evoxels.timesteppers

Classes

`ExponentialEuler`(problem, dt)	First-order exponential Euler (ETD1) method for semilinear problems.
`ForwardEuler`(problem, dt)	First order Euler forward scheme.
`PseudoSpectralIMEX`(problem, dt)	First‐order IMEX Fourier pseudo‐spectral scheme
`RKC1`(problem, dt[, polygrad, damping])	Runge-Kutta-Chebyshev Scheme of order 1.
`RKC2`(problem, dt[, polygrad, damping])	Runge-Kutta-Chebyshev Scheme of order 2.
`RungeKutta4`(problem, dt)	Classical explicit Runge-Kutta Scheme of order 4.
`TimeStepper`()	Abstract interface for single‐step timestepping schemes.

class evoxels.timesteppers.ExponentialEuler(problem: SemiLinearODE, dt: float)

First-order exponential Euler (ETD1) method for semilinear problems.

Implementation of the exponential time differencing method of order 1 (ETD1) described in Hochbruck, Lubich, Selhofer (1998), doi:10.1137/S1064827595295337 Update:

u_{n+1} = u_n + dt * varphi_1(dt L) * rhs(u_n)

where: varphi_1(z) = (exp(z) - 1) / z.

__init__(problem: SemiLinearODE, dt: float) → None

dt: float

property order: int: Temporal order of accuracy.

phi1(z)

Compute varphi_1(z) = (exp(z)-1)/(z)

with special handling for small v to avoid loss of significance. Coefficients for the degree-6 Padé approximation are taken from Hochbruck, Lubich, Selhofer (1998), doi:10.1137/S1064827595295337

phiPade(z, Q, Ncoeff, Dcoeff)

Evaluate (Q,Q)-Padé approximation of phi-function

This routine evaluates the exponential-integrator varphi_1(z) = (exp(z)-1)/z as the rational approximation

varphi_1(z) ≈ P_Q(z) / R_Q(z),

where P_Q and R_Q are degree-Q polynomials with coefficients given by Ncoeff and Dcoeff, respectively. It is used for arguments z near zero, where the direct formula suffers from loss of significance due to cancellation.

problem: SemiLinearODE

step(t: float, u: Any) → Any

Take one timestep from t to (t+dt).

Parameters:

t – Current time
u – Current state

Returns:

Updated state at t + dt.

step_dt(t: float, dt: float, u: Any) → Any: Take one timestep from t to t + dt.

class evoxels.timesteppers.ForwardEuler(problem: ODE, dt: float)

First order Euler forward scheme.

__init__(problem: ODE, dt: float) → None

dt: float

property order: int: Temporal order of accuracy.

problem: ODE

step_dt(t: float, dt: float, u: Any) → Any: Take one timestep from t to t + dt.

class evoxels.timesteppers.PseudoSpectralIMEX(problem: SemiLinearODE, dt: float)

First‐order IMEX Fourier pseudo‐spectral scheme

aka semi-implicit Fourier spectral method; see [Zhu and Chen 1999, doi:10.1103/PhysRevE.60.3564] for more details.

__init__(problem: SemiLinearODE, dt: float) → None

dt: float

property order: int: Temporal order of accuracy.

problem: SemiLinearODE

step(t: float, u: Any) → Any

Take one timestep from t to (t+dt).

Parameters:

t – Current time
u – Current state

Returns:

Updated state at t + dt.

step_dt(t: float, dt: float, u: Any) → Any: Take one timestep from t to t + dt.

class evoxels.timesteppers.RKC1(problem: ODE, dt: float, polygrad: int = 4, damping: float = 0.05)

Runge-Kutta-Chebyshev Scheme of order 1.

Based on the publication “Convergence properties of the Runge-Kutta-Chebyshev method” by Verwer, Hundsdorfer, Sommeijer (1990), doi: 10.1007/BF01386405

__init__(problem: ODE, dt: float, polygrad: int = 4, damping: float = 0.05) → None

damping: float = 0.05

dt: float

property order: int: Temporal order of accuracy.

polygrad: int = 4

problem: ODE

step_dt(t: float, dt: float, u: Any) → Any: Take one timestep from t to t + dt.

class evoxels.timesteppers.RKC2(problem: ODE, dt: float, polygrad: int = 4, damping: float = 0.15384615384615385)

Runge-Kutta-Chebyshev Scheme of order 2.

Based on the publication “Convergence properties of the Runge-Kutta-Chebyshev method” by Verwer, Hundsdorfer, Sommeijer (1990), doi: 10.1007/BF01386405

__init__(problem: ODE, dt: float, polygrad: int = 4, damping: float = 0.15384615384615385) → None

damping: float = 0.15384615384615385

dt: float

property order: int: Temporal order of accuracy.

polygrad: int = 4

problem: ODE

step_dt(t: float, dt: float, u: Any) → Any: Take one timestep from t to t + dt.

class evoxels.timesteppers.RungeKutta4(problem: ODE, dt: float)

Classical explicit Runge-Kutta Scheme of order 4.

__init__(problem: ODE, dt: float) → None

dt: float

property order: int: Temporal order of accuracy.

problem: ODE

step_dt(t: float, dt: float, u: Any) → Any: Take one timestep from t to t + dt.

class evoxels.timesteppers.TimeStepper

Abstract interface for single‐step timestepping schemes.

abstract property order: int: Temporal order of accuracy.

step(t: float, u: Any) → Any

Take one timestep from t to (t+dt).

Parameters:

t – Current time
u – Current state

Returns:

Updated state at t + dt.

abstractmethod step_dt(t: float, dt: float, u: Any) → Any: Take one timestep from t to t + dt.