Alternative Derivations of Euler's  Numerical O.D.E.'s  Euler's Method

Convergence and Error Analysis of Euler's Method

  When iterating Euler's formula, equation (5.11), we obtain a sequence of values  all of which are built upon the previous calculated values. In general, it is unlikely that  and  would completely agree, because Euler's method produces only an approximation to the true solution. Thus, in computing  from the already slightly-off point  , we're only compounding the effects of errors.

    The error that occurs in a single step only as a result of the numerical formula is referred to as the local truncation error or the discretization error.

    For Euler's method, this truncation error can be examined using Taylor's theorem. Using Taylor's theorem, write

  

for some  . Assuming that y(t) satisfies the differential equation, y'(t) = f(t,y(t)) so that in fact we have

for some  . Since we're just addressing the truncation error, which is introduced in a single step, we assert  , or

    The term

  

is the truncation error associated with a single step of Euler's method.

    To analyze the global truncation error of Euler's method, also referred to as the cumulative error, subtract

from equation (5.12), which yields

  

    Equation (5.14) shows that the error in computing  consists of two parts: (1) the truncation error  which is incurred in stepping from  to  , and (2) the contribution from the propagated error

  

    To further the analysis, the term inside of the brackets above can be viewed as simply a function of the single variable found in the second argument. I.e. let

Then, the Mean Value Theorem can be called upon to produce the result

where  belongs to the interval bounded by  and  . Placing this result in Equation (5.14) yields

  

for some  and some  in the interval bounded by  and  .

    This result can be used to rigorously produce a bound on the global error. Although the derivation is omitted, the result is summarized in the following theorem:

  

Note the first term on the right-hand side of this expression deals with the error in the initial value. If, in the ideal situation, the initial value used for Euler's method is equal to the true initial value, we have

  

    Note that a strong generalization can be made now between the global error and the truncation error. As shown by equation (5.13), the truncation error is  . In contrast, the global error of Euler's method as seen from the above corollary is  .

  
 Alternative Derivations of Euler's  Numerical O.D.E.'s  Euler's Method
Wed Oct 28 11:42:13 EST 1998