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 error of order one in time.

Description

Consider the ordinary differential equation is called the step size.

The backward Euler method computes the approximations using.

. For non-stiff problems, this can be done with fixed-point iteration.

Alternatively, one can use (some modification of) the Newton–Raphson method to solve the algebraic equation.

For a linear differential equation,

Derivation

Now approximate the integral on the right by the right-hand rectangle method and the formula for the backward Euler method follows.

The same reasoning leads to the (standard) Euler method if the left-hand rectangle rule is used instead of the right-hand one.

Analysis

The local truncation error, using the big O notation. It means that this method has order one LTE (local truncation error) is said to be of kth order.

The region of absolute stability for the backward Euler method is the complement in the complex plane of the disk with radius 1 centered at 1, depicted in the figure. This includes the whole left half of the complex plane, making it suitable for the solution of stiff equations. In fact, the backward Euler method is even L-stable.

The region for a discrete stable system by Backward Euler Method is a circle with radius 0.5 which is located at (0.5, 0) in the z-plane.

Extensions and modifications

The backward Euler method is a variant of the (forward) Euler method. Other variants are the semi-implicit Euler method and the exponential Euler method.

The backward Euler method can be seen as a Runge–Kutta method

The method can also be seen as a linear multistep method with one step. It is the first method of the family of Adams–Moulton methods, and also of the family of backward differentiation formulas.

See also

• Crank–Nicolson method

Notes

References

• .