# Finite Volume¶

class pybamm.FiniteVolume(options=None)[source]

A class which implements the steps specific to the finite volume method during discretisation.

Parameters

Add ghost nodes to a symbol.

For Dirichlet bcs, for a boundary condition “y = a at the left-hand 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 right-hand boundary condition.

For Neumann bcs no ghost nodes are added. Instead, the exact value provided by the boundary condition is used at the cell edge when calculating the gradient (see pybamm.FiniteVolume.add_neumann_values()).

Parameters
Returns

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

Return type

pybamm.Symbol

Dirichlet bcs are implemented using ghost nodes, see pybamm.FiniteVolume.add_ghost_nodes().

Parameters
• symbol (pybamm.SpatialVariable) – The variable to be discretised

• discretised_gradient (pybamm.Vector) – Contains the discretised gradient of symbol

• bcs (dict of tuples (pybamm.Scalar, str)) – Dictionary (with keys “left” and “right”) of boundary conditions. Each boundary condition consists of a value and a flag indicating its type (e.g. “Dirichlet”)

• domain (list of strings) – The domain of the gradient of the symbol (may include ghost nodes)

Returns

Matrix @ discretised_gradient + bcs_vector. When evaluated, this gives the discretised_gradient, with the values of the Neumann boundary conditions concatenated at each end (if given).

Return type

pybamm.Symbol

boundary_value_or_flux(symbol, discretised_child, bcs=None)[source]

Uses 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(child, vector_type='row', integration_dimension='primary')[source]

Matrix for finite-volume implementation of the definite integral in the primary dimension

$I = \int_{a}^{b}\!f(s)\,ds$

for where $$a$$ and $$b$$ are the left-hand and right-hand boundaries of the domain respectively

Parameters
• child (pybamm.Symbol) – The symbol being integrated

• vector_type (str, optional) – Whether to return a row or column vector in the primary dimension (default is row)

• integration_dimension (str, optional) – The dimension in which to integrate (default is “primary”)

Returns

The finite volume integral matrix for the domain

Return type

pybamm.Matrix

delta_function(symbol, discretised_symbol)[source]

Delta function. Implemented as a vector whose only non-zero 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]

Matrix-vector multiplication to implement the divergence operator. See pybamm.SpatialMethod.divergence()

divergence_matrix(domains)[source]

Divergence matrix for finite volumes in the appropriate domain. Equivalent to div(N) = (N[1:] - N[:-1])/dx

Parameters

domains (dict) – The domain(s) and auxiliary domain 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, method='arithmetic')[source]

Convert a discretised symbol evaluated on the cell edges to a discretised symbol evaluated on the cell nodes. See pybamm.FiniteVolume.shift()

Matrix-vector multiplication to implement the gradient operator. See pybamm.SpatialMethod.gradient()

Gradient matrix for finite volumes in the appropriate domain. Equivalent to grad(y) = (y[1:] - y[:-1])/dx

Parameters

domains (list) – The domain in which to compute the gradient matrix, including ghost nodes

Returns

The (sparse) finite volume gradient matrix for the domain

Return type

pybamm.Matrix

indefinite_integral(child, discretised_child, direction)[source]

Implementation of the indefinite integral operator.

indefinite_integral_matrix_edges(domains, direction)[source]

Matrix for finite-volume implementation of the indefinite integral where the integrand is evaluated on mesh edges (shape (n+1, 1)). The integral will then be evaluated on mesh nodes (shape (n, 1)).

Parameters
• domains (dict) – The domain(s) and auxiliary domains of integration

• direction (str) – The direction of integration (forward or backward). See notes.

Returns

The finite volume integral matrix for the domain

Return type

pybamm.Matrix

Notes

Forward integral

$F(x) = \int_0^x\!f(u)\,du$

The indefinite integral must satisfy the following conditions:

• $$F(0) = 0$$

• $$f(x) = \frac{dF}{dx}$$

or, in discrete form,

• BoundaryValue(F, “left”) = 0, i.e. $$3*F_0 - F_1 = 0$$

• $$f_{i+1/2} = (F_{i+1} - F_i) / dx_{i+1/2}$$

Hence we must have

• $$F_0 = du_{1/2} * f_{1/2} / 2$$

• $$F_{i+1} = F_i + du_{i+1/2} * f_{i+1/2}$$

Note that $$f_{-1/2}$$ and $$f_{end+1/2}$$ 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.

Backward integral

$F(x) = \int_x^{end}\!f(u)\,du$

The indefinite integral must satisfy the following conditions:

• $$F(end) = 0$$

• $$f(x) = -\frac{dF}{dx}$$

or, in discrete form,

• BoundaryValue(F, “right”) = 0, i.e. $$3*F_{end} - F_{end-1} = 0$$

• $$f_{i+1/2} = -(F_{i+1} - F_i) / dx_{i+1/2}$$

Hence we must have

• $$F_{end} = du_{end+1/2} * f_{end-1/2} / 2$$

• $$F_{i-1} = F_i + du_{i-1/2} * f_{i-1/2}$$

Note that $$f_{-1/2}$$ and $$f_{end+1/2}$$ 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.

indefinite_integral_matrix_nodes(domains, direction)[source]

Matrix for finite-volume implementation of the (backward) indefinite integral where the integrand is evaluated on mesh nodes (shape (n, 1)). The integral will then be evaluated on mesh edges (shape (n+1, 1)). This is just a straightforward (backward) cumulative sum of the integrand

Parameters
• domains (dict) – The domain(s) and auxiliary domains of integration

• direction (str) – The direction of integration (forward or backward)

Returns

The finite volume integral matrix for the domain

Return type

pybamm.Matrix

integral(child, discretised_child, integration_dimension)[source]

Vector-vector 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, method='arithmetic')[source]

Convert a discretised symbol evaluated on the cell nodes to a discretised symbol evaluated on the cell edges. See pybamm.FiniteVolume.shift()

preprocess_external_variables(var)[source]

For finite volumes, we need the boundary fluxes for discretising properly. Here, we extrapolate and then add them to the boundary conditions.

Parameters

var (pybamm.Variable or pybamm.Concatenation) – The external variable that is to be processed

Returns

new_bcs – A dictionary containing the new boundary conditions

Return type

dict

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. For this averaging we use the harmonic mean [1].

[1] Recktenwald, Gerald. “The control-volume finite-difference approximation to the diffusion equation.” (2012).

Parameters
Returns

Discretised binary operator

Return type

pybamm.BinaryOperator

shift(discretised_symbol, shift_key, method)[source]

Convert a discretised symbol evaluated at edges/nodes, to a discretised symbol evaluated at nodes/edges. Can be the arithmetic mean or the harmonic mean.

Note: when computing fluxes at cell edges it is better to take the harmonic mean based on [1].

[1] Recktenwald, Gerald. “The control-volume finite-difference approximation to the diffusion equation.” (2012).

Parameters
• discretised_symbol (pybamm.Symbol) – Symbol to be averaged. When evaluated, this symbol returns either a scalar or an array of shape (n,) or (n+1,), where n is the number of points in the mesh for the symbol’s domain (n = self.mesh[symbol.domain].npts)

• shift_key (str) – Whether to shift from nodes to edges (“node to edge”), or from edges to nodes (“edge to node”)

• method (str) – Whether to use the “arithmetic” or “harmonic” mean

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

pybamm.Symbol

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

upwind_or_downwind(symbol, discretised_symbol, bcs, direction)[source]

Implement an upwinding operator. Currently, this requires the symbol to have a Dirichlet boundary condition on the left side (for upwinding) or right side (for downwinding).

Parameters
• symbol (pybamm.SpatialVariable) – The variable to be discretised

• discretised_gradient (pybamm.Vector) – Contains the discretised gradient of symbol

• bcs (dict of tuples (pybamm.Scalar, str)) – Dictionary (with keys “left” and “right”) of boundary conditions. Each boundary condition consists of a value and a flag indicating its type (e.g. “Dirichlet”)

• direction (str) – Direction in which to apply the operator (upwind or downwind)