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Change to proper markdown
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@Ceyron
Ceyron committed Oct 22, 2024
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Showing 1 changed file with 79 additions and 76 deletions.
155 changes: 79 additions & 76 deletions exponax/stepper/generic/_linear.py
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Original file line number Diff line number Diff line change
Expand Up @@ -58,84 +58,87 @@ def __init__(
0`, `a_1 = -0.1`, and `a_2 = 0.01`.
**Arguments:**
- `num_spatial_dims`: The number of spatial dimensions `d`.
- `domain_extent`: The size of the domain `L`; in higher dimensions
the domain is assumed to be a scaled hypercube `Ω = (0, L)ᵈ`.
- `num_points`: The number of points `N` 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 is `Nᵈ`.
- `dt`: The timestep size `Δt` between two consecutive states.
- `linear_coefficients` (keyword-only): The list of coefficients `a_j`
corresponding to the derivatives. Default: `[0.0, -0.1, 0.01]`.
- `num_spatial_dims`: The number of spatial dimensions `d`.
- `domain_extent`: The size of the domain `L`; in higher dimensions
the domain is assumed to be a scaled hypercube `Ω = (0, L)ᵈ`.
- `num_points`: The number of points `N` 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 is `Nᵈ`.
- `dt`: The timestep size `Δt` between two consecutive states.
- `linear_coefficients` (keyword-only): The list of coefficients `a_j`
corresponding to the derivatives. Default: `[0.0, -0.1, 0.01]`.
**Notes:**
- There is a repeating pattern in the effect of orders of
derivatives:
- Even derivatives (i.e., 0, 2, 4, 6, ...) scale the
solution. Order 0 scales all wavenumbers/modes equally (if
its coefficient is negative, this is also called a drag).
Order 2 scales higher wavenumbers/modes stronger with the
dependence on the effect on the wavenumber being
quadratically. Order 4 also scales but stronger than order
4. Its dependency on the wavenumber is quartically. This
pattern continues for higher orders.
- Odd derivatives (i.e, 1, 3, 5, 7, ...) rotate the solution in
Fourier space. In state space, this is observed as
advection. Order 1 rotates all wavenumbers/modes equally. In
state space, this is observed in that the initial condition
just moves over the domain. Order 3 rotates higher
wavenumbers/modes stronger with the dependence on the
wavenumber being quadratic. If certain wavenumbers are
rotated at a different speed, there is still advection in
state space but certain patterns in the initial condition
are advected at different speeds. As such, it is observed
that the shape of the initial condition dissolves. The
effect continues for higher orders with the dependency on
the wavenumber becoming continuously stronger.
- Take care of the signs of coefficients. In contrast to the
indivial linear steppers ([`exponax.stepper.Advection`][],
[`exponax.stepper.Diffusion`][], etc.), the signs are not
automatically taken care of to produce meaningful coefficients.
For the general linear stepper all linear derivatives are on the
right-hand side of the equation. This has the following effect
based on the order of derivative (this a consequence of squaring
the imaginary unit returning -1):
- Zeroth-Order: A negative coeffcient is a drag and removes
energy from the system. A positive coefficient adds energy
to the system.
- First-Order: A negative coefficient rotates the solution
clockwise. A positive coefficient rotates the solution
counter-clockwise. Hence, negative coefficients advect
solutions to the right, positive coefficients advect
solutions to the left.
- Second-Order: A positive coefficient diffuses the solution
(i.e., removes energy from the system). A negative
coefficient adds energy to the system.
- Third-Order: A negative coefficient rotates the solution
counter-clockwise. A positive coefficient rotates the
solution clockwise. Hence, negative coefficients advect
solutions to the left, positive coefficients advect
solutions to the right.
- Fourth-Order: A negative coefficient diffuses the solution
(i.e., removes energy from the system). A positive
coefficient adds energy to the system.
- ...
- The stepper is unconditionally stable, no matter the choice of
any argument because the equation is solved analytically in
Fourier space. **However**, note if you have odd-order
derivative terms (e.g., advection or dispersion) and your
initial condition is **not** bandlimited (i.e., it contains
modes beyond the Nyquist frequency of `(N//2)+1`) there is a
chance spurious oscillations can occur.
- Ultimately, only the factors `a_j Δt / Lʲ` affect the
characteristic of the dynamics. See also
[`exponax.stepper.generic.NormalizedLinearStepper`][] with
`normalized_coefficients = [0, alpha_1, alpha_2, ...]` with
`alpha_j = coefficients[j] * dt / domain_extent**j`. You can use
the function [`exponax.stepper.generic.normalize_coefficients`][] to
obtain the normalized coefficients.
- There is a repeating pattern in the effect of orders of
derivatives:
- Even derivatives (i.e., 0, 2, 4, 6, ...) scale the
solution. Order 0 scales all wavenumbers/modes equally (if
its coefficient is negative, this is also called a drag).
Order 2 scales higher wavenumbers/modes stronger with the
dependence on the effect on the wavenumber being
quadratically. Order 4 also scales but stronger than order
4. Its dependency on the wavenumber is quartically. This
pattern continues for higher orders.
- Odd derivatives (i.e, 1, 3, 5, 7, ...) rotate the solution in
Fourier space. In state space, this is observed as
advection. Order 1 rotates all wavenumbers/modes equally. In
state space, this is observed in that the initial condition
just moves over the domain. Order 3 rotates higher
wavenumbers/modes stronger with the dependence on the
wavenumber being quadratic. If certain wavenumbers are
rotated at a different speed, there is still advection in
state space but certain patterns in the initial condition
are advected at different speeds. As such, it is observed
that the shape of the initial condition dissolves. The
effect continues for higher orders with the dependency on
the wavenumber becoming continuously stronger.
- Take care of the signs of coefficients. In contrast to the
indivial linear steppers ([`exponax.stepper.Advection`][],
[`exponax.stepper.Diffusion`][], etc.), the signs are not
automatically taken care of to produce meaningful coefficients.
For the general linear stepper all linear derivatives are on the
right-hand side of the equation. This has the following effect
based on the order of derivative (this a consequence of squaring
the imaginary unit returning -1):
- Zeroth-Order: A negative coeffcient is a drag and removes
energy from the system. A positive coefficient adds energy
to the system.
- First-Order: A negative coefficient rotates the solution
clockwise. A positive coefficient rotates the solution
counter-clockwise. Hence, negative coefficients advect
solutions to the right, positive coefficients advect
solutions to the left.
- Second-Order: A positive coefficient diffuses the solution
(i.e., removes energy from the system). A negative
coefficient adds energy to the system.
- Third-Order: A negative coefficient rotates the solution
counter-clockwise. A positive coefficient rotates the
solution clockwise. Hence, negative coefficients advect
solutions to the left, positive coefficients advect
solutions to the right.
- Fourth-Order: A negative coefficient diffuses the solution
(i.e., removes energy from the system). A positive
coefficient adds energy to the system.
- ...
- The stepper is unconditionally stable, no matter the choice of
any argument because the equation is solved analytically in
Fourier space. **However**, note if you have odd-order
derivative terms (e.g., advection or dispersion) and your
initial condition is **not** bandlimited (i.e., it contains
modes beyond the Nyquist frequency of `(N//2)+1`) there is a
chance spurious oscillations can occur.
- Ultimately, only the factors `a_j Δt / Lʲ` affect the
characteristic of the dynamics. See also
[`exponax.stepper.generic.NormalizedLinearStepper`][] with
`normalized_coefficients = [0, alpha_1, alpha_2, ...]` with
`alpha_j = coefficients[j] * dt / domain_extent**j`. You can use
the function
[`exponax.stepper.generic.normalize_coefficients`][] to obtain
the normalized coefficients.
"""
self.linear_coefficients = linear_coefficients
super().__init__(
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