pymgrit.heat package

Submodules

pymgrit.heat.heat_1d module

Vector and application class for the 1D heat equation

class pymgrit.heat.heat_1d.Heat1D(*args, **kwargs)

Bases: pymgrit.core.application.Application

Application class for the heat equation in 1D space,

u_t - a*u_xx = b(x,t), a > 0, x in [x_start,x_end], t in [0,T],

with homogeneous Dirichlet boundary conditions in space.

compute_matrix()

Define spatial discretization matrix for 1D heat equation

Second-order central finite differences with matrix stencil

(a / dx^2) * [-1 2 -1]

step(u_start: pymgrit.heat.heat_1d.VectorHeat1D, t_start: float, t_stop: float) → pymgrit.heat.heat_1d.VectorHeat1D

Time integration routine for 1D heat equation example problem:

Backward Euler (BDF1)

One-step method

u_i = (I + dt*L)^{-1} * (u_{i-1} + dt*b_i),

where L = self.space_disc is the spatial discretization operator

Parameters

• u_start – approximate solution for the input time t_start

• t_start – time associated with the input approximate solution u_start

• t_stop – time to evolve the input approximate solution to

Returns

approximate solution at input time t_stop

class pymgrit.heat.heat_1d.VectorHeat1D(size)

Bases: pymgrit.core.vector.Vector

Vector class for the 1D heat equation

clone()

Clone vector object

Return type

vector object with zero values

clone_rand()

Initialize vector object with random values

Return type

vector object with random values

clone_zero()

Initialize vector object with zeros

Return type

vector object with zero values

get_values()

Get vector data

Returns

values of vector object

norm()

Norm of a vector object

Returns

2-norm of vector object

pack()

Pack data

Returns

values of vector object

set_values(values)

Set vector data

Parameters

values – values for vector object

unpack(values)

Unpack and set data

Parameters

values – values for vector object

pymgrit.heat.heat_1d_2pts_bdf1 module

Application class for 1D heat problem using BDF1 time integration

Note: values at two consecutive time points are grouped as pairs

class pymgrit.heat.heat_1d_2pts_bdf1.Heat1DBDF1(*args, **kwargs)

Bases: pymgrit.core.application.Application

Application class for the heat equation in 1D space,

u_t - a*u_xx = b(x,t), a > 0, x in [x_start,x_end], t in [0,T],

with homogeneous Dirichlet boundary conditions in space

compute_matrix()

Define spatial discretization matrix for heat equation problem.

Discretization is centered finite differences with matrix stencil

(a / dx^2) * [-1 2 -1]

step(u_start: pymgrit.heat.vector_heat_1d_2pts.VectorHeat1D2Pts, t_start: float, t_stop: float) → pymgrit.heat.vector_heat_1d_2pts.VectorHeat1D2Pts

Time integration routine for 1D heat equation example problem:

Backward Euler (BDF1)

One-step method

u_i = (I + dt*L)^{-1} * (u_{i-1} + dt*b_i),

where L = self.space_disc is the spatial discretization operator

Note: step takes two BDF1 steps

• one step from (t_start + dtau) to t_stop

• one step from t_stop to (t_stop + dtau)

Parameters

• u_start – approximate solution for the input time t_start

• t_start – time associated with the input approximate solution u_start

• t_stop – time to evolve the input approximate solution to

Returns

approximate solution at input time t_stop

pymgrit.heat.heat_1d_2pts_bdf2 module

Application class for 1D heat problem using BDF2 time integration

Note: values at two consecutive time points are grouped as pairs

class pymgrit.heat.heat_1d_2pts_bdf2.Heat1DBDF2(*args, **kwargs)

Bases: pymgrit.core.application.Application

Application class for the heat equation in 1D space,

u_t - a*u_xx = b(x,t), a > 0, x in [x_start, x_end], t in [t_start, t_end],

with homogeneous Dirichlet boundary conditions in space

compute_matrix()

Define spatial discretization matrix for heat equation problem.

Discretization is centered finite differences with matrix stencil

(a / dx^2) * [-1 2 -1]

step(u_start: pymgrit.heat.vector_heat_1d_2pts.VectorHeat1D2Pts, t_start: float, t_stop: float) → pymgrit.heat.vector_heat_1d_2pts.VectorHeat1D2Pts

Time integration routine for 1D heat equation:

BDF2

Two-step method on variably spaced grid with spacing tau_i = t_i - t_{i-1}. In time-based stencil notation, we have at time point t_i

[r_i^2/(tau_i*(1+r_i))*I, -((1+r_i)/tau_i)*I, (1+2r_i)/(tau_i*(1+r_i))*I + L, 0, 0],

where L = self.space_disc is the spatial discretization operator and r_i = tau_i/tau_{i-1}

Note: For the pair associated with input time t_stop

• update at t_stop involves values at t_start and (t_start + dtau)

• update at t_stop + dtau involves values at (t_start + dtau) and t_stop

Parameters

• u_start – approximate solution for the input time t_start

• t_start – time associated with the input approximate solution u_start

• t_stop – time to evolve the input approximate solution to

Returns

approximate solution for the input time t_stop

pymgrit.heat.heat_2d module

Vector and application class for the 2D heat equation

Time integration methods:

Forward Euler Backward Euler Crank-Nicolson

class pymgrit.heat.heat_2d.Heat2D(*args, **kwargs)

Bases: pymgrit.core.application.Application

Application class for the heat equation in 2D space,

u_t - a(u_xx + u_yy) = b(x,y,t), a > 0, in [x_start, x_end] x [y_start, y_end] x (t_start, t_end],

with homogeneous Dirichlet boundary conditions in space.

compute_matrix()

Define spatial discretization matrix for 2D heat equation

Second-order central finite differences with matrix stencil

[ -f_y ] [-f_x 2(f_x + f_y) -f_x] [ -f_y ]

with f_x = (a / dx^2) and f_y = (a / dy^2)

compute_rhs(u_start: numpy.ndarray, t_start: float, t_stop: float)

Right-hand side of spatial system for implicit time integration methods

Parameters

• u_start – approximate solution for the input time t_start

• t_start – time associated with the input approximate solution u_start

• t_stop – time to evolve the input approximate solution to

Returns

right-hand side of spatial system at each time step in case of implicit time integration

step(u_start: pymgrit.heat.heat_2d.VectorHeat2D, t_start: float, t_stop: float) → pymgrit.heat.heat_2d.VectorHeat2D

Time integration routine for 2D heat equation example problem:

Backward Euler (BE) Forward Euler (FE) Crank-Nicolson (CN)

One-step method

(u_i - u_{i-1})/dt + (theta*L*u_i + (1-theta)*L*u_{i-1} = theta*b_i + (1-theta)*b_{i-1},

where L = self.space_disc is the spatial discretization operator and

theta = 0 -> FE theta = 1/2 -> CN theta = 1 -> BE

Parameters

• u_start – approximate solution for the input time t_start

• t_start – time associated with the input approximate solution u_start

• t_stop – time to evolve the input approximate solution to

Returns

approximate solution at input time t_stop

class pymgrit.heat.heat_2d.VectorHeat2D(nx, ny)

Bases: pymgrit.core.vector.Vector

Vector class for the 2D heat equation

clone()

Clone vector object

Return type

vector object with zero values

clone_rand()

Initialize vector object with random values

Return type

vector object with random values

clone_zero()

Initialize vector object with zeros

Return type

vector object with zero values

get_values()

Get vector data

Returns

values of vector object

norm()

Norm of a vector object

Returns

2-norm of vector object

pack()

Pack data

Returns

values of vector object

set_values(values)

Set vector data

Parameters

values – values for vector object

unpack(values)

Unpack and set data

Parameters

values – values for vector object

pymgrit.heat.vector_heat_1d_2pts module

Vector class for 1D heat problem

Note: values at two consecutive time points are grouped as pairs

class pymgrit.heat.vector_heat_1d_2pts.VectorHeat1D2Pts(size, dtau)

Bases: pymgrit.core.vector.Vector

Vector class for grouping values at two consecutive time points

clone()

Clone vector object

Returns

vector object with zero values

clone_rand()

Initialize vector object with random values

Returns

vector object with random values

clone_zero()

Initialize vector object with zeros

Returns

vector object with zero values

get_values()

Get vector data

Returns

tuple of values of member variables

norm()

Norm of a vector object

Returns

2-norm of vector object

pack()

Pack data

Returns

values of vector object

set_values(first_time_point, second_time_point, dtau)

Set vector data

Parameters

• first_time_point – values for first time point

• second_time_point – values for second time point

• dtau – time-step size within pair

unpack(values)

Unpack and set data

Parameters

values – values for vector object

Module contents