The Plurality of Numerical Methods in Computer Simulations and their Philosophical Analysis

Participation de Martin Carrier à la conférence internationale : "The Plurality of Numerical Methods in Computer Simulations and their Philosophical Analysis"

Numerical methods are preconditions of computer simulations: the latter would just be impossible without the former. Numerical methods are used for solving mathematical equations in computer simulations, especially when equations are not analytically tractable or take too long to be solved by other means. In other words, numerical methods are a necessary medium between the theoretical model and the simulation. Is this medium transparent or does it add a representational layer that would differ from the theoretical model?

If numerical methods are not transparent, does the plurality of methods mean that each one of them must be associated with a specific definition of computer simulations? Is it possible to provide a unique definition of computer simulation?

Besides, numerical methods must satisfy constraints that are specific to the computational architecture (parallel, sequential, digital or analog) and to the peculiar features of the machine (in terms of computational power, storage, system resource). To which extent do these constraints threaten the accuracy of the representation of the system under study that simulation models provide?

Another set of questions relates to the plurality of numerical methods, usually underestimated by philosophers. Let us mention some of them: methods for solving first-order or second-order differential equations, such as Euler’s, Runge-Kutta’s, Adams-Moulton’s and Numerov’s; the finite difference method; the finite element method; the Monte Carlo method; the Metropolis algorithm; the particle methods; etc. For a given problem, on what ground does one choose such or such numerical method? Does the choice depend on the nature of the problem? In some scientific disciplines, the Monte-Carlo method is preferably used for providing results of reference (benchmarks), and therefore for allowing the validation of differential equation-based simulations. How may the special functions attributed to some numerical methods be explained?