Continuous-Time Intgration Methods – An Overview
Available Solving Methods
- Forward Euler
Euler - Heun's Method
Heun - Forth-Order Runge-Kutta Method
RK4 - Runge-Kutta-Fehlberg-Method
RKF45
The list is ordered by fastest to slowest and at the same time by least to most precise.
Usage
Supported integration methods are exported by SimpleSim.jl as part of the enum SimpleSimIntegrator.
@enum SimpleSimIntegrator RK4 = 1 Euler = 2 Heun = 3 RKF45 = 4To choose an integration method for simulation, use the integrator keyword argument when running a simulation.
out = simulate(my_model, T = T_end, integrator = Heun)Fourth-order Runge-Kutta (RK4) is used by default.
Step Size Computation
The step size $\Delta t$ stays constant over the whole duration of the simulation. It is computed as the minimum of Δt_max, a keyword argument of simulate, and the greatest common divisor of Δt_max and all Δt values given for all models in the simulation.
All continuous-time models that are part of the same simulation are updated at the same frequency determined by the "fastest" model in the simulation. This sampling time is determined by computing the greatest common rational divisor of all sampling times in the simulation. Apart performance concerns it is always better to update continuous-time models more frequently.
Example:
Assume you have set up a model that should be updated at least every twelfth of a second.
my_model = (
# ...
Δt = 1 // 12,
)
out = simulate(my_model, T = T_end, Δt_max = 1 // 20)The fixed step size integrators will now be called with Δt = 1 // 60, which is the greatest common divisor of $\frac{1}{12}$ and $\frac{1}{20}$.
By default, Δt_max is set to 1 // 100 which is more than sufficient for most applications.