Euler method

The Euler Polygonzugverfahren or explicit Euler method (also Euler - Cauchy method or Euler - forward method) is the simplest method for the numerical solution of an initial value problem.

It was introduced by Leonhard Euler in 1768 in his book Institutiones calculi Integralis. Cauchy used it to prove some uniqueness results for ordinary differential equations.

The process is sometimes referred to in physics as a method of small steps.

The procedure

For the numerical solution of the initial value problem:

For an ordinary differential equation we choose a discretization step size, consider the discrete time points

And compute the iterated values

The calculated values ​​represent approximations to the actual values ​​of the exact solution of the initial value problem dar. The smaller the step size is chosen, the more computational work is necessary, but the more accurate the approximated values ​​.

A modification of the procedure here is that it selects the step size is variable. A meaningful change in the increment requires an algorithm for step-size control, which estimates the error in the current step, and then the step size for the next step selects accordingly.

If a procedure is defined above, we obtain the implicit Euler method. This is A-stable and therefore better suited for stiff initial value problems suitable.

Properties

The explicit Euler method has consistency and convergence of Procedure 1 The stability function and its stability region, therefore, the circle around -1 with radius 1 in the complex plane.

Generalizations

It can be generalized essentially by two different ideas on more efficient methods.

• The first idea is to include more than one of the previously calculated values ​​for the calculation of the next step. In this way, process yields higher order in the class of linear multistep methods.

• The second idea is to evaluate the function of the interval of a plurality of locations in the calculation of the next step. In this manner one obtains the class of the Runge- Kutta method.

The class of general linear method involves both ideas of generalization with one and contains the class of linear multistep methods as well as the class of Runge- Kutta methods as a special case.

• In addition, there is also an extension of the Euler method for stochastic differential equations: the Euler Maruyama method.