“Backward Euler method”的意思、由来-开放百科全书

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 in time.

Description

with initial value

Here the function

and the initial data

and

are known; the function

depends on the real variable

and is unknown. A numerical method produces a sequence

such that

approximates

, where

is called the step size.

This differs from the (forward) Euler method in that the latter uses

in place of

The backward Euler method is an implicit method: the new approximation

appears on both sides of the equation, and thus the method needs to solve an algebraic equation for the unknown

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

If this sequence converges (within a given tolerance), then the method takes its limit as the new approximation

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

Derivation

Now approximate the integral on the right by the right-hand rectangle method (with one rectangle):

Finally, use that

is supposed to approximate

and the formula for the backward Euler method follows.^[3]

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 backward Euler method has order one. This means that the local truncation error (defined as the error made in one step) is

, using the big O notation. The error at a specific time

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.^[4] This includes the whole left half of the complex plane, making it suitable for the solution of stiff equations.^[5] 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.^[6].

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 with one stage, described by the Butcher tableau:

The backward Euler 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.

Description

Derivation

Analysis

Extensions and modifications

See also

Notes

References