Collection of abstracts 13th GAMM-Seminar Kiel on Numerical Treatment of Multi-Scale Problems January 24th to 26th, 1997.

Saturday, January 25th, 1997

A Numerical Approach to Multiple Scale Problems Based on Asymptotic Analysis

Eric Sonnendrücker
Universit\'e Henri Poincar\'e Nancy I, Institut Elie Cartan
B.P. 239, F-54 506 Vandoeuvre les Nancy Cedex

Multiple scale asymptotic analysis is used as a guideline for developing numerical methods for multiple scale problems. These asymptotic results provide detailed insight into the structure of the solution and are used in the numerical framework. Standard operations of multiple scale asymptotic analysis are introduced as discretized versions of large scale differencing and averaging. This technique is applied to two different problems.

For the self-consistent motion of charged particles in electromagnetic fields described by the Maxwell-Vlasov equations a multiple time scale asymptotic analysis motivates a subcycling procedure to calculate the electron orbits. Two different time scales linked respectively to the electromagnetic wave propagation and the electron gyrofrequency are introduced.

The low Mach number regime of compressible fluid flow is characterized by a large discrepancy between the flow velocity and the speed of sound, leading to physical effects on different length scales and of different orders of magnitude. Acoustic waves with small amplitude but large wave length may interact with local small length scale flow structures while the global background pressure becomes nearly constant in space. A single time scale, multiple space scale asymptotic analysis is used for developing a semi-implicit scheme, which involves multiple pressure variables. Scale adaptive grids are used to capture large and small scale effects.