KPP Fisher Equation¶

The Fisher-KPP equation is a reaction-diffusion equation that can be used to model population growth and wave propagation. It is given by:

$$ \frac{\partial \phi}{\partial t} = \nu \nabla^2 \phi + r \phi (1 - \phi) $$

Solution in 1D¶

In [1]:

from torchfsm.operator import Operator, Laplacian, ImplicitSource

def KPPFisher(nu:float,r:float) -> Operator:
    return nu*Laplacian()+r*ImplicitSource(lambda phi:phi*(1-phi))

from torchfsm.operator import Operator, Laplacian, ImplicitSource def KPPFisher(nu:float,r:float) -> Operator: return nu*Laplacian()+r*ImplicitSource(lambda phi:phi*(1-phi))

In [2]:

import torch
from torchfsm.mesh import MeshGrid
from torchfsm.traj_recorder import AutoRecorder
from torchfsm.plot import plot_traj
from torchfsm.field import truncated_fourier_series
device='cuda' if torch.cuda.is_available() else 'cpu'
L=10.0; N=100; 

import torch from torchfsm.mesh import MeshGrid from torchfsm.traj_recorder import AutoRecorder from torchfsm.plot import plot_traj from torchfsm.field import truncated_fourier_series device='cuda' if torch.cuda.is_available() else 'cpu' L=10.0; N=100;

In [3]:

kpp_fisher = KPPFisher(nu=0.1, r=0.01)

mesh=MeshGrid([(0,L,N)],device=device)
x=mesh.bc_mesh_grid()
u_0=truncated_fourier_series(
    mesh=mesh,
    freq_threshold=5.0,
    normalize_mode="0_1",
)
traj=kpp_fisher.integrate(
    u_0=u_0,
    mesh=mesh,
    dt=0.01,
    step=200,
    trajectory_recorder=AutoRecorder(),
)

kpp_fisher = KPPFisher(nu=0.1, r=0.01) mesh=MeshGrid([(0,L,N)],device=device) x=mesh.bc_mesh_grid() u_0=truncated_fourier_series( mesh=mesh, freq_threshold=5.0, normalize_mode="0_1", ) traj=kpp_fisher.integrate( u_0=u_0, mesh=mesh, dt=0.01, step=200, trajectory_recorder=AutoRecorder(), )

In [4]:

plot_traj(traj,animation=False)

plot_traj(traj,animation=False)

Solution in 2D¶

In [5]:

kpp_fisher = KPPFisher(nu=0.01, r=10.0)

mesh=MeshGrid([(0,L,N)]*2,device=device)
x=mesh.bc_mesh_grid()
u_0=truncated_fourier_series(
    mesh=mesh,
    freq_threshold=5,
    normalize_mode="0_1",
)
traj=kpp_fisher.integrate(
    u_0=u_0,
    mesh=mesh,
    dt=0.01,
    step=50,
    trajectory_recorder=AutoRecorder(),
)

kpp_fisher = KPPFisher(nu=0.01, r=10.0) mesh=MeshGrid([(0,L,N)]*2,device=device) x=mesh.bc_mesh_grid() u_0=truncated_fourier_series( mesh=mesh, freq_threshold=5, normalize_mode="0_1", ) traj=kpp_fisher.integrate( u_0=u_0, mesh=mesh, dt=0.01, step=50, trajectory_recorder=AutoRecorder(), )

In [6]:

plot_traj(traj,animation=True)

plot_traj(traj,animation=True)

Out[6]:

Solution in 3D¶

In [7]:

kpp_fisher = KPPFisher(nu=0.01, r=10.0)

mesh=MeshGrid([(0,L,N)]*3,device=device)
x=mesh.bc_mesh_grid()
u_0=truncated_fourier_series(
    mesh=mesh,
    freq_threshold=2,
    normalize_mode="0_1",
)
traj=kpp_fisher.integrate(
    u_0=u_0,
    mesh=mesh,
    dt=0.01,
    step=50,
    trajectory_recorder=AutoRecorder(),
)

kpp_fisher = KPPFisher(nu=0.01, r=10.0) mesh=MeshGrid([(0,L,N)]*3,device=device) x=mesh.bc_mesh_grid() u_0=truncated_fourier_series( mesh=mesh, freq_threshold=2, normalize_mode="0_1", ) traj=kpp_fisher.integrate( u_0=u_0, mesh=mesh, dt=0.01, step=50, trajectory_recorder=AutoRecorder(), )

In [8]:

plot_traj(traj,animation=True)

plot_traj(traj,animation=True)

Out[8]: