Numerical Analysis/Truncation Errors

This page is about Truncation error of ODE methods.

DefinitionEdit

Truncation errors are defined as the errors that result from using an approximation in place of an exact mathematical procedure.

There are two ways to measure the errors:

  1. Local Truncation Error (LTE): the error,  , introduced by the approximation method at each step.
  2. Global Truncation Error (GTE): the error,  , is the absolute difference between the correct value and the approximate value.

Assume that our methods take the form:

Let yn+1 and yn be approximation values.

Then
 , where

  is the time step, equal to  , and
  is an increment function and is some algorithm for approximating the average slope  .

Three important examples of   are:

  • Euler’s method:  .
  • Modified Euler's method:  , where
 
  • Runge-Kutta method:  , where
 

Why do we care about truncation errors?Edit

In the case of one-step methods, the local truncation error provides us a measure to determine how the solution to the differential equation fails to solve the difference equation. The local truncation error for multistep methods is similar to that of one-step methods.

A one-step method with local truncation error   at the nth step:

  • This method is consistent with the differential equation it approximates if
 

Note that here we assume that the approximation values are exactly equal to the true solution at every step.

  • The method is convergent with respect to the differential equation it approximates if
 

where   denotes the approximation obtained from the method at the nth step, and   the exact value of the solution of the differential equation.

How do we avoid truncation errors?Edit

The truncation error generally increases as the step size increases, while the roundoff error decreases as the step size increases.

Relationship Between Local Truncation Error and Global Truncation ErrorEdit

The global truncation error (GTE) is one order lower than the local truncation error (LTE).
That is,

if  , then  .

ProofEdit

We assume that perfect knowledge of the true solution at the initial time step.
Let   be the exact solution of

 

The truncation error at step n+1 is defined as   Also, the global errors are defined as

 

According to the w:Triangle inequality, we obtain that

 

 

 

 

 

(1)

The second term on the right-hand side of (1 ) is the truncation error  . Here we assume

 

 

 

 

 

(2)

Thus,

 

 

 

 

 

(3)

The first term on the right-hand side of (1 ) is the difference between two exact solutions.

Both   and   satisfy   so

 

By subtracting one equation from the other, we can get that

 

Since   is w:Lipschitz continuous, then

  where  

By w:Gronwall's inequality,

 

where  

Setting  , we have that

 

 

 

 

 

(4)

Plugging equation (3 ) and (4 ) into (1 ), we can get that

 

 

 

 

 

(5)

Note that equation (5 ) is a recursive inequality valid for all values of  .

Next, we are trying to use it to estimate   where we assume  .

Let   Dividing both sides of (4 ) by   we get that

 

Summing over n = 0,1, 2,…, N-1,

 ,
 ,
 

and

 

Then we obtain

 

Since   we have

 

Using the inequality   we get

 

Therefore, we can obtain that

 

That is,

 

 

 

 

 

(6)

From equation (2 ) and (6 ),

 
 

so we can conclude that the global truncation error is one order lower than the local truncation error.

GraphEdit

In this graph,   The red line is the true value, the green line is the first step, and the blue line is the second step.

  is the local truncation error at step 1,  , equal to  
  is separation because after the first step we are on the wrong solution of the ODE.
  is  

Thus,   is the global truncation error at step 2,  

We can see from this,

 

Then,

 
 

ExerciseEdit

Find the order of the 2-steps Adams-Bashforth method. You need to show the order of truncation error.

ReferencesEdit

  1. Burden, R. L., & Faires, J. (2011). Numerical analysis ninth edition. Brooks/Cole, Cengage Learning.
  2. Materials from MATH 3600 Lecture 28 http://www.math.ohiou.edu/courses/math3600/lecture28.pdf.
  3. http://www.math.uiuc.edu/~ekirr/page/teaching/math385/handout2.pdf.
  4. http://users.soe.ucsc.edu/~hongwang/AMS147/Notes/Lecture09.pdf.
  5. http://livetoad.org/Courses/Documents/03e0/Notes/truncation_error.pdf.