Numerical Analysis/Truncation Errors

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This page is about Truncation error of ODE methods.

Definition[edit]

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 Error[edit]

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

if , then .

Proof[edit]

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.

Graph[edit]

File:W LTE and GTE.jpg
Relationship between LTE and GTE

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,

Exercise[edit]

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

References[edit]

  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.