Euler Method - Numerical Stability

Numerical Stability

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. This can be illustrated using the linear equation

The exact solution is, which decays to zero as . However, if the Euler method is applied to this equation with step size, then the numerical solution is qualitatively wrong: it oscillates and grows (see the figure). This is what it means to be unstable. If a smaller step size is used, for instance, then the numerical solution does decay to zero.

If the Euler method is applied to the linear equation, then the numerical solution is unstable if the product is outside the region

illustrated on the right. This region is called the (linear) instability region. In the example, equals −2.3, so if then which is outside the stability region, and thus the numerical solution is unstable.

This limitation—along with its slow convergence of error with h—means that the Euler method is not often used, except as a simple example of numerical integration.