Overview

Exponax comes with many pre-built dynamics. They are all based on the idea of solving semi-linear PDEs

\[ \partial_t u = \mathcal{L} u + \mathcal{N}(u) \]

where \(\mathcal{L}\) is a linear operator and \(\mathcal{N}\) is a non-linear differential operator.

Classification

We classify each equation/dynamic along the following properties:

• (L)inear: The PDE has no nonlinear term (\(\mathcal{N} = 0\)).
• (N)onlinear: The PDE includes a nonlinear term (\(\mathcal{N} \neq 0\)).
• (D)ecaying: The dynamics decay over time (energy is dissipated).
• (I)nfinitely running: The dynamics run indefinitely without decaying.
• (S)teady state: The dynamics reach a non-trivial steady state (not a domain-wide constant).
• (M)ulti-channel: The number of channels is greater than one.
• (C)haotic: The system is sensitive to the initial condition.

All (L)inear problems stay bandlimited if they start from a bandlimited initial condition. This might be different for (N)onlinear problems, which can produce higher modes and can become unresolved. This can lead to instability in the simulation, which might require stronger diffusion or substepping.

How the Equations Relate

The following diagram shows how the concrete PDEs in Exponax are composed from building blocks. Diffusion is the central ingredient that appears in almost every equation. Each arrow indicates what term is added (or changed) to obtain a more complex equation.

graph TD
    Adv["Advection<br/><small>c ∂ₓu</small>"]
    Diff["Diffusion<br/><small>ν ∂ₓₓu</small>"]
    Disp["Dispersion<br/><small>ξ ∂ₓₓₓu</small>"]
    Hyp["Hyper-Diffusion<br/><small>−μ ∂ₓₓₓₓu</small>"]

    Adv -->|"+ Diffusion"| AD["Advection-<br/>Diffusion"]

    Diff -->|"+ Convection"| B["Burgers"]
    B -->|"+ Dispersion<br/>+ Hyper-Diff."| KdV["Korteweg-<br/>de Vries"]

    KSC["KS<br/>Conservative"]
    KS["KS<br/>Combustion"]
    B -..->|"flip Diff. &<br/>Hyper-Diff. signs"| KSC
    KSC -..->|"Convection →<br/>Gradient Norm"| KS

    Diff -->|"+ Vort. Conv.<br/>(2D only)"| NS["Navier-Stokes"]
    NS -->|"+ Drag<br/>+ Forcing"| KF["Kolmogorov<br/>Flow"]

    Diff -->|"+ u(1−u)"| Fisher["Fisher-KPP"]
    Diff -->|"+ c₁u + c₃u³"| AC["Allen-Cahn"]
    Diff -->|"+ Δ(c₃u³ + c₁u)"| CH["Cahn-Hilliard"]
    Diff -->|"+ (k+Δ)²u + g(u)"| SH["Swift-<br/>Hohenberg"]
    Diff -->|"+ 2-species rxn"| GS["Gray-Scott"]

    classDef linear fill:#D4F2B6,stroke:#6a9c3a,color:#2e5a00
    classDef nonlinear fill:#F2D4D4,stroke:#c47070,color:#6b1a1a
    classDef reaction fill:#F1F2E1,stroke:#a0a170,color:#4a4b20
    classDef vorticity fill:#f3e5f5,stroke:#7b1fa2,color:#4a148c

    class Adv,Diff,Disp,Hyp,AD linear
    class B,KdV,KSC,KS nonlinear
    class NS,KF vorticity
    class Fisher,AC,CH,SH,GS reaction

Concrete Steppers

The following table provides an overview of all concrete PDE steppers available in Exponax. The Class column indicates the properties of the dynamics using the classification above. The Dims column shows in which spatial dimensions the stepper is available.

In higher spatial dimensions (\(d \geq 2\)), several linear steppers support anisotropic variants through their parameterization. These are listed as separate rows below. In 1D, the isotropic and anisotropic forms are identical.

Name	Stepper	Class	Dims	Channels	1D Equation	\(d\)D Equation
Advection	Advection	L-I	1,2,3	1	\(\partial_t u = - c \, \partial_x u\)	\(\partial_t u = - c \vec{1} \cdot \nabla u\)
Unbalanced Advection	Advection	L-I	2,3	1		\(\partial_t u = - \vec{c} \cdot \nabla u\)
Diffusion	Diffusion	L-D	1,2,3	1	\(\partial_t u = \nu \, \partial_{xx} u\)	\(\partial_t u = \nu \nabla \cdot \nabla u\)
Diagonal Diffusion	Diffusion	L-D	2,3	1		\(\partial_t u = \nabla \cdot (\vec{\nu} \odot \nabla u)\)
Anisotropic Diffusion	Diffusion	L-D	2,3	1		\(\partial_t u = \nabla \cdot (A \nabla u)\)
Advection-Diffusion	AdvectionDiffusion	L-D	1,2,3	1	\(\partial_t u = - c \, \partial_x u + \nu \, \partial_{xx} u\)	\(\partial_t u = - \vec{c} \cdot \nabla u + \nabla \cdot (A \nabla u)\)
Dispersion	Dispersion	L-I	1,2,3	1	\(\partial_t u = \xi \, \partial_{xxx} u\)	\(\partial_t u = \boldsymbol{\xi} \cdot (\nabla \odot \nabla \odot \nabla) u\)
Spatially-Mixed Dispersion	Dispersion	L-I	2,3	1		\(\partial_t u = \boldsymbol{\xi} \cdot \nabla(\nabla \cdot \nabla u)\)
Hyper-Diffusion	HyperDiffusion	L-D	1,2,3	1	\(\partial_t u = - \mu \, \partial_{xxxx} u\)	\(\partial_t u = - \mu \, ((\nabla \odot \nabla) \cdot (\nabla \odot \nabla)) u\)
Spatially-Mixed Hyper-Diffusion	HyperDiffusion	L-D	2,3	1		\(\partial_t u = - \mu \, (\nabla \cdot \nabla)(\nabla \cdot \nabla u)\)
Burgers	Burgers	N-D-M	1,2,3	\(d\)	\(\partial_t u = - b \frac{1}{2} \partial_x u^2 + \nu \, \partial_{xx} u\)	\(\partial_t u = - b \frac{1}{2} \nabla \cdot (\mathbf{u} \otimes \mathbf{u}) + \nu \nabla \cdot \nabla \mathbf{u}\)
Burgers (single-channel)	Burgers	N-D	2,3	1		\(\partial_t u = - b \frac{1}{2} (\vec{1} \cdot \nabla) u^2 + \nu \nabla \cdot \nabla u\)
Korteweg-de Vries	KortewegDeVries	N-D-M	1,2,3	\(d\)	\(\partial_t u = - b \frac{1}{2} \partial_x u^2 + a_3 \, \partial_{xxx} u + \nu \, \partial_{xx} u - \mu \, \partial_{xxxx} u\)	\(\partial_t u = - b \frac{1}{2} \nabla \cdot (\mathbf{u} \otimes \mathbf{u}) + a_3 \nabla(\Delta \mathbf{u}) + \nu \nabla \cdot \nabla \mathbf{u} - \mu \, \Delta(\Delta \mathbf{u})\)
Kuramoto-Sivashinsky	KuramotoSivashinsky	N-I-C	1,2,3	1	\(\partial_t u = - b \frac{1}{2} (\partial_x u)^2 - \psi_1 \partial_{xx} u - \psi_2 \partial_{xxxx} u\)	\(\partial_t u = - b \frac{1}{2} \lVert \nabla u \rVert_2^2 - \psi_1 \nabla \cdot \nabla u - \psi_2 \Delta(\Delta u)\)
KS (conservative)	KuramotoSivashinskyConservative	N-I-C-M	1,2,3	\(d\)	\(\partial_t u = - b \frac{1}{2} \partial_x u^2 - \psi_1 \partial_{xx} u - \psi_2 \partial_{xxxx} u\)	\(\partial_t u = - b \frac{1}{2} \nabla \cdot (\mathbf{u} \otimes \mathbf{u}) - \psi_1 \nabla \cdot \nabla \mathbf{u} - \psi_2 \Delta(\Delta \mathbf{u})\)
Navier-Stokes	NavierStokesVorticity	N-D	2	1		\(\partial_t u = - b \left(\begin{bmatrix}1 \\ -1\end{bmatrix} \odot \nabla(\Delta^{-1} u)\right) \cdot \nabla u + \nu \nabla \cdot \nabla u\)
Kolmogorov Flow	KolmogorovFlowVorticity	N-I-C	2	1		\(\partial_t u = - b \left(\begin{bmatrix}1 \\ -1\end{bmatrix} \odot \nabla(\Delta^{-1} u)\right) \cdot \nabla u + \lambda u + \nu \nabla \cdot \nabla u + f\)
Fisher-KPP	FisherKPP	N-S	1,2,3	1	\(\partial_t u = \nu \, \partial_{xx} u + r \, u(1 - u)\)	\(\partial_t u = \nu \nabla \cdot \nabla u + r \, u(1 - u)\)
Allen-Cahn	AllenCahn	N-S	1,2,3	1	\(\partial_t u = \nu \, \partial_{xx} u + c_1 u + c_3 u^3\)	\(\partial_t u = \nu \nabla \cdot \nabla u + c_1 u + c_3 u^3\)
Cahn-Hilliard	CahnHilliard	N-S	1,2,3	1	\(\partial_t u = \nu \, \partial_{xx}(c_3 u^3 + c_1 u - \gamma \, \partial_{xx} u)\)	\(\partial_t u = \nu \Delta(c_3 u^3 + c_1 u - \gamma \Delta u)\)
Swift-Hohenberg	SwiftHohenberg	N-S	1,2,3	1	\(\partial_t u = r \, u - (k + \partial_{xx})^2 u + g(u)\)	\(\partial_t u = r \, u - (k + \Delta)^2 u + g(u)\)
Gray-Scott	GrayScott	N-S-M	1,2,3	2		\(\partial_t u_0 = \nu_0 \Delta u_0 - u_0 u_1^2 + f(1 - u_0)\); \(\partial_t u_1 = \nu_1 \Delta u_1 + u_0 u_1^2 - (f + k) u_1\)

Anisotropic and variant notes

• Unbalanced Advection: Use Advection with a vector velocity (different speed per spatial axis). In the isotropic case (scalar velocity), all directions use the same speed \(c \vec{1} \cdot \nabla u\).
• Diagonal/Anisotropic Diffusion: Use Diffusion (or AdvectionDiffusion) with a vector diffusivity for diagonal diffusion (\(\vec{\nu}\)) or a full \(d \times d\) SPD matrix for fully anisotropic diffusion (\(A\)).
• Spatially-Mixed Dispersion: Use Dispersion(advect_on_diffusion=True). The default (False) computes the third-order derivative element-wise without spatial mixing. The mixed variant applies gradient to the Laplacian, introducing cross-dimensional coupling.
• Spatially-Mixed Hyper-Diffusion: Use HyperDiffusion(diffuse_on_diffuse=True). The default (False) computes the fourth-order derivative element-wise. The mixed variant applies the Laplacian twice, introducing spatial mixing.
• Burgers (single-channel): Use Burgers(single_channel=True). The default multi-channel Burgers has \(d\) channels growing with the spatial dimension. The single-channel variant keeps 1 channel regardless of dimension.
• The KortewegDeVries and KuramotoSivashinskyConservative steppers also support single_channel=True.

Generic Steppers

Beyond the concrete steppers, Exponax provides generic steppers that allow flexibly defining a wide range of dynamics by providing arbitrary coefficients. Each generic stepper family generalizes a group of concrete steppers by replacing specific coefficients with arbitrary ones.

graph LR
    subgraph GL["<b>GeneralLinearStepper</b>"]
        direction LR
        A1["Advection"]
        A2["Diffusion"]
        A3["Adv.-Diff."]
        A4["Dispersion"]
        A5["Hyper-Diff."]
    end

    subgraph GC["<b>GeneralConvectionStepper</b>"]
        direction LR
        B1["Burgers"]
        B2["KdV"]
        B3["KS Cons."]
    end

    subgraph GGN["<b>GeneralGradientNormStepper</b>"]
        C1["KS Combustion"]
    end

    subgraph GP["<b>GeneralPolynomialStepper</b>"]
        direction LR
        D1["Fisher-KPP"]
        D2["Allen-Cahn"]
        D3["Swift-Hohenberg"]
    end

    subgraph GVC["<b>GeneralVorticityConvectionStepper</b><br/>(2D only)"]
        direction LR
        E1["Navier-Stokes"]
        E2["Kolmogorov Flow"]
    end

    GN["<b>GeneralNonlinearStepper</b><br/><small>encompasses single-channel, isotropic<br/>versions of Linear, Convection,<br/>and Gradient Norm families</small>"]

    GN -.-> GL
    GN -.-> GC
    GN -.-> GGN

There are three interfaces for each generic stepper family:

1. Physical (General...Stepper): Uses physical coefficients directly.
2. Normalized (Normalized...Stepper): Coefficients are normalized based on domain extent and resolution for better comparability.
3. Difficulty (Difficulty...Stepper): Uses a "difficulty"-based parameterization relevant for learning tasks.

Family	Nonlinearity	Channels	Dims	Concrete Special Cases
Linear	\(\mathcal{N} = 0\)	1	1,2,3	Advection, Diffusion, Advection-Diffusion, Dispersion, Hyper-Diffusion
Convection	\(\frac{b}{2} \nabla \cdot (\mathbf{u} \otimes \mathbf{u})\)	\(d\) (or 1)	1,2,3	Burgers, KdV, KS (conservative)
Gradient Norm	\(\frac{b}{2} \lVert \nabla u \rVert_2^2\)	1	1,2,3	KS (combustion)
Polynomial	\(\sum_i c_i u^i\)	1	1,2,3	Fisher-KPP, Allen-Cahn, Swift-Hohenberg
Nonlinear	\(b_0 u^2 + \frac{b_1}{2}(\vec{1} \cdot \nabla) u^2 + \frac{b_2}{2} \lVert \nabla u \rVert_2^2\)	1	1,2,3	Combines quadratic, single-channel convection, and gradient norm
Vorticity Convection	\(b \left(\begin{bmatrix}1 \\ -1\end{bmatrix} \odot \nabla(\Delta^{-1} u)\right) \cdot \nabla u\)	1	2	Navier-Stokes, Kolmogorov Flow

All generic steppers (except vorticity convection) combine their respective nonlinearity with an arbitrary number of isotropic linear terms:

\[ \mathcal{L} u = \sum_j a_j (\vec{1} \cdot \nabla)^j u \]

Note that the generic steppers only support the isotropic linear operator. For anisotropic linear dynamics, use the concrete steppers with their respective anisotropic parameterizations (see table above).