Euler method

Lore & Background

It is a first-order method, meaning the error per step is proportional to the square of the step size, and the global error is proportional to the step size. The method often serves as the basis to construct more complex methods, such as the predictor–corrector method.

Reader's Guide

The Euler method is a foundational technique in numerical analysis for approximating solutions to ordinary differential equations. Its significance lies in its simplicity and its role as the simplest explicit Runge–Kutta method. The method works by taking small steps along the tangent line of the unknown curve, starting from a known initial point. While the Euler method itself has limited accuracy (first-order), it provides the conceptual basis for more advanced numerical integration techniques. The method is explicit, meaning the solution at each step is a direct function of previous values. Any higher-order ODE can be reduced to a system of first-order ODEs and then solved using the Euler method. Its legacy endures as a teaching tool and a building block for more sophisticated algorithms in computational science.

Did You Know?

• The Euler method is the simplest Runge–Kutta method.
• The global error of the Euler method is proportional to the step size.
• The Euler method often serves as the basis to construct more complex methods, e.g., predictor–corrector method.