It is important to use a stable method when solving a stiff equation.

For more details, see the section on stiff equations and multistep methods.

The advantage of implicit Runge-Kutta methods above explicit ones is their greater stability, especially when applied to stiff equations.

See Stiff equation.

The stability of numerical methods for solving stiff equations is indicated by their region of absolute stability.

The latter result is known as the second Dahlquist barrier; it restricts the usefulness of linear multistep methods for stiff equations.

This includes the whole left half of the complex plane, so the backward Euler method is A-stable, making it suitable for the solution of stiff equations.

The advantage of implicit methods such as (6) is that they are usually more stable for solving a stiff equation, meaning that a larger step size h can be used.

Unfortunately, explicit Runge-Kutta methods are generally unsuitable for the solution of stiff equations because their region of absolute stability is small; in particular, it is bounded.

Euler contains libraries for statistics, exact numerical computations with interval inclusions, differential equations and stiff equations, astronomical functions, geometry, and more.

In (Tiwari et al. 2000) we have performed simulations of incompressible flows as the limit of the compressible Navier-Stokes equations with some stiff equation of state.

The Euler method can also be numerically unstable, especially for stiff equations, meaning that the numerical solution grows very large for equations where the exact solution does not.

The well-known Bernoulli's equation can be derived by integrating Euler's equation along a streamline, under the assumption of constant density and a sufficiently stiff equation of state.

In mathematics, a stiff equation is a differential equation for which certain numerical methods for solving the equation are numerically unstable, unless the step size is taken to be extremely small.