implementations.problem_classes.AllenCahn_1D_FD module

class allencahn_front_finel(nvars=127, dw=-0.04, eps=0.04, newton_maxiter=100, newton_tol=1e-12, interval=(-0.5, 0.5), stop_at_nan=True, stop_at_maxiter=False)

Bases: allencahn_front_fullyimplicit

This class implements the one-dimensional Allen-Cahn equation with driving force using inhomogeneous Dirichlet boundary conditions

\[\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} - \frac{2}{\varepsilon^2} u (1 - u) (1 - 2u) - 6 d_w u (1 - u)\]

for \(u \in [0, 1]\). Centered finite differences are used for discretization of the Laplacian. The exact solution is given by

\[u(x, t) = 0.5 \left(1 + \tanh\left(\frac{x - vt}{\sqrt{2}\varepsilon}\right)\right)\]

with \(v = 3 \sqrt{2} \varepsilon d_w\).

Let \(A\) denote the finite difference matrix to discretize \(\frac{\partial^2 u}{\partial x^2}\). Here, Finel’s trick is used. Let

\[a = \tanh\left(\frac{\Delta x}{\sqrt{2}\varepsilon}\right)^2,\]

then, the right-hand side of the problem can be written as

\[\frac{\partial u}{\partial t} = A u - \frac{1}{\Delta x^2} \left[ \frac{1 - a}{1 - a (2u - 1)^2} - 1 \right] (2u - 1).\]

For time-stepping, this problem will be treated in a fully-implicit way. The nonlinear system is solved using Newton.

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 (e.g. for time-dependent BCs).

Returns:

me – The solution as mesh.

Return type:

dtype_u

class allencahn_front_fullyimplicit(nvars=127, dw=-0.04, eps=0.04, newton_maxiter=100, newton_tol=1e-12, interval=(-0.5, 0.5), stop_at_nan=True, stop_at_maxiter=False)

Bases: ptype

Example implementing the one-dimensional Allen-Cahn equation with driving force using inhomogeneous Dirichlet boundary conditions

\[\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} - \frac{2}{\varepsilon^2} u (1 - u) (1 - 2u) - 6 d_w u (1 - u)\]

for \(u \in [0, 1]\). The second order spatial derivative is approximated using centered finite differences. The exact solution is given by

\[u(x, t)= 0.5 \left(1 + \tanh\left(\frac{x - vt}{\sqrt{2}\varepsilon}\right)\right)\]

with \(v = 3 \sqrt{2} \varepsilon d_w\). For time-stepping, this problem is implemented to be treated fully-implicit using Newton to solve the nonlinear system.

Parameters:

• nvars (int) – Number of unknowns in the problem.

• dw (float) – Driving force.

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

• newton_maxiter (int) – Maximum number of iterations for Newton’s method.

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

• interval (list) – Interval of spatial domain.

• stop_at_nan (bool, optional) – Indicates that the Newton solver should stop if nan values arise.

A

Second-order FD discretization of the 1D laplace operator.

Type:

scipy.diags

dx

Distance between two spatial nodes.

Type:

float

xvalues

Spatial grid values.

Type:

np.1darray

uext

Contains additionally the external values of the boundary.

Type:

dtype_u

work_counters

Counter for statistics. Here, number of Newton calls and number of evaluations of right-hand side are counted.

Type:

WorkCounter

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.

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)

Routine to return initial condition or the exact solution

Parameters:

t (float) – Time at which the exact solution is computed.

Returns:

me – The exact solution (in space and time).

Return type:

dtype_u

class allencahn_front_semiimplicit(nvars=127, dw=-0.04, eps=0.04, newton_maxiter=100, newton_tol=1e-12, interval=(-0.5, 0.5), stop_at_nan=True, stop_at_maxiter=False)

Bases: allencahn_front_fullyimplicit

This class implements the one-dimensional Allen-Cahn equation with driving force using inhomogeneous Dirichlet boundary conditions

\[\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} - \frac{2}{\varepsilon^2} u (1 - u) (1 - 2u) - 6 d_w u (1 - u)\]

for \(u \in [0, 1]\). Centered finite differences are used for discretization of the second order spatial derivative. The exact solution is given by

\[u(x, t) = 0.5 \left(1 + \tanh\left(\frac{x - vt}{\sqrt{2}\varepsilon}\right)\right)\]

with \(v = 3 \sqrt{2} \varepsilon d_w\). For time-stepping, this problem will be treated in a semi-implicit way, i.e., the Laplacian is treated implicitly, and the rest of the right-hand side will be handled explicitly.

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 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_periodic_fullyimplicit(nvars=128, dw=-0.04, eps=0.04, newton_maxiter=100, newton_tol=1e-12, interval=(-0.5, 0.5), radius=0.25, stop_at_nan=True)

Bases: ptype

Example implementing the one-dimensional Allen-Cahn equation with driving force and periodic boundary conditions

\[\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} - \frac{2}{\varepsilon^2} u (1 - u) (1 - 2u) - 6 d_w u (1 - u)\]

for \(u \in [0, 1]\). Centered finite differences are used for discretization of the Laplacian. The exact solution is

\[u(x, t) = 0.5 \left(1 + \tanh\left(\frac{r - |x| - vt}{\sqrt{2}\varepsilon}\right)\right)\]

with \(v = 3 \sqrt{2} \varepsilon d_w\) and radius \(r\) of the circles. For time-stepping, the problem is treated fully-implicitly, i.e., the nonlinear system is solved by Newton.

Parameters:

• nvars (int) – Number of unknowns in the problem.

• dw (float) – Driving force.

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

• newton_maxiter (int) – Maximum number of iterations for Newton’s method.

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

• interval (list) – Interval of spatial domain.

• radius (float) – Radius of the circles.

• stop_at_nan (bool, optional) – Indicates that the Newton solver should stop if nan values arise.

A

Second-order FD discretization of the 1D laplace operator.

Type:

scipy.diags

dx

Distance between two spatial nodes.

Type:

float

xvalues

Spatial grid points.

Type:

np.1darray

work_counters

Counter for statistics. Here, number of Newton calls and number of evaluations of right-hand side are counted.

Type:

WorkCounter

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.

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:

u – The solution as mesh.

Return type:

dtype_u

u_exact(t)

Routine to return initial condition or the exact solution.

Parameters:

t (float) – Time at which the approximated exact solution is computed.

Returns:

me – The approximated exact solution.

Return type:

dtype_u

class allencahn_periodic_multiimplicit(nvars=128, dw=-0.04, eps=0.04, newton_maxiter=100, newton_tol=1e-12, interval=(-0.5, 0.5), radius=0.25, stop_at_nan=True)

Bases: allencahn_periodic_fullyimplicit

This class implements the one-dimensional Allen-Cahn equation with driving force and periodic boundary conditions

\[\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} - \frac{2}{\varepsilon^2} u (1 - u) (1 - 2u) - 6 d_w u (1 - u)\]

for \(u \in [0, 1]\). For discretization of the second order spatial derivative, centered finite differences are used. The exact solution is then given by

\[u(x, t) = 0.5 \left(1 + \tanh\left(\frac{r - |x| - vt}{\sqrt{2}\varepsilon}\right)\right)\]

with \(v = 3 \sqrt{2} \varepsilon d_w\) and radius \(r\) of the circles. For time-stepping, the problem is treated in a multi-implicit fashion, i.e., the nonlinear system containing the part with the Laplacian is solved with a linear solver provided by a SciPy routine, and the nonlinear system including the rest of the right-hand side is 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 (not used here).

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 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_periodic_semiimplicit(nvars=128, dw=-0.04, eps=0.04, newton_maxiter=100, newton_tol=1e-12, interval=(-0.5, 0.5), radius=0.25, stop_at_nan=True)

Bases: allencahn_periodic_fullyimplicit

This class implements the one-dimensional Allen-Cahn equation with driving force and periodic boundary conditions

\[\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} - \frac{2}{\varepsilon^2} u (1 - u) (1 - 2u) - 6 d_w u (1 - u)\]

for \(u \in [0, 1]\). For discretization of the Laplacian, centered finite differences are used. The exact solution is

\[u(x, t) = 0.5 \left(1 + \tanh\left(\frac{r - |x| - vt}{\sqrt{2}\varepsilon}\right)\right)\]

with \(v = 3 \sqrt{2} \varepsilon d_w\) and radius \(r\) of the circles. For time-stepping, the problem is treated in semi-implicit way, i.e., the part containing the Laplacian is treated implicitly, and 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.

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