Backward Euler method: Difference between revisions

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{{Short description|Numerical method for ordinary differential equations}}

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

== Description ==

Consider the [[ordinary differential equation]]

Consider the [[ordinary differential equation]] <math> y' = f(t,y) </math> with initial value <math> y(t_0) = y_0. </math> A numerical method produces a sequence <math> y_0, y_1, y_2, \ldots </math> such that <math> y_k </math> approximates <math> y(t_0+kh) </math>, where <math> h </math> is called the step size.

:<math> \frac{\mathrm{d} y}{\mathrm{d} t} = f(t,y) </math>

Considerwith theinitial [[ordinary differential equation]]value <math> y'(t_0) = f(t,y)y_0. </math> withHere initialthe valuefunction <math>f</math> y(and the initial data <math>t_0)</math> =and <math>y_0.</math> are known; the function <math>y</math> depends on the real variable <math>t</math> and is unknown. A numerical method produces a sequence <math> y_0, y_1, y_2, \ldots </math> such that <math> y_k </math> approximates <math> y(t_0+kh) </math>, where <math> h </math> is called the step size.

The backward Euler method computes the approximations using

:<math> y_{k+1} = y_k + h f(t_{k+1}, y_{k+1}). </math> <ref>{{harvnb|Butcher|2003|p=57}}</ref>

This differs from the (forward) Euler method in that the latterforward method uses <math> f(t_k, y_k) </math> in place of <math>f(t_{k+1}, y_{k+1})</math>.

The backward Euler method is an implicit method: the new approximation <math> y_{k+1} </math> appears on both sides of the equation, and thus the method needs to solve an algebraic equation for the unknown <math> y_{k+1} </math>. SometimesFor non-[[Stiff equation|stiff]] problems, this can be done bywith [[fixed-point iteration]]:

:<math> y_{k+1}^{[0]} = y_k, \quad y_{k+1}^{[i+1]} = y_k + h f(t_{k+1}^{[i]}, y_{k+1}^{[i]}). </math>

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

<math> y_{k+1} </math>. <ref>{{harvnb|Butcher|2003|p=57}}</ref>

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

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== Derivation ==

Integrating the differential equation <math> \frac{\mathrm{d} y'}{\mathrm{d} t} = f(t,y) </math> from <math> t_kt_n </math> to <math> t_{kn+1} = t_kt_n + h </math> yields

: <math> y(t_{kn+1}) - y(t_kt_n) = h \int_{t_kt_n}^{t_{kn+1}} f(\taut, y(\taut)) \,\mathrm{d}\taut. </math>

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

: <math> y(t_{kn+1}) - y(t_kt_n) \approx h f(t_{kn+1}, y(t_{kn+1})) \,\mathrm{d}\tau. </math>

Finally, use that <math> y_ky_n </math> is supposed to approximate <math> y(t_kt_n) </math> and the formula for the backward Euler method follows.<ref>{{harvnb|Butcher|2003|p=57}}</ref>

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

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[[File:Stability region for BDF1.svg|thumb|The pink region outside the disk shows the stability region of the backward Euler method.]]

The [[local truncation error]] (defined as the error made in one step) of the backward Euler Method is <math> O(h^2) </math>, using the [[big O notation]]. The error at a specific time <math> t </math> is <math> O(h^2) </math>. It means that this method has '''order one'''. In general, a method with <math> O(h^{k+1}) </math> LTE (local truncation error) is said to be of ''k''th order.

The backward Euler method has order one. It is [[A-stability|A-stable]] and even [[L-stability|L-stable]], making it suitable for the solution of [[stiff equation]]s.

The [[Stiff_equation#Runge%E2%80%93Kutta_methods|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.<ref>{{harvnb|Butcher|2003|p=70}}</ref> This includes the whole left half of the complex plane, making it suitable for the solution of [[stiff equation]]s.<ref>{{harvnb|Butcher|2003|p=71}}</ref> In fact, the backward Euler method is even [[L-stability|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.<ref>Wai-Kai Chen, Ed., Analog and VLSI Circuits The Circuits and Filters Handbook, 3rd ed. Chicago, USA: CRC Press, 2009.</ref>

== Extensions and modifications ==

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</math>

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 method]]s, and also of the family of [[backward differentiation formula]]s.

==See also==

*[[Crank–Nicolson method]]

== Notes ==