implementations.problem_classes.AllenCahn_2D_FD module

class allencahn_fullyimplicit(nvars=(128, 128), nu=2, eps=0.04, newton_maxiter=200, newton_tol=1e-12, lin_tol=1e-08, lin_maxiter=100, inexact_linear_ratio=None, radius=0.25, order=2)

Bases: ptype

Example implementing the two-dimensional Allen-Cahn equation with periodic boundary conditions \(u \in [-1, 1]^2\)

\[\frac{\partial u}{\partial t} = \Delta u + \frac{1}{\varepsilon^2} u (1 - u^\nu)\]

for constant parameter \(\nu\). Initial condition are circles of the form

\[u({\bf x}, 0) = \tanh\left(\frac{r - \sqrt{x_i^2 + y_j^2}}{\sqrt{2}\varepsilon}\right)\]

for \(i, j=0,..,N-1\), where \(N\) is the number of spatial grid points. For time-stepping, the problem is treated fully-implicitly, i.e., the nonlinear system is solved by Newton.

Parameters:

• nvars (tuple of int, optional) – Number of unknowns in the problem, e.g. nvars=(128, 128).

• nu (float, optional) – Problem parameter \(\nu\).

• eps (float, optional) – Scaling parameter \(\varepsilon\).

• newton_maxiter (int, optional) – Maximum number of iterations for the Newton solver.

• newton_tol (float, optional) – Tolerance for Newton’s method to terminate.

• lin_tol (float, optional) – Tolerance for linear solver to terminate.

• lin_maxiter (int, optional) – Maximum number of iterations for the linear solver.

• radius (float, optional) – Radius of the circles.

• order (int, optional) – Order of the finite difference matrix.

A

Second-order FD discretization of the 2D laplace operator.

Type:

scipy.spdiags

dx

Distance between two spatial nodes (same for both directions).

Type:

float

xvalues

Spatial grid points, here both dimensions have the same grid points.

Type:

np.1darray

newton_itercount

Number of iterations of Newton solver.

Type:

int

lin_itercount

Number of iterations of linear solver.

newton_ncalls

Number of calls of Newton solver.

Type:

int

lin_ncalls

Number of calls of linear solver.

Type:

int

dtype_f

alias of mesh

dtype_u

alias of mesh

eval_f(u, t)

Routine to evaluate the right-hand side of the problem.

Parameters:

• u (dtype_u) – Current values of the numerical solution.

• t (float) – Current time of the numerical solution is computed (not used here).

Returns:

f – The right-hand side of the problem.

Return type:

dtype_f

solve_system(rhs, factor, u0, t)

Simple Newton solver.

Parameters:

• rhs (dtype_f) – Right-hand side for the nonlinear system

• factor (float) – Abbrev. for the node-to-node stepsize (or any other factor required).

• u0 (dtype_u) – Initial guess for the iterative solver.

• t (float) – Current time (required here for the BC).

Returns:

me – The solution as mesh.

Return type:

dtype_u

u_exact(t, u_init=None, t_init=None)

Routine to compute the exact solution at time \(t\).

Parameters:

t (float) – Time of the exact solution.

Returns:

me – The exact solution.

Return type:

dtype_u

class allencahn_multiimplicit(nvars=(128, 128), nu=2, eps=0.04, newton_maxiter=200, newton_tol=1e-12, lin_tol=1e-08, lin_maxiter=100, inexact_linear_ratio=None, radius=0.25, order=2)

Bases: allencahn_fullyimplicit

Example implementing the two-dimensional Allen-Cahn equation with periodic boundary conditions \(u \in [-1, 1]^2\)

\[\frac{\partial u}{\partial t} = \Delta u + \frac{1}{\varepsilon^2} u (1 - u^\nu)\]

for constant parameter \(\nu\). Initial condition are circles of the form

\[u({\bf x}, 0) = \tanh\left(\frac{r - \sqrt{x_i^2 + y_j^2}}{\sqrt{2}\varepsilon}\right)\]

for \(i, j=0,..,N-1\), where \(N\) is the number of spatial grid points. For time-stepping, the problem is treated in multi-implicit fashion, i.e., the linear system containing the Laplacian is solved by the conjugate gradients method, and the system containing the rest of the right-hand side will be solved by Newton’s method.

dtype_f

alias of comp2_mesh

eval_f(u, t)

Routine to evaluate the right-hand side of the problem.

Parameters:

• u (dtype_u) – Current values of the numerical solution.

• t (float) – Current time of the numerical solution is computed.

Returns:

f – The right-hand side of the problem.

Return type:

dtype_f

solve_system_1(rhs, factor, u0, t)

Simple linear solver for \((I-factor\cdot A)\vec{u}=\vec{rhs}\).

Parameters:

• rhs (dtype_f) – Right-hand side for the linear system.

• factor (float) – Abbrev. for the local stepsize (or any other factor required).

• u0 (dtype_u) – Initial guess for the iterative solver.

• t (float) – Current time (e.g. for time-dependent BCs).

Returns:

me – The solution as mesh.

Return type:

dtype_u

solve_system_2(rhs, factor, u0, t)

Simple Newton solver.

Parameters:

• rhs (dtype_f) – Right-hand side for the nonlinear system.

• factor (float) – Abbrev. for the node-to-node stepsize (or any other factor required).

• u0 (dtype_u) – Initial guess for the iterative solver.

• t (float) – Current time (required here for the BC).

Returns:

me – The solution as mesh.

Return type:

dtype_u

class allencahn_multiimplicit_v2(nvars=(128, 128), nu=2, eps=0.04, newton_maxiter=200, newton_tol=1e-12, lin_tol=1e-08, lin_maxiter=100, inexact_linear_ratio=None, radius=0.25, order=2)

Bases: allencahn_fullyimplicit

This class implements the two-dimensional Allen-Cahn (AC) equation with periodic boundary conditions \(u \in [-1, 1]^2\)

\[\frac{\partial u}{\partial t} = \Delta u + \frac{1}{\varepsilon^2} u (1 - u^\nu)\]

for constant parameter \(\nu\). The initial condition has the form of circles

\[u({\bf x}, 0) = \tanh\left(\frac{r - \sqrt{x_i^2 + y_j^2}}{\sqrt{2}\varepsilon}\right)\]

for \(i, j=0,..,N-1\), where \(N\) is the number of spatial grid points. For time-stepping, a special AC-splitting is used here to get another kind of semi-implicit treatment of the problem: The term \(\Delta u - \frac{1}{\varepsilon^2} u^{\nu + 1}\) is handled implicitly and the nonlinear system including this part will be solved by Newton. \(\frac{1}{\varepsilon^2} u\) is solved by a linear solver provided by a SciPy routine.

dtype_f

alias of comp2_mesh

eval_f(u, t)

Routine to evaluate the right-hand side of the problem.

Parameters:

• u (dtype_u) – Current values of the numerical solution.

• t (float) – Current time of the numerical solution is computed.

Returns:

f – The right-hand side of the problem.

Return type:

dtype_f

solve_system_1(rhs, factor, u0, t)

Simple Newton solver.

Parameters:

• rhs (dtype_f) – Right-hand side for the nonlinear system.

• factor (float) – Abbrev. for the node-to-node stepsize (or any other factor required).

• u0 (dtype_u) – Initial guess for the iterative solver.

• t (float) – Current time (required here for the BC).

Returns:

me – The solution as mesh.

Return type:

dtype_u

solve_system_2(rhs, factor, u0, t)

Simple linear solver for \((I-factor\cdot A)\vec{u}=\vec{rhs}\).

Parameters:

• rhs (dtype_f) – Right-hand side for the linear system.

• factor (float) – Abbrev. for the local stepsize (or any other factor required).

• u0 (dtype_u) – Initial guess for the iterative solver.

• t (float) – Current time (e.g. for time-dependent BCs).

Returns:

me – The solution as mesh.

Return type:

dtype_u

class allencahn_semiimplicit(nvars=(128, 128), nu=2, eps=0.04, newton_maxiter=200, newton_tol=1e-12, lin_tol=1e-08, lin_maxiter=100, inexact_linear_ratio=None, radius=0.25, order=2)

Bases: allencahn_fullyimplicit

This class implements the two-dimensional Allen-Cahn equation with periodic boundary conditions \(u \in [-1, 1]^2\)

\[\frac{\partial u}{\partial t} = \Delta u + \frac{1}{\varepsilon^2} u (1 - u^\nu)\]

for constant parameter \(\nu\). Initial condition are circles of the form

\[u({\bf x}, 0) = \tanh\left(\frac{r - \sqrt{x_i^2 + y_j^2}}{\sqrt{2}\varepsilon}\right)\]

for \(i, j=0,..,N-1\), where \(N\) is the number of spatial grid points. For time-stepping, the problem is treated in a semi-implicit way, i.e., the linear system containing the Laplacian is solved by the conjugate gradients method, and the system containing the rest of the right-hand side is only evaluated at each time.

dtype_f

alias of imex_mesh

eval_f(u, t)

Routine to evaluate the right-hand side of the problem.

Parameters:

• u (dtype_u) – Current values of the numerical solution.

• t (float) – Current time of the numerical solution is computed (not used here).

Returns:

f – The right-hand side of the problem.

Return type:

dtype_f

solve_system(rhs, factor, u0, t)

Simple linear solver for \((I-factor\cdot A)\vec{u}=\vec{rhs}\).

Parameters:

• rhs (dtype_f) – Right-hand side for the linear system.

• factor (float) – Abbrev. for the local stepsize (or any other factor required).

• u0 (dtype_u) – Initial guess for the iterative solver.

• t (float) – Current time (e.g. for time-dependent BCs).

Returns:

me – The solution as mesh.

Return type:

dtype_u

u_exact(t, u_init=None, t_init=None)

Routine to compute the exact solution at time t.

Parameters:

t (float) – Time of the exact solution.

Returns:

me – The exact solution.

Return type:

dtype_u

class allencahn_semiimplicit_v2(nvars=(128, 128), nu=2, eps=0.04, newton_maxiter=200, newton_tol=1e-12, lin_tol=1e-08, lin_maxiter=100, inexact_linear_ratio=None, radius=0.25, order=2)

Bases: allencahn_fullyimplicit

This class implements the two-dimensional Allen-Cahn (AC) equation with periodic boundary conditions \(u \in [-1, 1]^2\)

\[\frac{\partial u}{\partial t} = \Delta u + \frac{1}{\varepsilon^2} u (1 - u^\nu)\]

for constant parameter \(\nu\). Initial condition are circles of the form

\[u({\bf x}, 0) = \tanh\left(\frac{r - \sqrt{x_i^2 + y_j^2}}{\sqrt{2}\varepsilon}\right)\]

for \(i, j=0,..,N-1\), where \(N\) is the number of spatial grid points. For time-stepping, a special AC-splitting is used to get a semi-implicit treatment of the problem: The term \(\Delta u - \frac{1}{\varepsilon^2} u^{\nu + 1}\) is handled implicitly and the nonlinear system including this part will be solved by Newton. \(\frac{1}{\varepsilon^2} u\) is only evaluated at each time.

dtype_f

alias of imex_mesh

eval_f(u, t)

Routine to evaluate the right-hand side of the problem.

Parameters:

• u (dtype_u) – Current values of the numerical solution.

• t (float) – Current time of the numerical solution is computed.

Returns:

f – The right-hand side of the problem.

Return type:

dtype_f

solve_system(rhs, factor, u0, t)

Simple Newton solver.

Parameters:

• rhs (dtype_f) – Right-hand side for the nonlinear system.

• factor (float) – Abbrev. for the node-to-node stepsize (or any other factor required).

• u0 (dtype_u) – Initial guess for the iterative solver.

• t (float) – Current time (required here for the BC).

Returns:

me – The solution as mesh.

Return type:

dtype_u