TABLE OF CONTENTS / Summary / Existence and Uniqueness / Accuracy and Precision / Time Integration Errors / Short-Time Runge-Kutta Analysis / Long-Time Lyapunov Analysis / Expectations from Molecular Dynamics / Smooth-Particle Spatial Integration Errors / Lattice Instability / Even-Odd Instability /
/ 8 Figures /
Example Problem :
[ Shear-Flow Convergence ]
The computational methods we develop in this book are designed to give approximate, but useful, solutions of the partial differential evolution equations of mathematical physics. Our goal is to predict the future by solving these equations. In this Chapter we explore the quality of smooth-particle integration algorithms both the convergence of the solutions to "true" solutions and the stability of the results to small perturbations. The straightforward approach to these questions is to perform series of simulations, comparing results for different numbers of particles, different mesh structures, different boundary treatments, different forms of the weight function, and for different ranges h . Though reliable and educational, and sometimes absolutely necessary, this approach can be time consuming.
Though there is no substitute for the experience gained in solving real, nonlinear problems, one can prepare oneself, and become a better predictor, by first analyzing simple linear differential equations. Runge-Kutta techniques provide straightforward short-time analyses of the linear cases. At long times nonlinearity dominates. Long-time convergence and stability require a discussion of the Lyapunov spectrum
Spatial discretization plays a r le too. Molecular dynamics provides useful guidance to understanding the influence of irregular meshes. Regular mesh instability can be...
