It has become a cliché to say that convection lies at the foundation of the activities, magnetism, and dynamos of the Sun and cool stars – the title of this colloquium. However, it has not been possible, until recently, to study theoretically these phenomena and their connections with quantitative rigor. The new development owes much to the rapid growth in the power of computers and the spread of sophisticated numerical techniques.
Since the last decade numerical studies of solar and stellar convection problems thrive. The trend towards a greater use of computers in these studies will likely become stronger. Now is a good time to reflect on the numerical approach itself, so that we may understand it better and make better use of it in the future. Here we look at two aspects of this methodology. First (Section 2), we examine its relationships to the analytical approach and observation, for which we make a differentiation between numerical experimentation and simulation. It is obvious that the ultimate test of any theory, numerical or not, is to confront observations, but not generally realized is that the connection to analytical studies is an important requisite for the healthy development of numerical studies. This point is to be elaborated. Second (Section 3), we discuss the prevalence of pitfalls in the numerical approach; two examples of current interest in convection computations are analysed in detail. In Section 4, we go on to suggest that one of these pitfalls might have caused the discrepancy between the results of numerical modelling of the solar differential rotation and helioseismology.