Backward Euler method について

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Euler's backward method ：ウィキペディア英語版

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.
== Description ==

Consider the ordinary differential equation
:

\frac = f(t,y)

with initial value

y(t_0) = y_0.

Here the function

f

and the initial data

t_0

and

y_0

are known; the function

y

depends on the real variable

t

and is unknown. A numerical method produces a sequence

y_0, y_1, y_2, \ldots

such that

y_k

approximates

y(t_0+kh)

, where

h

is called the step size.
The backward Euler method computes the approximations using
:

y_ = y_k + h f(t_, y_).

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

f(t_k, y_k)

in place of

f(t_, y_)

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

y_

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

y_

. Sometimes, this can be done by fixed-point iteration:
:

y_^ = y_k, \quad y_^ = y_k + h f(t_, y_^).

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

y_

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

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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