fv_operators_conv_explicit_engquist_osher.m File Reference

computes convection contribution to finite volume time evolution matrices, or their Frechet derivative More...

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Functions
function [ L_E_conv , bdir_E_conv ] =	fv_operators_conv_explicit_engquist_osher (model, model_data, U, NU_ind)
	computes convection contribution to finite volume time evolution matrices, or their Frechet derivative More...

Detailed Description

computes convection contribution to finite volume time evolution matrices, or their Frechet derivative

Definition in file fv_operators_conv_explicit_engquist_osher.m.

Function Documentation

function [ L_E_conv , bdir_E_conv ] = fv_operators_conv_explicit_engquist_osher	(		model,
			model_data,
			U,
			NU_ind
	)

computes convection contribution to finite volume time evolution matrices, or their Frechet derivative

This function computes a convection operator \(L_{\text{eo}}\) and a corresponding offset vector \(b_{\text{eo}}\) that can be used by fv_operators_implicit_explicit() to build evolution matrices for a finite volume time step \(L_I U^{k+1} = L_E U^k + b_E + b_I\).

(L_E_conv)il = 1/|Ti| *
                   (sum j (NB(i) cup NB_dir(i): v_ij*n_ij>=0 )
                |S_ij| (v(c_ij)*n_ij))    for l=i
               1/|Ti| * |S_il| ( - v(c_il)*n_il)
                                          for l NB(i) and v_il * n_il < 0
                      0                   else

\[ (L_{E,conv})_{il} = \begin{cases} \frac{1}{|T_i|} \sum_{ j \in \{ NB(i) \cup NB_{dir}(i): v_{ij} \cdot n_{ij}>=0 \} } |S_{ij}| (v(c_{ij}) \cdot n_{ij}) & \text{for } l=i \\ - \frac{1}{|T_i|} |S_{il}| v(c_{il}) \cdot n_{il} & \text{for } l \in NB(i) \text{ and }v_{il} \cdot n_{il} < 0 \\ 0 & \text{else} \end{cases} \]

The analytical term inspiring this operator looks like \( v \cdot \nabla u \) or in the non-linear case, where the Frechet derivative is computed \( \nabla \cdot f(u). \) Here, \(v\) is a space dependent velocity field and \(f\) some smooth function in \(C^2(\mathbb{R}, \mathbb{R}^d)\).

The result are a sparse matrix L_E_conv and an offset vector bdir_E_conv, the latter containing dirichlet value contributions

See also

fv_num_conv_flux_engquist_osher()

Parameters

model	model
model_data	model data
U	U
NU_ind	NU ind

Return values

L_E_conv	sparse matrix \(L_{\text{eo}}\)
bdir_E_conv	offset vector \(b_{\text{eo}}\)

Required fields of model:

• decomp_mode —  flag indicating the operation mode of the function:
  • 0 (complete) : no affine parameter dependence or decomposition is performed.
  • 1 (components) : for each output argument a cell array of output matrices is returned representing the \(q\)-th component independent of the parameters given in mu_names.
  • 2 (coefficients) : returns a vector where each coordinate represents the \(q\)-the coefficient \(\sigma_{\cdot}^{q}(\mu)\) dependent on the parameters given in mu_names.
• verbose —  flag indicating the verbosity level of informative output
• flux_linear —  flux linear
• conv_flux_derivative_ptr —  conv flux derivative ptr
• debug —  flag indicating wether debug output shall be turned on
• dirichlet_values_ptr —  dirichlet values ptr

Required fields of model_data:

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

Definition at line 17 of file fv_operators_conv_explicit_engquist_osher.m.

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