Choosing an Algorithm

CuBIE ships several integration algorithm families. This page helps you pick the right one.

Decision Guide

Problem type	Recommended family	Notes
Non-stiff	ERK	Fast per step; Dormand–Prince 5(4) is a good default.
Mildly stiff	DIRK or Rosenbrock-W	DIRK is robust; Rosenbrock-W avoids Newton iteration.
Very stiff	FIRK	Radau IIA 5 handles extreme stiffness well.
Fixed step required	`euler` or non-adaptive tableau	Forward Euler for explicit, Backward Euler for implicit.

Explicit Runge–Kutta (ERK)

Name	Order	Adaptive	Notes
`heun_21`	2	Yes	Heun’s method.
`bogacki_shampine_32`	3(2)	Yes	Low-order, cheap.
`ralston_33`	3	No	Ralston’s method.
`dormand_prince_54` / `rk45`	5(4)	Yes	Industry standard; good default.
`tsitouras_54` / `tsit5`	5(4)	Yes	Often slightly more efficient than Dormand–Prince.
`dverk_65`	6(5)	Yes	Verner’s method.
`fehlberg_78` / `rk78`	7(8)	Yes	High order; useful for smooth, high-accuracy problems.

Diagonally Implicit Runge–Kutta (DIRK)

Name	Order	Adaptive	Notes
`implicit_midpoint`	2	No	Symmetric, energy-preserving.
`trapezoidal_dirk`	2	No	Trapezoidal rule.
`sdirk_2_2`	2	Yes	L-stable SDIRK.
`l_stable_dirk3`	3	Yes	L-stable, 3 stages.
`l_stable_sdirk4`	4	Yes	L-stable, 5 stages.
`lobatto_iiic_3`	4	Yes	Default DIRK tableau.

Fully Implicit Runge–Kutta (FIRK)

Name	Order	Adaptive	Notes
`firk_gauss_legendre_2`	4	Yes	2-stage Gauss–Legendre; default FIRK.
`radau_iia_5` / `radau`	5	Yes	3-stage Radau IIA; excellent for stiff problems.

Rosenbrock-W

Name	Order	Adaptive	Notes
`ros3p`	3	Yes	Default Rosenbrock; no Newton iteration.
`rodas3p`	3	Yes	Stiffly accurate variant.
`rosenbrock_23_sciml`	3	Yes	SciML-compatible tableau.

Simple methods

Name	Order	Adaptive	Notes
`euler`	1	No	Forward Euler; explicit.
`backwards_euler`	1	No	Backward Euler; implicit, L-stable.
`backwards_euler_pc`	1	No	Predictor-corrector backward Euler.
`crank_nicolson`	2	No	Implicit trapezoidal rule.

Adaptive algorithms use an error controller to adjust the step size. CuBIE supports several controllers:

PI controller: The most common choice. Balances responsiveness with stability.
PID controller: Adds a derivative term for smoother step-size histories; can reduce oscillations in the step size on problems with sharp transients.
Gustafsson controller: Predictive controller that accounts for the previous step’s error ratio. Useful when step rejections are frequent.

For most problems the default controller works well. Adjust via step_control_settings on the Solver.

For the mathematical background behind these algorithms, see Numerical Integration.