status of the MooseObject.

• save_inThe name of auxiliary variables to save this Kernel's residual contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)

C++ Type:std::vector

Description:The name of auxiliary variables to save this Kernel's residual contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)

• use_displaced_meshFalseWhether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.

Default:False

C++ Type:bool

Description:Whether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.

• control_tagsAdds user-defined labels for accessing object parameters via control logic.

C++ Type:std::vector

Description:Adds user-defined labels for accessing object parameters via control logic.

• seed0The seed for the master random number generator

Default:0

C++ Type:unsigned int

Description:The seed for the master random number generator

• diag_save_inThe name of auxiliary variables to save this Kernel's diagonal Jacobian contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)

C++ Type:std::vector

Description:The name of auxiliary variables to save this Kernel's diagonal Jacobian contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)

• implicitTrueDetermines whether this object is calculated using an implicit or explicit form

Default:True

C++ Type:bool

Description:Determines whether this object is calculated using an implicit or explicit form

### Advanced Parameters

• vector_tagstimeThe tag for the vectors this Kernel should fill

Default:time

C++ Type:MultiMooseEnum

Description:The tag for the vectors this Kernel should fill

• extra_vector_tagsThe extra tags for the vectors this Kernel should fill

C++ Type:std::vector

Description:The extra tags for the vectors this Kernel should fill

• matrix_tagssystem timeThe tag for the matrices this Kernel should fill

Default:system time

C++ Type:MultiMooseEnum

Description:The tag for the matrices this Kernel should fill

• extra_matrix_tagsThe extra tags for the matrices this Kernel should fill

C++ Type:std::vector

Description:The extra tags for the matrices this Kernel should fill