Advection¤
In 1D:
\[ \frac{\partial u}{\partial t} + c \frac{\partial u}{\partial x} = 0 \]
In higher dimensions:
\[ \frac{\partial u}{\partial t} + \vec{c} \cdot \nabla u = 0 \]
(often just \(\vec{c} = c \vec{1}\))
exponax.stepper.Advection
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Bases: BaseStepper
Source code in exponax/stepper/_linear.py
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__init__
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__init__(
num_spatial_dims: int,
domain_extent: float,
num_points: int,
dt: float,
*,
velocity: Union[Float[Array, D], float] = 1.0
)
Timestepper for the d-dimensional (d ∈ {1, 2, 3}
) advection equation
on periodic boundary conditions.
In 1d, the advection equation is given by
uₜ + c uₓ = 0
with c ∈ ℝ
being the velocity/advection speed.
In higher dimensions, the advection equation can written as the inner product between velocity vector and gradient
uₜ + c ⋅ ∇u = 0
with c ∈ ℝᵈ
being the velocity/advection vector.
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.velocity
(keyword-only): The advection speedc
. In higher dimensions, this can be a scalar (=float) or a vector of lengthd
. If a scalar is given, the advection speed is assumed to be the same in all spatial dimensions. Default:1.0
.
Notes:
- The stepper is unconditionally stable, not matter the choice of
any argument because the equation is solved analytically in Fourier
space. However, note that initial conditions with modes higher
than the Nyquist freuency (
(N//2)+1
withN
being thenum_points
) lead to spurious oscillations. - Ultimately, only the factor
c Δt / L
affects the characteristic of the dynamics. See alsoexponax.normalized.NormalizedLinearStepper
withnormalized_coefficients = [0, alpha_1]
with `alpha_1 = - velocity- dt / domain_extent`.
Source code in exponax/stepper/_linear.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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