Introduction to Scientific Computing

Abstract

This chapter introduces five basic steps in scientific computing applied to an initial value problem. The first step constructs a mathematical model, consisting of an ordinary differential equation and an initial value. The second step examines the mathematical well-posedness of the model problem, to see if the a solution exists, is unique, and depends continuously on the data. In the third step, we construct a simple numerical method for this problem, and in the fourth step we develop computer programs to execute this method. In the final step, we perform a mathematical analysis of the numerical method to determine its stability and convergence properties, even in the presence of computer roundoff errors. This analysis helps us to choose appropriate parameters for the numerical method, such as the time step size, and to compare the relative efficiency of competing methods.

There is nothing more difficult to take in hand, more perilous to conduct, or more uncertain in its success, than to take the lead in the introduction of a new order of things.

Nicolò Machiavelli, The Prince

Additional Material: The details of the computer programs referred in the text are available in the Springer website (http://extras.springer.com/2018/978-3-319-69105-3) for authorized users.