Backward Euler method

In numerical analysis and scientific computing, the backward Euler method (or implicit Euler method) is one of the most basic numerical methods for the solution of ordinary differential equations. It is similar to the (standard) Euler method, but differs in that it is an implicit method. The backward Euler method has order one and is A-stable.

Consider the ordinary differential equation

with initial value ${\displaystyle y(t_{0})=y_{0}.}$ Here the function ${\displaystyle f}$ and the initial data ${\displaystyle t_{0}}$ and ${\displaystyle y_{0}}$ are known; the function ${\displaystyle y}$ depends on the real variable ${\displaystyle t}$ and is unknown. A numerical method produces a sequence ${\displaystyle y_{0},y_{1},y_{2},\ldots }$ such that ${\displaystyle y_{k}}$ approximates ${\displaystyle y(t_{0}+kh)}$, where ${\displaystyle h}$ is called the step size.

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