Partial Differential Equations

This module contains class for solving Partial Differential Equations (PDE) with Dirichlet, Neumann and Robin Boundary Conditions.

class findiff.pde.BoundaryConditions(shape)

Represents boundary conditions for a PDE.

Supports Dirichlet, Neumann and Robin (mixed) boundary conditions.

Dirichlet:: bc[index] = value
Neumann:: bc[index] = (diff_operator, value)
Robin (\(\alpha u + \beta \partial u / \partial n = g\)):: bc[index] = (alpha, diff_operator, beta, value)

Alternatively, Robin BCs can be specified using the operator-tuple form bc[index] = (alpha * Identity() + beta * diff_operator, value).

Initializes the BoundaryCondition object.

The BoundaryCondition objects needs information about the grid on which to solve the PDE, specifically the shape of the (equidistant) grid.

Parameters

class findiff.pde.PDE(lhs, rhs, bcs)

Representation of a partial differential equation.

Initializes the PDE.

You need to specify the left hand side (lhs) in terms of derivatives as well as the right hand side in terms of an array.

lhs: FinDiff object or combination of FinDiff objects: the left hand side of the PDE
rhs: numpy.ndarray: the right hand side of the PDE
bcs: BoundaryConditions: the boundary conditions for the PDE

solve(solver=None, **solver_options)

Solves the PDE.

solverstr, callable, or None

Which linear solver to use.

None or 'direct': use scipy.sparse.linalg.spsolve (default, backward compatible).
'cg': Conjugate Gradient (requires symmetric positive definite system).
'gmres': Generalized Minimal Residual (general systems).
'bicgstab': BiConjugate Gradient Stabilized (general non-symmetric systems).
'lgmres': LGMRES variant of GMRES.
'minres': Minimal Residual (symmetric indefinite systems).
A callable with signature f(A, b, **kw) -> x or f(A, b, **kw) -> (x, info). If the callable returns a 2-tuple the second element is treated as a convergence flag (0 = success).

**solver_options

Additional keyword arguments passed to the iterative solver. Common options include:

rtol (float): Relative convergence tolerance.
maxiter (int): Maximum number of iterations.
x0 (ndarray): Initial guess for the solution.
preconditioner (str or LinearOperator): If 'ilu', an incomplete LU preconditioner is built automatically. Otherwise pass a scipy.sparse.linalg.LinearOperator to use as the preconditioner M.

RuntimeError: If an iterative solver fails to converge.
ValueError: If an unknown solver name is given or if solver_options are passed with the direct solver.