fv_num_diff_flux_gradient_tensor.m File Reference

computes a numerical diffusive flux for a diffusion problem including a tensor More...

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Functions
function num_flux =	fv_num_diff_flux_gradient_tensor (model, model_data, U, NU_ind)
	computes a numerical diffusive flux for a diffusion problem including a tensor More...

Detailed Description

computes a numerical diffusive flux for a diffusion problem including a tensor

Definition in file fv_num_diff_flux_gradient_tensor.m.

Function Documentation

function num_flux = fv_num_diff_flux_gradient_tensor	(		model,
			model_data,
			U,
			NU_ind
	)

computes a numerical diffusive flux for a diffusion problem including a tensor

This function computes a numerical diffusive flux matrix for a diffusion problem. Dirichlet boundary treatment is performed, but neuman values are set to NaN. These must be handled by the calling function.

For discretization of a gradient_tensor the parameter NU_ind is necessary, if the gradient shall only be approximated on the local grid coordinates given by NU_ind.

The analytical term inspiring this flux looks like

\[ - \nabla \cdot \left( d(x) K(x,u) \nabla u \right),\]

with a function \(d: R \to R\) and a tensor \(K: R \times R \to \text{Lin}(R^2, R^2)\).

The implemented flux then can be written for inner boundaries as

\[g_{ij}(u,v) = d(x_{ij}) \tilde{K}(x,u,v) \tilde{\nabla}(u,v) |e_{ij}|\]

and for Dirichlet boundaries as

\[g_{dir,ij}(u) = d(x_{ij}) \tilde{K}_{dir}(x,u) \tilde{\nabla}(u) |e_{ij}|\]

where the discretizations \(\tilde{K} and \tilde{\nabla}\) of \(K\) respectively \(\nabla\) are given by the function pointer model.diffusivity_tensor_ptr respectively the function gradient_approx(). Note, that the latter is only available for rectangular grids.

if grid is empty, it is generated

Parameters

model	model
model_data	model data
U	U
NU_ind	NU ind

Return values

num_flux

num flux

Required fields of model:

• diffusivity_ptr —  function pointer to diffusivity function \(d\)
• diffusivity_tensor_ptr —  function pointer to a tensor implementation for \(K\)
• verbose —  flag indicating the verbosity level of informative output
• debug —  flag indicating wether debug output shall be turned on
• gridtype —  gridtype

Required fields of model_data:

• grid —  a structure containing geometry information of a mesh used for discretizations

Generated fields of num_flux:

• epsilon —  diffusion coefficient bound
• G —  the matrix of numerical flux values across the edges. boundary treatment is performed as follows: in dirichlet-boundary edges, the neighbour-values are simply set to this value and the flux is computed.

Definition at line 17 of file fv_num_diff_flux_gradient_tensor.m.

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