MATHEMATICA TUTORIAL

"FixedStep" Method for NDSolve

Introduction

It is often useful to carry out a numerical integration using fixed step sizes.

For example, certain methods such as and carry out a sequence of fixed-step integrations before combining the solutions to obtain a more accurate method with an error estimate that allows adaptive step sizes to be taken.

The method allows any one-step integration method to be invoked using fixed step sizes.

This loads a package with some example problems and a package with some utility functions.

In[3]:=
Click for copyable input

Examples

Define an example problem.
In[5]:=
Click for copyable input
Out[5]=
This integrates a differential system using the method with a fixed step size of 1/10.
In[6]:=
Click for copyable input
Out[6]=
Actually the method has no adaptive step size control. Therefore, the integration is already carried out using fixed step sizes so the specification of is unnecessary.
In[7]:=
Click for copyable input
Out[8]=
Here are the step sizes taken by the method for this problem.
In[9]:=
Click for copyable input
Out[10]=
This specifies that fixed step sizes should be used for the method .
In[11]:=
Click for copyable input
Out[12]=

The option MaxStepFraction provides an absolute bound on the step size that depends on the integration interval.

Since the default value of MaxStepFraction is 1/10, the step size in this example is bounded by one-tenth of the integration interval, which leads to using a constant step size of 1/20.
In[13]:=
Click for copyable input
Out[15]=
By setting the value of MaxStepFraction to a different value, the dependence of the step size on the integration interval can be relaxed or removed entirely.
In[16]:=
Click for copyable input
Out[17]=

Option Summary

option name
default value
MethodNonespecify the method to use with fixed step sizes

Option of the method .

New to Mathematica? Find your learning path »
Have a question? Ask support »