Korteweg-de Vries¤
exponax.stepper.KortewegDeVries
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Bases: BaseStepper
Source code in exponax/stepper/_korteweg_de_vries.py
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__init__
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__init__(
num_spatial_dims: int,
domain_extent: float,
num_points: int,
dt: float,
*,
convection_scale: float = -6.0,
dispersivity: float = 1.0,
advect_over_diffuse: bool = False,
single_channel: bool = False,
diffusivity: float = 0.0,
order: int = 2,
dealiasing_fraction: float = 2 / 3,
num_circle_points: int = 16,
circle_radius: float = 1.0
)
Timestepper for the d-dimensional (d ∈ {1, 2, 3}
) Korteweg-de Vries
equation on periodic boundary conditions.
In 1d, the Korteweg-de Vries equation is given by
uₜ + b₁ 1/2 (u²)ₓ + a₃ uₓₓₓ = ν uₓₓ
with b₁
the convection coefficient, a₃
the dispersion coefficient
and ν
the diffusivity. Oftentimes b₁ = -6
and ν = 0
. The
nonlinearity is similar to the Burgers equation and the number of
channels grow with the number of spatial dimensions. In higher
dimensions, the equation reads (using vector format for the channels)
uₜ + b₁ 1/2 ∇ ⋅ (u ⊗ u) + a₃ 1 ⋅ (∇⊙∇⊙(∇u)) = ν Δu
or
uₜ + b₁ 1/2 ∇ ⋅ (u ⊗ u) + a₃ ∇ ⋅ ∇(Δu) = ν Δu
if advect_over_diffuse
is True
.
In 1d, the expected temporal behavior is that the initial condition breaks into soliton waves that propagate at a speed depending on their height. They interact with other soliton waves by being spatially displaced but having an unchanged shape and propagation speed. If the diffusivity is non-zero, the solution decays to a constant state. Otherwise, the soliton interaction continues indefinitely.
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.convection_scale
: The convection coefficientb₁
. Note that the convection is already scaled by 1/2 to account for the conservative evaluation. The value ofb₁
scales it further. Oftentimesb₁ = -6
to match the analytical soliton solutions. See also https://en.wikipedia.org/wiki/Korteweg%E2%80%93De_Vries_equation#One-soliton_solutiondispersivity
: The dispersion coefficienta₃
. Dispersion refers to wavenumber-dependent advection, i.e., higher wavenumbers are advected faster. Default1.0
,advect_over_diffuse
: IfTrue
, the dispersion is computed as advection over diffusion. This adds spatial mixing. Default isFalse
.diffusivity
: The rate at which the solution decays.single_channel
: Whether to use the single channel mode in higher dimensions. In this case the the convection isb₁ (∇ ⋅ 1)(u²)
. In this case, the state always has a single channel, no matter the spatial dimension. Default: False.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. The default 2/3 corresponds to Orszag's 2/3 rule. To fully eliminate aliasing, use 1/2. Default: 2/3.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.
Notes:
-
Good Values:
- There is an anlytical solution to the (inviscid,
ν = 0
) KdV ofu(t, x) = - 1/2 c^2 sech^2(c/2 (x - ct - a))
with the hyperbolic secantsech
and arbitrarily selected speedc
and shifta
. - For a nice simulation with an initial condition that breaks into
solitons choose
domain_extent=20.0
and an initial condition with the first 5-10 modes. Set dt=0.01, num points in the range of 50-200 are sufficient.
Source code in exponax/stepper/_korteweg_de_vries.py
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__call__
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__call__(
u: Float[Array, "C ... N"]
) -> Float[Array, "C ... N"]
Performs a check
Source code in exponax/_base_stepper.py
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