Why is forward euler unstable

The Forward Euler Method is the conceptually simplest method for solving the initial-value problem. For simplicity, let us discretize time, with equal spacings:. The Forward Euler Method consists of the approximation. The first two terms are precisely equal to the right-hand side of the Forward Euler Method formula.

For the Forward Euler Method, the local truncation error leads to a profound problem known as instability. Because the method involves repeatedly applying a formula with a local truncation error at each step, it is possible for the errors on successive steps to progressively accumulate, until the solution itself blows up. To see this, consider the differential equation. Using Eq. These results can be better perceived from Figures 1 and 2.

The stability criterion for the forward Euler method requires the step size h to be less than 0. As seen from there, the method is numerically stable for these values of h and becomes more accurate as h decreases. The numerical instability which occurs for is shown in Figure 2.

The convergence of the solution can be analyzed quantitatively. We know that the local truncation error LTE at any given step for the Euler method scales with h 2. Hence, the global error g n is expected to scale with nh 2. So the global error g n at the n th Euler step is proportional to h. The conditional stability , i.