ABSTRACT
The rapid, almost exponential, growth of the power of the silicon computing chip, made available to the user at ever-decreasing cost, has made the computational approach a viable and practical alternative for a variety of problems in physics and engineering, in particular the complex nonlinear problems of fluid mechanics. Nearly a half century earlier, Von Neumann, the inventor of the modern electronic computing machine, carried out the first numerical calculations for the inviscid, nonlinear problem of gas dynamics involving shock waves. The concept of artificial viscosity, proposed first by Von Neumann and Richtmeyer (1950) and refined subsequently to a high degree of perfection by many others, has proved to be a powerful tool that made the numerical algorithms for such problems not only possible but also accurate, reliable and robust. The progress in the efficiency and accuracy of solution algorithms during the last five decades, keeping pace with the power of the computing chip, has proved beyond doubt the truth of Von Neumann’s forecast in 1945: “Really efficient high speed computing devices may, in the field of nonlinear partial differential equations as well as in many other fields which are now difficult or are entirely denied of access, provide us with those heuristic hints which are needed in all parts of mathematics for genuine progress”. Needless to say progress in numerical methods is equally dependent on the sharpening of the analytical tools of applied mathematics.
