Euler method

A very simple finite-difference method for the numerical solution of an ordinary differential equation. Let a differential equation

$$ \tag{1 } y ^ \prime = f ( x , y ) $$

with initial condition

$$ y ( x _ {0} ) = y _ {0} $$

be given. A sufficiently small step $ h $ on the $ x $- axis is chosen and points $ x _ {i} = x _ {0} + ih $, $ i= 0 , 1 \dots $ are constructed. One then replaces the desired integral curve $ y ( x) $ by a polygonal line (Euler's polygonal line) whose segments are rectilinear on the intervals $ [ x _ {i} , x _ {i+} 1 ] $; and the ordinates of the end points of these segments are determined from the formulas

$$ \tag{2 } y _ {i+} 1 = y _ {i} + h f ( x _ {i} , y _ {i} ) ,\ \ i = 0 , 1 ,\dots . $$

If the right-hand side $ f ( x , y ) $ of (1) is continuous, then the sequence of Euler polygonal lines converges uniformly, as $ h \rightarrow 0 $, to the unknown curve $ y ( x) $ on a sufficiently small interval $ [ x _ {0} , x _ {0} + H ] $.

The Euler method consists in representing the integral of the differential equation (1) on every interval $ [ x _ {i} , x _ {i+} 1 ] $ by the first two terms of its Taylor series:

$$ y ( x _ {i} + h ) = y ( x _ {i} ) + h y ^ \prime ( x _ {i} ) ,\ \ i = 0 , 1 ,\dots . $$

The error of this method is of order $ h ^ {2} $ for every step.

The Euler method can be refined by means of various modifications. For example, an improved method of polygonal lines is obtained by replacing the formula (2) for the computation of the ordinates by the relation

$$ \tag{3 } y _ {i+} 1 = y _ {i} + h f ( x _ {i+} 1/2 , y _ {i+} 1/2 ) ,\ \ i = 0 , 1 \dots $$

where

$$ \tag{4 } x _ {i+} 1/2 = x _ {i} + \frac{h}{2} ,\ \ y _ {i+} 1/2 = y _ {i} + \frac{h}{2} f ( x _ {i} , y _ {i} ) , $$

that is, by taking into account the direction of the field of integral curves at the midpoints (4) of the segments of the polygonal line.

Another modification leads to the improved Euler–Cauchy method:

$$ \tag{5 } y _ {i+} 1 = y _ {i} + h \frac{f ( x _ {i} , y _ {i} ) + f ( x _ {i+} 1 , \overline{y}\; _ {i+} 1 ) }{2} ,\ \ i = 0 , 1 \dots $$

where

$$ \overline{y}\; _ {i+} 1 = y _ {i} + h f ( x _ {i} , y _ {i} ) . $$

The last method can be refined even further by using an iterative scheme to improve the value $ \overline{y}\; _ {i+} 1 $:

$$ \tag{6 } y _ {i+} 1 ^ {(} k) = y _ {i} + \frac{h}{2} [ f ( x _ {i} , y _ {i} ) + f ( x _ {i+} 1 , y _ {i+} 1 ^ {(} k- 1) ) ] ,\ \ k = 1 , 2 \dots $$

where the zero- $ th $ approximation is

$$ y _ {i+} 1 ^ {(} 0) = y _ {i} + h f ( x _ {i} , y _ {i} ) . $$

The iterative computation by means of (6) is continued until two consecutive approximations $ y _ {i+} 1 ^ {(} k) $ and $ y _ {i+} 1 ^ {(} k+ 1) $ coincide in a prescribed number of decimal places. If after three or four iterations this does not happen, the step $ h $ should be made smaller. The error of the Euler method with iterative computation of the ordinates is of order $ h ^ {3} $ for every step.

Euler's method and its modifications carry over to the more general case of solving a system of $ n $ ordinary differential equations

$$ y _ {k} ^ \prime = f _ {k} ( x , y _ {1} \dots y _ {n} ) ,\ \ k = 1 \dots n , $$

with given initial conditions

$$ y _ {k} ( x _ {0} ) = y _ {k _ {0} } . $$

The numerical algorithm of the Euler method can easily be programmed on a computer.

This method was proposed by L. Euler (1768).

References

[1]	B.P. Demidovich, I.A. Maron, "Foundations of computational mathematics" , Moscow (1960) (In Russian)

Comments

In the West, only the method defined by (2) is called the Euler method, more precisely, Euler's forward method. The implicit analogue

$$ y _ {i+} 1 = y _ {i} + h f ( x _ {i+} 1 , y _ {i+} 1 ) $$

is called Euler's backward method. The method defined by (3) is usually called the midpoint method, while (3) and (4) together are known as the Runge method [a4], or modified Euler method, which is considered as the oldest method of Runge–Kutta type (Runge–Kutta methods are characterized by the property that each step involves a multiplicity of evaluations of the right-hand side function $ f $, cf. Runge–Kutta method). Method (5) is sometimes called Heun's second-order method if $ y _ {i+} 1 $ is predicted by Euler's forward method, and it is called the trapezoidal rule otherwise.

Method (6) may be considered as the iterative solution of the trapeziodal rule.

Although the Euler method (2) is not recommended in actual computation, it serves as a model for theoretical considerations and facilitates comparison with more complicated methods (cf. [a1], [a2]).

References

[a1]	J.C. Butcher, "The numerical analysis of ordinary differential equations. Runge–Kutta and general linear methods" , Wiley (1987)
[a2]	L. Euler, "Institutionum calculi integralis Vol. Primum (1768)" , Opera Omnia Series Prima , 11 , Teubner (1913)
[a3]	P. Henrici, "Discrete variable methods in ordinary differential equations" , Wiley (1962)
[a4]	C. Runge, "Ueber die numerische Auflösung von Differentialgleichungen" Math. Ann. , 46 (1895) pp. 167–178
[a5]	J.M. Watt (ed.) , Modern numerical methods for ordinary differential equations , Clarendon Press (1976)