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2. pde

torchfsm.pde.Burgers ¤

Burgers(nu: float) -> Operator
Burgers equation
\[\frac{\partial \mathbf{u}}{\partial t} =-\mathbf{u} \cdot \nabla \mathbf{u} + \nu \nabla^2 \mathbf{u}\]

Parameters:

Name Type Description Default
nu float

Viscosity coefficient.

required

Returns:

Name Type Description
Operator Operator

The operator representing the Burgers equation.
Source code in torchfsm/pde.py 
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def Burgers(nu: float) -> Operator:
    r"""
    Burgers equation:
        $$\frac{\partial \mathbf{u}}{\partial t} =-\mathbf{u} \cdot \nabla \mathbf{u} + \nu \nabla^2 \mathbf{u}$$

    Args:
        nu (float): Viscosity coefficient.    

    Returns:
        Operator: The operator representing the Burgers equation.

    """
    return nu * Laplacian() - Convection()

torchfsm.pde.KortewegDeVries ¤

KortewegDeVries(
    dispersivity: Union[float, Tensor] = -1,
    convection_coef: Union[float, Tensor] = 6.0,
) -> Operator
Korteweg-De Vries equation
\[\frac{\partial \mathbf{u}}{\partial t} = d \nabla \cdot (\nabla^2\mathbf{u}) +c \mathbf{u} \cdot \nabla \mathbf{u}\]

Parameters:

Name Type Description Default
dispersivity Union[float, Tensor]

Dispersion coefficient. Default is -1.

 -1
convection_coef Union[float, Tensor]

Convection coefficient. Default is 6.0.

6.0

Returns:

Name Type Description
Operator Operator

The operator representing the Korteweg-De Vries equation.
Source code in torchfsm/pde.py 
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def KortewegDeVries(dispersivity:Union[float,Tensor]=-1,convection_coef:Union[float,Tensor]=6.0) -> Operator:
    r""" Korteweg-De Vries equation:
         $$\frac{\partial \mathbf{u}}{\partial t} = d \nabla \cdot (\nabla^2\mathbf{u}) +c \mathbf{u} \cdot \nabla \mathbf{u}$$

    Args:
        dispersivity (Union[float,Tensor]): Dispersion coefficient. Default is -1.
        convection_coef (Union[float,Tensor]): Convection coefficient. Default is 6.0.    

    Returns:
        Operator: The operator representing the Korteweg-De Vries equation.
    """

    return dispersivity*Dispersion()+convection_coef*Convection()

torchfsm.pde.KuramotoSivashinsky ¤

KuramotoSivashinsky() -> Operator
1D Kuramoto-Sivashinsky equation
\[\frac{\partial \phi}{\partial t}=-\frac{\partial^2 \phi}{\partial x^2} -\frac{\partial^4 \phi}{\partial x^4} - \phi\frac{\partial\phi}{\partial x}\]

Returns:

Name Type Description
Operator Operator

The operator representing the Kuramoto-Sivashinsky equation.
Source code in torchfsm/pde.py 
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def KuramotoSivashinsky() -> Operator:
    r"""
    1D Kuramoto-Sivashinsky equation:
        $$\frac{\partial \phi}{\partial t}=-\frac{\partial^2 \phi}{\partial x^2} -\frac{\partial^4 \phi}{\partial x^4} - \phi\frac{\partial\phi}{\partial x}$$

    Returns:
        Operator: The operator representing the Kuramoto-Sivashinsky equation.
    """
    ks_eqn = -Laplacian() - Biharmonic() - Convection()
    ks_eqn.register_additional_check(
        lambda dim_value, dim_mesh: dim_value == 1 and dim_mesh == 1
    )
    return ks_eqn

torchfsm.pde.KuramotoSivashinskyHighDim ¤

KuramotoSivashinskyHighDim() -> Operator
High dimensional Kuramoto-Sivashinsky equation
\[\frac{\partial \mathbf{\phi}}{\partial t}=-\nabla^2 \phi- \nabla^4 \phi - \frac{1}{2}|\nabla \phi|^2\]

Returns:

Name Type Description
Operator Operator

The operator representing the Kuramoto-Sivashinsky equation.
Source code in torchfsm/pde.py 
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def KuramotoSivashinskyHighDim() -> Operator:
    r"""
    High dimensional Kuramoto-Sivashinsky equation:
        $$\frac{\partial \mathbf{\phi}}{\partial t}=-\nabla^2 \phi- \nabla^4 \phi - \frac{1}{2}|\nabla \phi|^2$$

    Returns:
        Operator: The operator representing the Kuramoto-Sivashinsky equation.
    """
    return -Laplacian() - Biharmonic() - KSConvection()

torchfsm.pde.NavierStokesVorticity ¤

NavierStokesVorticity(
    Re: Union[float, Tensor],
    force: Optional[Operator] = None,
) -> Operator
Navier-Stokes equation in vorticity form
\[\frac{\partial \omega}{\partial t} + (\mathbf{u}\cdot\nabla) \omega = \frac{1}{Re} \nabla^2 \omega + \nabla \times \mathbf{f}\]

Parameters:

Name Type Description Default
Re float

Reynolds number.

 required
force Optional[Operator]

Optional external force term. Default is None. If provided, it will be added to the vorticity equation. Note that the provided force should be \(\nabla \times \mathbf{f}\) rather than \(\mathbf{f}\) itself.

 None

Returns:

Name Type Description
Operator Operator

The operator representing the Navier-Stokes equation in vorticity form.
Source code in torchfsm/pde.py 
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def NavierStokesVorticity(Re:Union[float,Tensor],force:Optional[Operator]=None)->Operator:
    r"""
    Navier-Stokes equation in vorticity form:
        $$\frac{\partial \omega}{\partial t} + (\mathbf{u}\cdot\nabla) \omega = \frac{1}{Re} \nabla^2 \omega + \nabla \times \mathbf{f}$$

    Args:
        Re (float): Reynolds number.
        force (Optional[Operator]): Optional external force term. Default is None.
            If provided, it will be added to the vorticity equation. Note that the provided force should be $\nabla \times \mathbf{f}$ rather than $\mathbf{f}$ itself.

    Returns:
        Operator: The operator representing the Navier-Stokes equation in vorticity form.

    """
    ns_vorticity=-VorticityConvection() + 1/Re*Laplacian()
    if force is not None:
        ns_vorticity+=force
    return ns_vorticity

torchfsm.pde.NavierStokes ¤

NavierStokes(
    Re: Union[float, Tensor],
    force: Optional[Operator] = None,
) -> Operator
Navier-Stokes equation
\[\frac{\partial\mathbf{u}}{\partial t}=-\nabla (\nabla^{-2} \nabla \cdot (\left(\mathbf{u}\cdot\nabla\right)\mathbf{u}-f))-\left(\mathbf{u}\cdot\nabla\right)\mathbf{u}+\nu \nabla^2 \mathbf{u} + \mathbf{f}\]

Parameters:

Name Type Description Default
Re float

Reynolds number.

 required
force Optional[Operator]

Optional external force term. Default is None. If provided, it will be added to the vorticity equation.

None

Returns:

Name Type Description
Operator Operator

The operator representing the Navier-Stokes equation.
Source code in torchfsm/pde.py 
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def NavierStokes(Re:Union[float,Tensor],force:Optional[Operator]=None)->Operator:
    r"""
    Navier-Stokes equation:
        $$\frac{\partial\mathbf{u}}{\partial t}=-\nabla (\nabla^{-2} \nabla \cdot (\left(\mathbf{u}\cdot\nabla\right)\mathbf{u}-f))-\left(\mathbf{u}\cdot\nabla\right)\mathbf{u}+\nu \nabla^2 \mathbf{u} + \mathbf{f}$$

    Args:
        Re (float): Reynolds number.
        force (Optional[Operator]): Optional external force term. Default is None.
            If provided, it will be added to the vorticity equation. 

    Returns:
        Operator: The operator representing the Navier-Stokes equation.

    """

    return NSPressureConvection(force)+1/Re*Laplacian()

torchfsm.pde.GrayScott ¤

GrayScott(
    nu_0: Union[float, Tensor],
    nu_1: Union[float, Tensor],
    feed_rate: Union[float, Tensor],
    kill_rate: Union[float, Tensor],
    de_aliasing_rate: float = 0.5,
) -> Operator
Gray-Scott equation
\[\begin{aligned} \frac{\partial \phi_0}{\partial t} &= \nu_0 \Delta \phi_0 - \phi_0 \phi_1^2 + f (1 - \phi_0) \\ \frac{\partial \phi_1}{\partial t} &= \nu_1 \Delta \phi_1 + \phi_0 \phi_1^2 - (f + k) \phi_1 \end{aligned} \]

Parameters:

Name Type Description Default
nu_0 Union[float, Tensor]

Diffusion coefficient for \(\phi_0\).

 required
nu_1 Union[float, Tensor]

Diffusion coefficient for \(\phi_1\).

 required
feed_rate Union[float, Tensor]

Feed rate \(f\).

 required
kill_rate Union[float, Tensor]

Kill rate \(k\).

 required
de_aliasing_rate float

De-aliasing rate for the operator. Default is 0.5 due to the cubic nonlinearity.

 0.5

Returns:

Name Type Description
Operator Operator

The operator representing the Gray-Scott equation.
Source code in torchfsm/pde.py 
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def GrayScott(nu_0: Union[float,Tensor], 
              nu_1:Union[float,Tensor], 
              feed_rate:Union[float,Tensor], 
              kill_rate:Union[float,Tensor], 
              de_aliasing_rate:float=0.5) -> Operator:
    r"""
    Gray-Scott equation:
        $$\begin{aligned}
        \frac{\partial \phi_0}{\partial t} &= \nu_0 \Delta \phi_0 - \phi_0 \phi_1^2 + f (1 - \phi_0) \\
        \frac{\partial \phi_1}{\partial t} &= \nu_1 \Delta \phi_1 + \phi_0 \phi_1^2 - (f + k) \phi_1
        \end{aligned}
        $$

    Args:
        nu_0 (Union[float,Tensor]): Diffusion coefficient for \(\phi_0\).
        nu_1 (Union[float,Tensor]): Diffusion coefficient for \(\phi_1\).
        feed_rate (Union[float,Tensor]): Feed rate \(f\).
        kill_rate (Union[float,Tensor]): Kill rate \(k\).
        de_aliasing_rate (float): De-aliasing rate for the operator. Default is 0.5 due to the cubic nonlinearity.

    Returns:
        Operator: The operator representing the Gray-Scott equation.
    """

    op = ChannelWisedDiffusion([nu_0,nu_1])+ GrayScottSource(feed_rate, kill_rate)
    op.set_de_aliasing_rate(de_aliasing_rate) # GrayScottSource requires lower de-aliasing rate due to the cubic nonlinearity
    return op

torchfsm.pde.KPPFisher ¤

KPPFisher(
    nu: Union[float, Tensor], r: Union[float, Tensor]
) -> Operator
Fisher-KPP equation
\[\frac{\partial u}{\partial t} = r u (1 - u) + \nu \nabla^2 u\]

Parameters:

Name Type Description Default
nu Union[float, Tensor]

Diffusion coefficient.

 required
r Union[float, Tensor]

Growth rate.

 required

Returns:

Name Type Description
Operator Operator

The operator representing the Fisher-KPP equation.
Source code in torchfsm/pde.py 
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def KPPFisher(nu:Union[float,Tensor],r:Union[float,Tensor]) -> Operator:
    r"""
    Fisher-KPP equation:
        $$\frac{\partial u}{\partial t} = r u (1 - u) + \nu \nabla^2 u$$

    Args:
        nu (Union[float,Tensor]): Diffusion coefficient.
        r (Union[float,Tensor]): Growth rate.

    Returns:
        Operator: The operator representing the Fisher-KPP equation.

    """
    return nu*Laplacian()+r*ImplicitSource(lambda phi:phi*(1-phi))

torchfsm.pde.SwiftHohenberg ¤

SwiftHohenberg(r: Union[float, Tensor]) -> Operator
Swift-Hohenberg equation
\[\frac{\partial \phi}{\partial t} = \nu \nabla^2 \phi + r \phi (1 - \phi)\]

Parameters:

Name Type Description Default
r Union[float, Tensor]

Control parameter.

 required

Returns:

Name Type Description
Operator Operator

The operator representing the Swift-Hohenberg equation.
Source code in torchfsm/pde.py 
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def SwiftHohenberg(r:Union[float,Tensor]) -> Operator:
    r"""
    Swift-Hohenberg equation:
        $$\frac{\partial \phi}{\partial t} = \nu \nabla^2 \phi + r \phi (1 - \phi)$$

    Args:
        r (Union[float,Tensor]): Control parameter.

    Returns:
        Operator: The operator representing the Swift-Hohenberg equation.
    """
    return -HyperDiffusion()-2*Laplacian()+ImplicitSource(lambda phi:r*phi-phi+phi**2-phi**3)