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, 2-stage IMEX Runge-Kutta.

RK222()

Properties:

Order: 2
Stages: 2
Implicit part: 2-stage DIRK for linear terms
Explicit part: 2-stage explicit RK for nonlinear terms
Memory: 2 state copies

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
Memory: 4 state copies

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 |

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

Explicit Methods

CFL condition limits timestep:

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

RK111: C ≈ 1.0
RK222: C ≈ 1.0
RK443: C ≈ 1.4

IMEX Methods

Diffusive stability removed:

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

Only advective CFL, not diffusive.

Method Selection Guide

Problem Type	Method	Reason
General purpose	RK222	Balance of cost/accuracy
High accuracy	RK443	More stages, higher order
Diffusion-dominated	SBDF2	Implicit diffusion
Very stiff	SBDF3/SBDF4	Strong stability

Performance

Computational Cost

Method	RHS Evaluations	Linear Solves
RK111	1	0
RK222	2	0
RK443	4	0
CNAB2	1	1
SBDF2	1	1

Memory Requirements

Method	State Copies
RK111	1
RK222	2
RK443	4
SBDF2	2 (history)
SBDF4	4 (history)