implementations.problem_classes.GeneralizedFisher_1D_PETSc module

class Fisher_full(da, prob, factor, dx)

Bases: object

Helper class to generate residual and Jacobian matrix for PETSc’s nonlinear solver SNES.

Parameters:

• da (DMDA object) – Object of PETSc.

• prob (problem instance) – Contains problem information for PETSc.

• factor (float) – Temporal factor \(\Delta t Q_\Delta\).

• dx (float) – Grid spacing in x direction.

localX

Local vector for PETSc.

Type:

PETSc vector object

xs, xe

Defines the ranges for spatial domain.

Type:

int

mx

Get sizes for the vector containing the spatial points.

Type:

tuple

row

Row for a matrix.

Type:

PETSc matrix stencil object

col

Column for a matrix.

Type:

PETSc matrix stencil object

formFunction(snes, X, F)

Function to evaluate the residual for the Newton solver. This function should be equal to the RHS in the solution.

Parameters:

• snes (PETSc solver object) – Nonlinear solver object.

• X (PETSc vector object) – Input vector.

• F (PETSc vector object) – Output vector \(F(X)\).

Returns:

Overwrites F.

Return type:

None

formJacobian(snes, X, J, P)

Function to return the Jacobian matrix.

Parameters:

• snes (PETSc solver object) – Nonlinear solver object.

• X (PETSc vector object) – Input vector.

• J (PETSc matrix object) – Current Jacobian matrix.

• P (PETSc matrix object) – New Jacobian matrix.

Returns:

Matrix status.

Return type:

PETSc.Mat.Structure.SAME_NONZERO_PATTERN

class Fisher_reaction(da, prob, factor)

Bases: object

Helper class to generate residual and Jacobian matrix for PETSc’s nonlinear solver SNES.

Parameters:

• da (DMDA object) – Object of PETSc.

• prob (problem instance) – Contains problem information for PETSc.

• factor (float) – Temporal factor \(\Delta t Q_\Delta\).

• dx (float) – Grid spacing in x direction.

localX

Local vector for PETSc.

Type:

PETSc vector object

formFunction(snes, X, F)

Function to evaluate the residual for the Newton solver. This function should be equal to the RHS in the solution.

Parameters:

• snes (PETSc solver object) – Nonlinear solver object.

• X (PETSc vector object) – Input vector.

• F (PETSc vector object) – Output vector \(F(X)\).

Returns:

Overwrites F.

Return type:

None

formJacobian(snes, X, J, P)

Function to return the Jacobian matrix.

Parameters:

• snes (PETSc solver object) – Nonlinear solver object.

• X (PETSc vector object) – Input vector.

• J (PETSc matrix object) – Current Jacobian matrix.

• P (PETSc matrix object) – New Jacobian matrix.

Returns:

Matrix status.

Return type:

PETSc.Mat.Structure.SAME_NONZERO_PATTERN

class petsc_fisher_fullyimplicit(nvars, lambda0, nu, interval, comm=petsc4py.PETSc.COMM_WORLD, lsol_tol=1e-10, nlsol_tol=1e-10, lsol_maxiter=None, nlsol_maxiter=None)

Bases: petsc_fisher_multiimplicit

The following one-dimensional problem is an example of a reaction-diffusion equation with traveling waves, and can be seen as a generalized Fisher equation. This class implements a special case of the Kolmogorov-Petrovskii-Piskunov problem [1] using periodic boundary conditions

\[\frac{\partial u}{\partial t} = \Delta u + \lambda_0^2 u (1 - u^\nu)\]

with exact solution

\[u(x, 0) = \left[ 1 + \left(2^{\nu / 2} - 1\right) \exp\left(-(\nu / 2)\sigma_1 x + 2 \lambda_1 t\right) \right]^{-2 / \nu}\]

for \(x \in \mathbb{R}\), and

\[\sigma_1 = \lambda_1 - \sqrt{\lambda_1^2 - \lambda_0^2},\]

\[\lambda_1 = \frac{\lambda_0}{2} \left[ \left(1 + \frac{\nu}{2}\right)^{1/2} + \left(1 + \frac{\nu}{2}\right)^{-1/2} \right].\]

This class is implemented to be solved in spatial using PETSc [2], [3]. For time-stepping, the problem is treated fully-implicitly.

dtype_f

alias of petsc_vec

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 the numerical solution is computed.

Returns:

f – Right-hand side of the problem.

Return type:

dtype_f

solve_system(rhs, factor, u0, t)

Nonlinear solver for \((I - factor \cdot F)(\vec{u}) = \vec{rhs}\).

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 – Solution as mesh.

Return type:

dtype_u

class petsc_fisher_multiimplicit(nvars, lambda0, nu, interval, comm=petsc4py.PETSc.COMM_WORLD, lsol_tol=1e-10, nlsol_tol=1e-10, lsol_maxiter=None, nlsol_maxiter=None)

Bases: ptype

The following one-dimensional problem is an example of a reaction-diffusion equation with traveling waves, and can be seen as a generalized Fisher equation. This class implements a special case of the Kolmogorov-Petrovskii-Piskunov problem [1] using periodic boundary conditions

\[\frac{\partial u}{\partial t} = \Delta u + \lambda_0^2 u (1 - u^\nu)\]

with exact solution

\[u(x, 0) = \left[ 1 + \left(2^{\nu / 2} - 1\right) \exp\left(-(\nu / 2)\sigma_1 x + 2 \lambda_1 t\right) \right]^{-2 / \nu}\]

for \(x \in \mathbb{R}\), and

\[\sigma_1 = \lambda_1 - \sqrt{\lambda_1^2 - \lambda_0^2},\]

\[\lambda_1 = \frac{\lambda_0}{2} \left[ \left(1 + \frac{\nu}{2}\right)^{1/2} + \left(1 + \frac{\nu}{2}\right)^{-1/2} \right].\]

This class is implemented to be solved in spatial using PETSc [2], [3]. For time-stepping, the problem will be solved multi-implicitly.

Parameters:

• nvars (int, optional) – Spatial resolution.

• lambda0 (float, optional) – Problem parameter : math:lambda_0.

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

• interval (tuple, optional) – Defines the spatial domain.

• comm (PETSc.COMM_WORLD, optional) – Communicator for PETSc.

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

• nlsol_tol (float, optional) – Tolerance for nonlinear solver to terminate.

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

• nlsol_maxiter (int, optional) – Maximum number of iterations for nonlinear solver.

dx

Mesh grid width.

Type:

float

xs, xe

Define the ranges.

Type:

int

A

Discretization matrix.

Type:

PETSc matrix object

localX

Local vector for solution.

Type:

PETSc vector object

ksp

Linear solver object.

Type:

PETSc solver object

snes

Nonlinear solver object.

Type:

PETSc solver object

F

Global vector.

Type:

PETSc vector object

J

Jacobi matrix.

Type:

PETSc matrix object

References

[1] (1,2,3)

Z. Feng. Traveling wave behavior for a generalized fisher equation. Chaos Solitons Fractals 38(2), 481–488 (2008).

[2] (1,2,3)

PETSc Web page. Satish Balay et al. https://petsc.org/ (2023).

[3] (1,2,3)

Parallel distributed computing using Python. Lisandro D. Dalcin, Rodrigo R. Paz, Pablo A. Kler, Alejandro Cosimo. Advances in Water Resources (2011).

dtype_f

alias of petsc_vec_comp2

dtype_u

alias of petsc_vec

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 the numerical solution is computed.

Returns:

f – Right-hand side of the problem.

Return type:

dtype_f

get_sys_mat(factor)

Helper function to assemble the system matrix of the linear problem.

Parameters:

factor (float) – Factor to define the system matrix.

Returns:

A – Matrix for the system to solve.

Return type:

PETSc matrix object

solve_system_1(rhs, factor, u0, t)

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 – Solution as mesh.

Return type:

dtype_u

solve_system_2(rhs, factor, u0, t)

Nonlinear solver for \((I - factor \cdot F)(\vec{u}) = \vec{rhs}\).

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 – Solution as mesh.

Return type:

dtype_u

u_exact(t)

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

Parameters:

t (float) – Time of the exact solution.

Returns:

me – Exact solution.

Return type:

dtype_u

class petsc_fisher_semiimplicit(nvars, lambda0, nu, interval, comm=petsc4py.PETSc.COMM_WORLD, lsol_tol=1e-10, nlsol_tol=1e-10, lsol_maxiter=None, nlsol_maxiter=None)

Bases: petsc_fisher_multiimplicit

The following one-dimensional problem is an example of a reaction-diffusion equation with traveling waves, and can be seen as a generalized Fisher equation. This class implements a special case of the Kolmogorov-Petrovskii-Piskunov problem [1] using periodic boundary conditions

\[\frac{\partial u}{\partial t} = \Delta u + \lambda_0^2 u (1 - u^\nu)\]

with exact solution

\[u(x, 0) = \left[ 1 + \left(2^{\nu / 2} - 1\right) \exp\left(-(\nu / 2)\sigma_1 x + 2 \lambda_1 t\right) \right]^{-2 / \nu}\]

for \(x \in \mathbb{R}\), and

\[\sigma_1 = \lambda_1 - \sqrt{\lambda_1^2 - \lambda_0^2},\]

\[\lambda_1 = \frac{\lambda_0}{2} \left[ \left(1 + \frac{\nu}{2}\right)^{1/2} + \left(1 + \frac{\nu}{2}\right)^{-1/2} \right].\]

This class is implemented to be solved in spatial using PETSc [2], [3]. For time-stepping, the problem here will be solved in a semi-implicit way.

dtype_f

alias of petsc_vec_imex

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 the numerical solution is computed.

Returns:

f – 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 – Solution as mesh.

Return type:

dtype_u