Modeling and Simulation of Continuous Systems - 07

Question 1

What are the two kinds of errors numerical techniques suffer from? What do they consist of?

Answer 1

Numerical techniques suffer from two kinds of errors:
* Discretization error: Refers to the error due to time discretization. Smaller step sizes h decrease the discretization
error.
* Rounding error: Refers to the error due to the finite number of digits to represent numbers in computers. Larger
step sizes h increase the number of time steps and thus reduce the number of computations with rounding errors.
There exists an optimal step size for the total error.

Question 2

Describe the adaptive step size approach.

Answer 2

The discretization error varies depending on the characteristics of the ODE. In some areas, f(x) is sensitive with respect
to changes of x, i.e., ∂
∂xf(x) is large, while it is small elsewhere.
A popular method based on the Runge-Kutta family is the Runge-Kutta-Fehlberg method (implemented in MATLAB/
Simulink as ode45 ). The basic idea of the Runge-Kutta-Fehlberg method:
* Compute one-step solution with lower order Runge-Kutta method (order 4).
* Compute one-step solution with higher order Runge-Kutta method (order 5).
* If the error is large, the step size is shortened. Otherwise, the step size is enlarged.