Numerical simulation of non-uniform processes

In logistics as well as in other branches of science, engineering or transport, one frequently observes nonhomogeneous spatial or temporal patterns. It is a challenge for mathematical modeling and computer simulation to preserve such non-uniform behavior. Typically, the type of solutions to model equations depends on certain characteristic parameters. It is important to determine critical values of those.

NUMERICAL SIMULATION OF NON-UNIFORM PROCESSES

In logistics, transport, as well as other scientific fields, non-homogeneous phenomena in time or space occur frequently. Correctly reflecting such behavior poses a challenge for mathematical modeling and computer simulations. It is a typical circumstance that the solutions to the corresponding model equations depend on characteristic parameters. It is of significant importance to calculate the critical values of such parameters.

1. INTRODUCTION

Many processes in nature, engineering or economy can be easily described by means of differential equations, and often it is not very difficult to find certain special solutions to these model equations.

In particular, if intuition suggests e.g. symmetries or invariances of the studied phenomenon, numerical calculations of that type of solutions may be fast and effective.

However, in many applications, solutions turn out to be non-unique, and in certain ranges of parameters non-trivial behavior has to be studied.

Let us consider some examples. We start with a dynamical system, described by an ordinary differential equation. The equations of motion of a rigid wheelset on a straight track under a constant driving moment admit a solution with vanishing lateral motion and constant speed along the middle line between the rails. However, if a certain critical speed is exceeded, the uniform solution becomes unstable and so-called hunting occurs, cf. [7, 2, 4]. (…)