Numerical methods

XPP Lab offers XPPAUT's fifteen integrators. Choose one in the toolbar, in Workspace > Numerics, or with @ meth=... in the model file.

Method	Kind	Use it for
rungekutta	fixed-step RK4	A good default for smooth, non-stiff systems.
5dp	adaptive Dormand-Prince 5(4)	Accurate results without tuning dt.
83dp	adaptive Dormand-Prince 8(5,3)	High accuracy.
stiff	adaptive Rosenbrock	Stiff systems (fast and slow time scales).
cvode	CVODE (BDF or Adams)	Large or stiff systems.
2rb	Rosenbrock 2(3)	Stiff systems, with a banded Jacobian option.
gear	Gear	Older stiff solver.
adams	Adams-Bashforth 4	Smooth problems, fixed step.
euler	forward Euler	Noise (wiener), quick checks.
modeuler	modified Euler (Heun)	Noise, a little more accurate.
backeul	backward Euler	Very stiff problems, noise.
qualrk	quality-controlled RK4	Adaptive RK4.
volterra	for integral equations	Models with int{...}.
ymp	symplectic	Mechanical (Hamiltonian) systems, long runs.
discrete	maps	Difference equations x(t+1) = f(x(t)).

The settings that matter

• Total: how long to run.
• dt: the step of fixed-step methods. For adaptive methods it is the interval between stored points.
• Output every (nout): store one point every n steps. Use it to keep long runs small.
• Transient: drop output before this time.
• Bound: stop when a variable grows larger than this, so a blow-up is caught instead of filling the screen with infinities. aux quantities count too: an aux in degrees passes the default of 100 easily, and the run then ends early with "out of bounds". Raise it with @ bound=1000000 in that case.
• Tolerances (tol, atol): accuracy of adaptive methods.

Is my system stiff?

If the solution needs a tiny dt to stay stable, or an adaptive method crawls, the system is probably stiff: it mixes very fast and very slow processes. Switch to stiff or cvode. Neuron models with fast spikes are a common example.

Delays and noise

Delay equations (delay(x,tau) or x(t-tau)) need @ delay= at least as large as the longest delay. For noise (wiener), use euler, modeuler or backeul: the adaptive methods do not handle random terms.