Methods of integration
Choosing the method of integration
- For many problems an Adams type of method may prove to be
quite adequate, and this will not require a Jacobian.
- However, if the problem is stiff, i.e. there are components
with widely differing time constants, such methods may fail
giving misleading results.
Such stiffness is common and it
can be estimated by the condition number (ratio of largest
eigenvalue to smallest eigenvalue) of the related Jacobian.
- To get round this problem, a BDF method can be used,
but this also requires a Jacobian matrix to be supplied.
Such Jacobians can either be be supplied explicitly, or approximated
internally by numerical differentiation.
- For n equations dy(i)/dx = f(i)(x,y) the n^2 (df(i)/dy(j))
Jacobian can be coded explicitly or approximated.
Problems
occur when a faulty Jacobian is supplied, so results using
coded Jacobians should always be compared with results from
numerically approximated Jacobians.
To create a model file
with no Jacobian, just consult deqmod3.tf1 and deqmod3.tf2.
A file ending with two consecutive percentage signs (like
deqmod3.tf2) will cause model parsing to stop without
checking the code for a Jacobian.
- Only by experimenting with a particular problem will it be
clear which method to use but the general rule is that, if
you can supply an accurate Jacobian, use a BDF method.
- If you do not supply a Jacobian on the model file,
but just use percentages as escape sequences in the model
file, then the numerical approximation will be used.
If you do want to create a model with a Jacobian
you should always make two versions,
one without, and one with a Jacobian, to compare the behaviour.
After using a model file with no Jacobian, you have to alter
the configuration to restore use of a Jacobian.
- Program DEQSOL will not allow you to integrate a user supplied
model file with no Jacobian using a method such as BDF with
an explicit Jacobian.
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