pybamm.
FiniteVolume
(mesh)[source]¶A class which implements the steps specific to the finite volume method during discretisation.
For broadcast and mass_matrix, we follow the default behaviour from SpatialMethod.
Parameters: 


add_ghost_nodes
(symbol, discretised_symbol, bcs)[source]¶Add ghost nodes to a symbol.
For Dirichlet bcs, for a boundary condition “y = a at the lefthand boundary”, we concatenate a ghost node to the start of the vector y with value “2*a  y1” where y1 is the value of the first node. Similarly for the righthand boundary condition.
For Dirichlet bcs, for a boundary condition “y = a at the lefthand boundary”, we concatenate a ghost node to the start of the vector y with value “2*a  y1” where y1 is the value of the first node. Similarly for the righthand boundary condition.
For Neumann bcs, for a boundary condition “dy/dx = b at the lefthand boundary”, we concatenate a ghost node to the start of the vector y with value “b*h + y1” where y1 is the value of the first node and h is the mesh size. Similarly for the righthand boundary condition.
Parameters: 


Returns:  Matrix @ discretised_symbol + bcs_vector. When evaluated, this gives the discretised_symbol, with appropriate ghost nodes concatenated at each end. 
Return type: 

boundary_value_or_flux
(symbol, discretised_child)[source]¶Uses linear extrapolation to get the boundary value or flux of a variable in the Finite Volume Method.
See pybamm.SpatialMethod.boundary_value()
concatenation
(disc_children)[source]¶Discrete concatenation, taking edge_to_node for children that evaluate on
edges.
See pybamm.SpatialMethod.concatenation()
definite_integral_matrix
(domain, vector_type='row')[source]¶Matrix for finitevolume implementation of the definite integral in the primary dimension
for where \(a\) and \(b\) are the lefthand and righthand boundaries of the domain respectively
Parameters:  domain (list) – The domain(s) of integration 

Returns: 

delta_function
(symbol, discretised_symbol)[source]¶Delta function. Implemented as a vector whose only nonzero element is the first (if symbol.side = “left”) or last (if symbol.side = “right”), with appropriate value so that the integral of the delta function across the whole domain is the same as the integral of the discretised symbol across the whole domain.
divergence
(symbol, discretised_symbol, boundary_conditions)[source]¶Matrixvector multiplication to implement the divergence operator.
See pybamm.SpatialMethod.divergence()
divergence_matrix
(domain)[source]¶Divergence matrix for finite volumes in the appropriate domain. Equivalent to div(N) = (N[1:]  N[:1])/dx
Parameters:  domain (list) – The domain(s) in which to compute the divergence matrix 

Returns:  The (sparse) finite volume divergence matrix for the domain 
Return type:  pybamm.Matrix 
edge_to_node
(discretised_symbol)[source]¶Convert a discretised symbol evaluated on the cell edges to a discretised symbol
evaluated on the cell nodes.
See pybamm.FiniteVolume.shift()
gradient
(symbol, discretised_symbol, boundary_conditions)[source]¶Matrixvector multiplication to implement the gradient operator.
See pybamm.SpatialMethod.gradient()
gradient_matrix
(domain)[source]¶Gradient matrix for finite volumes in the appropriate domain. Equivalent to grad(y) = (y[1:]  y[:1])/dx
Parameters:  domain (list) – The domain(s) in which to compute the gradient matrix 

Returns:  The (sparse) finite volume gradient matrix for the domain 
Return type:  pybamm.Matrix 
indefinite_integral
(child, discretised_child)[source]¶Implementation of the indefinite integral operator.
indefinite_integral_matrix_edges
(domain)[source]¶Matrix for finitevolume implementation of the indefinite integral where the integrand is evaluated on mesh edges
The indefinite integral must satisfy the following conditions:
or, in discrete form,
Hence we must have
Note that \(f_{1/2}\) and \(f_{n+1/2}\) are included in the discrete integrand vector f, so we add a column of zeros at each end of the indefinite integral matrix to ignore these.
Parameters:  domain (list) – The domain(s) of integration 

Returns:  The finite volume integral matrix for the domain 
Return type:  pybamm.Matrix 
indefinite_integral_matrix_nodes
(domain)[source]¶Matrix for finitevolume implementation of the indefinite integral where the integrand is evaluated on mesh nodes. This is just a straightforward cumulative sum of the integrand
Parameters:  domain (list) – The domain(s) of integration 

Returns:  The finite volume integral matrix for the domain 
Return type:  pybamm.Matrix 
integral
(child, discretised_child)[source]¶Vectorvector dot product to implement the integral operator.
internal_neumann_condition
(left_symbol_disc, right_symbol_disc, left_mesh, right_mesh)[source]¶A method to find the internal neumann conditions between two symbols on adjacent subdomains.
Parameters: 


laplacian
(symbol, discretised_symbol, boundary_conditions)[source]¶Laplacian operator, implemented as div(grad(.))
See pybamm.SpatialMethod.laplacian()
node_to_edge
(discretised_symbol)[source]¶Convert a discretised symbol evaluated on the cell nodes to a discretised symbol
evaluated on the cell edges.
See pybamm.FiniteVolume.shift()
process_binary_operators
(bin_op, left, right, disc_left, disc_right)[source]¶Discretise binary operators in model equations. Performs appropriate averaging of diffusivities if one of the children is a gradient operator, so that discretised sizes match up.
Parameters: 


Returns:  Discretised binary operator 
Return type: 
shift
(discretised_symbol, shift_key)[source]¶Convert a discretised symbol evaluated at edges/nodes, to a discretised symbol evaluated at nodes/edges. For now we just take the arithemtic mean, though it may be better to take the harmonic mean based on [1].
[1] Recktenwald, Gerald. “The controlvolume finitedifference approximation to the diffusion equation.” (2012).
Parameters: 


Returns:  Averaged symbol. When evaluated, this returns either a scalar or an array of shape (n+1,) (if shift_key = “node to edge”) or (n,) (if shift_key = “edge to node”) 
Return type: 
spatial_variable
(symbol)[source]¶Creates a discretised spatial variable compatible with the FiniteVolume method.
Parameters:  symbol (pybamm.SpatialVariable ) – The spatial variable to be discretised. 

Returns:  Contains the discretised spatial variable 
Return type:  pybamm.Vector 