This chapter provides an introduction to numerical methods by presenting the numerical counterparts of some analytical models introduced earlier in the book. We will discuss a particular type of solution method for a selection of problems, explain its most important properties and present the resulting numerical code down to the level of the individual program statements. The use of these program scripts will be demonstrated by means of computational examples. In the wake of each method treated we will also summarize some other methods, providing the reader with an overview of alternative solution strategies.

Introduction

The preceding chapters dealt with various types of unsteady flow, resulting in analytical solutions of the governing mathematical equations. These solutions generally provide relations between the characteristics of a water system and its forcing, on the one hand, and the system's response, on the other. In terms of readiness and the overall insight they provide, analytical approaches are still unmatched. However, for practical applications involving complex geometries, arbitrary forcings, and possibly nonlinear effects, they are also somewhat limited due to the assumptions and simplifications on which they rely.

Since the 1950s computer models have therefore gradually, and decisively, replaced the former analytical approaches for computing solutions to real-world problems. The use of advanced numerical tools to compute flow problems in today's engineering practice is ubiquitous and has witnessed a tremendous increase in possibilities. Yet however sophisticated these models may be, they are based on the same theoretical concepts as their analytical predecessors, the study of which is still relevant to understand the results provided by modern computer tools.

In this chapter we will take a first step into the realm of numerical computing by working out some simple examples from previous chapters using the Python programming language. For this purpose some basic knowledge of Python will be needed (see for instance Hetland (2005) or Langtangen (2009)), but this may also be learned as we proceed along the examples in this chapter.

Canal-Basin System

Our first numerical endeavor concerns a simple model of a small tidal basin that is connected to the sea by a narrow gap or entrance channel. The imposed tidal water level at sea (hs) leads to a time-varying water level in the basin (hb) that does not depend on the position within the basin.