Error in Euler's Method

Numerical Analysis · Axiom Academy

LESSON Error Analysis in Euler's Method Understanding, bounding, and estimating numerical errors The local truncation error (LTE) measures the error introduced in a single step, assuming we start with the exact solution. It comes from truncating the Taylor series: Euler's method uses only the first two terms, so the LTE is the remainder: The global error is the total error at the end of the integration. Over N steps from t=a to t=b: Since N = (b-a)/h, we have N local errors each of size O(h^2): We can derive rigorous bounds on the global error using the Lipschitz condition : L = Lipschitz constant (measures how fast f changes with y) M = bound on |y''(t)| (determines LTE magnitude) Larger L means errors amplify faster Stability determines whether errors grow or decay. For the test equation y' = lambda*y: Euler gives y_ n+1 = (1 + h*lambda)*y_n. Errors decay when: Richardson extrapolation and step doubling provide practical error estimates without knowing the exact solution: Compare solutions with step sizes h and h/2: Compute y_ h/2 with step size h/2 Estimate error as |y_ h/2 - y_h| Improve: y_improved = 2*y_ h/2 - y_h (for first-order methods)

This is the written version of the interactive lesson above. See the full Numerical Analysis course.