Gray-Scott¤
exponax.stepper.reaction.GrayScott
¤
Bases: BaseStepper
Source code in exponax/stepper/reaction/_gray_scott.py
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__init__
¤
__init__(
num_spatial_dims: int,
domain_extent: float,
num_points: int,
dt: float,
*,
diffusivity_1: float = 2e-05,
diffusivity_2: float = 1e-05,
feed_rate: float = 0.04,
kill_rate: float = 0.06,
order: int = 2,
dealiasing_fraction: float = 1 / 2,
num_circle_points: int = 16,
circle_radius: float = 1.0
)
Timestepper for the d-dimensional (d ∈ {1, 2, 3}
) Gray-Scott reaction
diffusion equation on periodic boundary conditions. This
reaction-diffusion models the interaction of two chemical species u & v.
In 1d, the Gray-Scott equation is given by
uₜ = ν₁ uₓₓ + f(1 - u) - u v²
vₜ = ν₂ vₓₓ - (f + k) v + u v²
with ν₁
and ν₂
the diffusivities, f
the feed rate, and k
the
kill rate. No matter the spatial dimension, this dynamics always has two
channels, refering to the two chemical species. In higher dimensions,
the equations read
uₜ = ν₁ Δu + f(1 - u) - u v²
vₜ = ν₂ Δv - (f + k) v + u v²
with Δ
the Laplacian.
The Gray-Scott equation is known to produce a variety of patterns, such as spots, stripes, and spirals. The expected temporal behavior is highly dependent on the values of the feed and kill rates, see also this paper: https://www.ljll.fr/hecht/ftp/ff++/2015-cimpa-IIT/edp-tuto/Pearson.pdf
IMPORTANT: Both channels are expected to have values in the range [0,
1]
.
Arguments:
num_spatial_dims
: The number of spatial dimensionsd
.domain_extent
: The size of the domainL
; in higher dimensions the domain is assumed to be a scaled hypercubeΩ = (0, L)ᵈ
.num_points
: The number of pointsN
used to discretize the domain. This includes the left boundary point and excludes the right boundary point. In higher dimensions; the number of points in each dimension is the same. Hence, the total number of degrees of freedom isNᵈ
.dt
: The timestep sizeΔt
between two consecutive states.diffusivity_1
: The diffusivityν₁
of the first species. Default is2e-5
.diffusivity_2
: The diffusivityν₂
of the second species. Default is1e-5
.feed_rate
: The feed ratef
. Default is0.04
.kill_rate
: The kill ratek
. Default is0.06
.order
: The order of the Exponential Time Differencing Runge Kutta method. Must be one of {0, 1, 2, 3, 4}. The option0
only solves the linear part of the equation. Use higher values for higher accuracy and stability. The default choice of2
is a good compromise for single precision floats.dealiasing_fraction
: The fraction of the wavenumbers to keep before evaluating the nonlinearity. Default: 1/2.num_circle_points
: How many points to use in the complex contour integral method to compute the coefficients of the exponential time differencing Runge Kutta method. Default: 16.circle_radius
: The radius of the contour used to compute the coefficients of the exponential time differencing Runge Kutta method. Default: 1.0.
TODO: Translate the different configurations of https://www.ljll.fr/hecht/ftp/ff++/2015-cimpa-IIT/edp-tuto/Pearson.pdf
Source code in exponax/stepper/reaction/_gray_scott.py
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__call__
¤
__call__(
u: Float[Array, "C ... N"]
) -> Float[Array, "C ... N"]
Perform one step of the time integration for a single state.
Arguments:
u
: The state vector, shape(C, ..., N,)
.
Returns:
u_next
: The state vector after one step, shape(C, ..., N,)
.
Tip
Use this call method together with exponax.rollout
to efficiently
produce temporal trajectories.
Info
For batched operation, use jax.vmap
on this function.
Source code in exponax/_base_stepper.py
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