Adaptive stepsize

In numerical analysis, some methods for the numerical solution of ordinary differential equations (including the special case of numerical integration) use an adaptive stepsize in order to control the errors of the method and to ensure stability properties such as A-stability. Romberg's method is an example of a numerical integration method which uses an adaptive stepsize.

For simplicity, the following example uses the simplest integration method, the Euler method; in practice, higher-order methods such as Runge–Kutta methods are preferred due to their superior convergence and stability properties.

Consider the initial value problem

where y and f may denote vectors (in which case this equation represents a system of coupled ODEs in several variables).

We are given the function f(t,y) and the initial conditions (a, y_a), and we are interested in finding the solution at t=b. Let y(b) denote the exact solution at b, and let y_b denote the solution that we compute. We write ${\displaystyle y_{b}+\epsilon =y(b)}$, where ${\displaystyle \epsilon }$ is the error in the numerical solution.

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