Timesteppers API

Time integration schemes for evolving PDEs in time.

TimeStepper Types

Abstract Type

abstract type TimeStepper end

All timesteppers inherit from this abstract type.

IMEX Runge-Kutta

RK111

1st-order IMEX Runge-Kutta.

RK111()

Properties:

• Order: 1
• Stages: 1
• Implicit part: Backward Euler for linear terms
• Explicit part: Forward Euler for nonlinear terms
• Memory: Minimal

RK222

2nd-order IMEX Runge-Kutta with three stored stages: an explicit first stage and two diagonally implicit stages.

RK222()

Properties:

• Order: 2
• Stages: 3 (c = [0, γ, 1])
• Implicit part: ESDIRK with γ = 1 - 1/sqrt(2) on stages 2 and 3
• Explicit part: matched three-row explicit tableau
• Workspace field sets: 6 (stage state, explicit RHS, and implicit RHS storage)

Recommended for: General purpose problems.

RK443

3rd-order, 4-stage IMEX Runge-Kutta.

RK443()

Properties:

• Order: 3
• Stages: 4
• Implicit part: 4-stage DIRK for linear terms
• Explicit part: 4-stage explicit RK for nonlinear terms
• Workspace field sets: 12 (three field sets per stage)

Recommended for: High accuracy requirements.

IMEX Multistep Methods

CNAB (Crank-Nicolson Adams-Bashforth)

CNAB1()  # 1st order
CNAB2()  # 2nd order

Treatment:

• Linear terms: Crank-Nicolson (implicit)
• Nonlinear terms: Adams-Bashforth (explicit)

Properties:

• CNAB1: Order 1, 1 history level
• CNAB2: Order 2, 2 history levels

SBDF (Semi-implicit BDF)

SBDF1()  # 1st order
SBDF2()  # 2nd order
SBDF3()  # 3rd order
SBDF4()  # 4th order

Treatment:

• Linear terms: BDF (implicit)
• Nonlinear terms: Extrapolation (explicit)

Properties: | Method | Order | History Levels | Stability | |––––|–––-|––––––––|–––––-| | SBDF1 | 1 | 1 | A-stable | | SBDF2 | 2 | 2 | A-stable | | SBDF3 | 3 | 3 | A(α)-stable | | SBDF4 | 4 | 4 | A(α)-stable |

Exponential Time Differencing (ETD)

Exponential integrators that treat the linear term exactly via the matrix exponential, which is well suited to very stiff linear operators.

ETD_RK222()  # 2nd-order exponential Runge-Kutta
ETD_CNAB2()  # 2nd-order exponential Crank-Nicolson Adams-Bashforth
ETD_SBDF2()  # 2nd-order exponential semi-implicit BDF

Properties:

• Order: 2
• Linear term: exact exponential propagation (via φ-functions)
• Nonlinear term: explicit (RK / Adams-Bashforth / BDF extrapolation)
• Best for problems where the linear part dominates the stiffness.

Specialized IMEX-RK

RKSMR

RKSMR()

The Spalart–Moser–Rogers scheme is represented as a four-stage additive RK tableau. Its nonlinear/explicit part is third order and its linear/implicit part is second order. It uses the same generic IMEX runtime paths as RK222 and RK443.

Diagonal IMEX

DiagonalIMEX_RK222()  # 2nd order
DiagonalIMEX_RK443()  # 3rd order
DiagonalIMEX_SBDF2()  # 2nd order multistep

Properties: the implicit solve is diagonal in spectral space — the linear operator is applied per Fourier mode ((I + γ·dt·L̂)⁻¹ per wavenumber) rather than through a global matrix solve. Much cheaper for pure-Fourier / diagonalizable linear terms; requires the implicit operator to be diagonal in the chosen basis.

Usage with Solver

# Create solver with timestepper
solver = InitialValueSolver(problem, RK222(); dt=0.001)

# Or IMEX
solver = InitialValueSolver(problem, SBDF2(); dt=0.01)

# Time stepping
while solver.sim_time < t_end
    step!(solver)
end

Timestepper State

TimestepperState

Internal state for multi-step methods.

struct TimestepperState
    timestepper::TimeStepper
    dt::Float64
    history::Vector  # Previous time levels
end

Managed automatically by the solver.

Stability Regions

Advective CFL

Explicitly treated advection/nonlinear terms still limit the timestep:

\[\Delta t < C \frac{\Delta x}{u_{max}}\]

The admissible CFL factor depends on the PDE, spatial discretization, and chosen scheme; Tarang does not define universal constants for individual steppers.

IMEX Methods

Implicit linear treatment removes the corresponding explicit diffusive limit, but it does not remove the advective CFL limit:

\[\Delta t < C \frac{\Delta x}{u_{max}}\]

Only advective CFL, not diffusive.

Method Selection Guide

Performance

Computational Cost

Memory Requirements

Workspace storage is preallocated by timestepper type. The RK implementations reserve three field sets per stage (X_stage, explicit RHS, implicit RHS), while multistep methods use history rings and two general workspace field sets. These are reusable buffers, not fresh state copies allocated on every step.

See Also

• Solvers: Using timesteppers
• User Guide: Timesteppers: Conceptual guide