OrdinaryDiffEqLowOrderRK

Low-order explicit Runge-Kutta methods for non-stiff differential equations. Most of the time, you should use Tsit5, which is the most common and efficient low-order RK method. The alternative methods provided here are for special circumstances where Tsit5 is not suitable.

Key Properties

Low-order explicit RK methods offer:

Computational efficiency at higher tolerances (>1e-6)
Robust error control for difficult non-stiff problems
Specialized interpolation properties for applications requiring dense output
Lower-order derivatives requirements for non-smooth functions
Good performance for specific problem types

When to Use Alternative Low-Order RK Methods

Choose these methods instead of Tsit5 when:

ODE function f is not differentiable to 5th order - use lower-order methods (the more discontinuous, the lower the order needed)
Heavy use of interpolations - OwrenZen methods have superior interpolation convergence
Delay differential equations - OwrenZen methods are most efficient (see SciMLBenchmarks)
Very high tolerances (>1e-3) - BS3 is more efficient than Tsit5
Quadratic polynomial ODEs - SIR54 is optimized for these systems
Educational purposes - simpler methods for understanding algorithms

Solver Selection Guide

Primary recommendation

For most problems, use Tsit5 instead of these methods.

High tolerances (>1e-3)

BS3: Third-order Bogacki-Shampine method, most efficient for very high tolerances

Superior interpolation needs

OwrenZen3: Third-order with excellent interpolation convergence
OwrenZen5: Fifth-order with excellent interpolation, optimal for DDEs
OwrenZen4: Fourth-order interpolation-optimized method

Non-smooth or discontinuous ODEs

BS3: Third-order for mildly non-smooth functions
Heun: Second-order for more discontinuous functions (not generally recommended)
Euler: First-order for highly discontinuous problems

Robust error control alternatives

BS5: Fifth-order with very robust error estimation
DP5: Fifth-order Dormand-Prince method, classical alternative to Tsit5

Specialized applications

RK4: Fourth-order with special residual error control, good for DDEs. Note: Uses adaptive timestepping by default - set adaptive=false in solve() for traditional fixed-step RK4
SIR54: Fifth-order optimized for ODEs defined by quadratic polynomials (e.g., SIR-type epidemiological models)
Stepanov5: Fifth-order method with enhanced stability properties and optimized error constants
Ralston: Second-order with optimized error constants

Periodic and oscillatory problems

Anas5: Fifth-order optimized for periodic problems with minimal phase error
FRK65: Sixth-order zero dissipation method for oscillatory problems

Advanced specialized methods

RKO65: Sixth-order optimized method
MSRK5, MSRK6: Multi-stage methods for specific applications
PSRK4p7q6, PSRK3p5q4, PSRK3p6q5: Pseudo-symplectic methods
Alshina2, Alshina3, Alshina6: Methods with optimized parameters

Installation

To be able to access the solvers in OrdinaryDiffEqLowOrderRK, you must first install them use the Julia package manager:

using Pkg
Pkg.add("OrdinaryDiffEqLowOrderRK")

This will only install the solvers listed at the bottom of this page. If you want to explore other solvers for your problem, you will need to install some of the other libraries listed in the navigation bar on the left.

Example usage

using OrdinaryDiffEqLowOrderRK

function lorenz!(du, u, p, t)
    du[1] = 10.0 * (u[2] - u[1])
    du[2] = u[1] * (28.0 - u[3]) - u[2]
    du[3] = u[1] * u[2] - (8 / 3) * u[3]
end
u0 = [1.0; 0.0; 0.0]
tspan = (0.0, 100.0)
prob = ODEProblem(lorenz!, u0, tspan)
sol = solve(prob, BS3())

Full list of solvers

OrdinaryDiffEqLowOrderRK.Euler — Type

Euler()

Explicit Runge-Kutta Method. The canonical forward Euler method. Fixed timestep only.

Keyword Arguments

References

E. Hairer, S.P. Norsett, G. Wanner, (1993) Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edition. Springer Series in Computational Mathematics, Springer-Verlag.