Robert Piché
Tampere
University of Technology
After you've formed a model of a physical system, you'll want to compute the solution. Usually, the model is so large and complex that you'll want to use numerical solvers rather than attempt an ``exact'' solution. You can use the numerical solvers available in your simulation software or use algorithms available in general-purpose mathematical software packages. In either case, you will be faced with a range of choices: which method to use and what solution parameters to give it. This lesson provides some guidance to help you make the right choices.
A model of the dynamics of a physical system can have many different kinds of mathematical object, including ordinary differential equations, partial differential equations, algebraic equations, and difference equations (``discrete-time'' models). In this lesson, attention is restricted to models that are ordinary differential equation initial value problems. We start by giving a precise description of this mathematical problem.
We then look at the general ideas behind numerical solution algorithms. We explain the difference between algorithms that are constant or variable time step, one-step or multistep, explicit or implicit, low-order or high-order.
The lesson closes with a discussion of the importance of choosing appropriate error tolerances for the solution algorithm.