Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
why does the local error increase in proportion to the square of the step size but the global error in proportion to the step size in euler's method for solving ODE’s? I am curious and have not been able to find a satisfactory answer

why does the local error increase in proportion to the square of the step size but the global error in proportion to the step size in euler's method for solving ODE’s? I am curious and have not been able to find a satisfactory answer

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question
why does the local error increase in proportion to the square of the step size but the global error in proportion to the step size in euler's method for solving ODE’s? I am curious and have not been able to find a satisfactory answer
Expert Solution
Step 1: Explanation of Local errors

To understand why the local error increases in proportion to the square of the step size, we need to look at the Taylor series expansion of the exact solution around each point. Suppose we have an ODE of the form

                y′(x)=f(x,y(x))  with initial condition  y(x0​)=y0​


Then the exact solution at x0​+h,where h is the step size given by 

                         y(x0+h)=y(x0)+hy′(x0)+h22y″(x0)+O(h3)

               where    o(h3) represents the higher order term that are negligible for small h.


On the other hand approximation by Euler method at  x0+h  is given by

                                      y1=y0+hf(x0,y0)

which is same as first two term of Taylor series ,therefore local error at x0+h  is

                                                   y(x0+h)−y1=h22y″(x0)+O(h3)

   which is proportional to h2  for small.h.

                            

steps

Step by step

Solved in 3 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Similar questions
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS