spectral.KdVSolver#

class spectral.KdVSolver(N: int, L: float, dealias: bool = False)[source]#

Bases: object

Korteweg-de Vries equation solver using Fourier spectral methods.

Solves the KdV equation u_t + 6u*u_x + u_xxx = 0 on a periodic domain using Fourier collocation for spatial discretization. The nonlinear term can optionally be dealiased using the 3/2-rule.

Notes

The Fourier spectral method provides exponential convergence for smooth periodic solutions. Spatial derivatives are computed in Fourier space via multiplication by ik for first derivatives and (ik)^3 for third derivatives, where k is the wavenumber.

The 3/2-rule dealiasing prevents aliasing errors in the nonlinear convolution product u*u_x by padding the Fourier coefficients to 3/2 times the original resolution.

Methods

`__init__`	Initialize the KdV solver.
`compute_conserved_quantities`	Compute conserved quantities for KdV equation.
`compute_eigenvalues`	Compute eigenvalues of frozen-coefficient linearized KdV operator.
`get_spectrum`	Compute Fourier spectrum for spectral analysis.
`rhs`	Compute right-hand side of semi-discrete KdV equation.
`solve`	Solve KdV equation from t=0 to t=t_final.
`stable_dt`	Compute stable time step via absolute-stability & frozen coefficients.

static compute_conserved_quantities(u: ndarray, dx: float) → tuple[float, float, float][source]#

Compute conserved quantities for KdV equation.

Mass: M = ∫ u dx Momentum: V = ∫ u² dx Energy: E = ∫ (½u_x² - u³) dx

Parameters:

unp.ndarray: Solution field
dxfloat: Grid spacing

Returns:

Mfloat: Mass
Vfloat: Momentum
Efloat: Energy

compute_eigenvalues(u_max: float) → ndarray[source]#

Compute eigenvalues of frozen-coefficient linearized KdV operator.

For the KdV equation u_t = -6u*u_x - u_xxx, the linearization around a frozen coefficient u_max gives: L = -6*u_max*D1 - D3

where D1 and D3 are the first and third derivative operators. This is useful for stability analysis.

Parameters:

u_maxfloat: Maximum amplitude for frozen-coefficient approximation

Returns:

np.ndarray: Complex eigenvalues of the linearized operator

get_spectrum(u: ndarray) → tuple[ndarray, ndarray, ndarray][source]#

Compute Fourier spectrum for spectral analysis.

Parameters:

unp.ndarray: Solution field

Returns:

knp.ndarray: Wave numbers
magnitudenp.ndarray: Magnitude \(|\hat{u}_k|\) of Fourier coefficients
phasenp.ndarray: Phase angle of Fourier coefficients

rhs(u: ndarray, t: float, source_term: Callable | None = None) → ndarray[source]#

Compute right-hand side of semi-discrete KdV equation.

RHS = \(-6u u_x - u_{xxx} + f(x,t)\)

Parameters:

unp.ndarray: Solution at current time
tfloat: Current time
source_termCallable[[np.ndarray, float], np.ndarray] | None, optional: Optional source term function f(x, t) for manufactured solutions

Returns:

np.ndarray: Time derivative du/dt

solve(u0: ndarray, t_final: float, dt: float, save_every: int = 1, integrator: TimeIntegrator = None, measure_performance: bool = False) → tuple[ndarray, ndarray] | tuple[ndarray, ndarray, dict][source]#

Solve KdV equation from t=0 to t=t_final.

Parameters:

u0np.ndarray: Initial condition
t_finalfloat: Final time
dtfloat: Time step
save_everyint, optional: Save solution every N steps, by default 1
integratorTimeIntegrator, optional: Time integration method, by default RK4()
measure_performancebool, optional: Measure and return performance metrics, by default False

Returns:

t_savednp.ndarray: Times at which solution was saved
u_savednp.ndarray: Saved solutions (shape: [n_saves, N])
performancedict, optional: Performance metrics (returned if measure_performance=True): - ‘wall_time_s’: Total wall time in seconds - ‘mean_step_time_ms’: Mean time per step in milliseconds - ‘std_step_time_ms’: Standard deviation of step times - ‘nsteps’: Total number of time steps

static stable_dt(N: int, L: float, u_max: float, *, integrator_name: str = 'rk4', dealiased: bool = False) → float[source]#

Compute stable time step via absolute-stability & frozen coefficients.

Semi-discrete KdV eigenvalues (frozen u): \(\lambda_k = ik(k^2 - 6u_{max})\), so \(|\lambda_k| = |k| \cdot |k^2 - 6u_{max}|\). Choose \(\Delta t\) so that \(\Delta t |\lambda_{max}| \leq s(method)\) on the imaginary axis.

Parameters:

N, Lgrid size and half-domain for [-L, L]
u_maxmax \(|u|\) expected (for 1-soliton with parameter c, use u_max = c/2)
integrator_nameone of {“rk4”, “rk3”}
dealiasedretained for API compatibility; the returned Δt is conservative: across both aliased and de-aliased configurations.

Returns:

float: Suggested stable time step.