Numerical "Particle-in-Cell" Methods theory and applications NUMERICAL "PARTICLE-IN-CELL" METHODS THEORY AND APPLICATIONS Yu.N. Grigoryev, V.A. Vshivkov and M.P. Fedoruk ///VSP/// Utrecht · Boston, 2002 VSP BV Tel: +31 30 692 5790 P.O. Box 346 Fax: +31 30 693 2081 3700 AH Zeist [email protected] The Netherlands www.vsppub.com © VSP BV 2002 First published in 2002 ISBN 90-6764-368-8 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. Printed in The Netherlands by Ridderprint bv, Ridderkerk. Preface i Preface Algorithms known as "particle" methods axe becoming more and more widespread in modern mathematical modeling. The characteristic feature of these methods is the discretization technique when the set of discrete objects is introduced that are model "particles" considered as some mesh of moving nodes. Numerical models based on particle methods are widely used for full-scale computational experiments. The results of such experiments are of great significance for science and practice. For example, the modifications of these methods are applied in the simulation of com- plex physical processes in Tocomacs and other unique plasma facilities of thermonuclear fusion. The effect on continuum media of powerful explosions causing great volume deformations, strong shock waves, s- hears and discontinuities was successfully modeled by particle methods in nuclear centers of USA and Russia. The aerodynamics of rarefied gases is an extensive field for the ap- plication of such methods, which were used for the calculations of the flow past complex spacecrafts, such as "Shuttle" and "Buran" Until recently, particle methods have been developed mainly by spe- cialists in applied calculations. They were looking for an alternative to classical numerical methods, first of all, to solve problems of plasma physics, deformed continuum media, rarefied gases, etc. As a result no specialized monographs and textbooks on particle methods are avail- able, as for finite-difference or finite-element methods. There are certainly some manuals on the application of "particle" methods in different areas of physics and mechanics. Most of them deal with plasma physics problems. All of these books contain much method- ological information that is useful for specialists in computations, but, strictly speaking, they are not books on computational mathematics. From this point of view, their common drawback is dealing too deeply with the specifics of the field. Besides, the mathematical part often suffers from verbal descriptiveness, and is presented as certain "know- how" . It is clear that all this makes grasping the essence of computa- tional methods difficult. This book deals with combined Lagrangian-Eulerian schemes of the "particle-in-cell" type, most widespread among particle methods. The authors describe a universal approach to the construction of such al- gorithms. The approach is based on splitting the initial problem by which the auxiliary problem with a hyperbolic (divergent) operator is separated. After special discretization of the solution, such a splitting naturally leads to the well-known schemes of "particle-in-cell" methods. The Introduction gives a general characteristic of particle method- s. It contains characteristic properties of this group of numerical algo- rithms. It briefly shows their development in retrospective, and contains ii Yu. Ν. Grigoryev, V. A. Vshivkov, M. P. Fedoruk a review of supplements. Chapter 1 describes a general scheme of the construction of algo- rithms of the particle-in-cell type, the particle models used, and their interpolating and approximating properties. Chapter 2 deals with the implementation of particle-in-cell methods on unstructured grids the development of which is started only during the last ten years. Chapter 3 presents particle-in-cell methods for problems of inviscid gas dynamics problems, that originated from the well-known paper of F. H. Harlow. Chapter 4 is devoted to schemes that can be described as "vortex-in- cell" methods used in the calculations of vortex flows. Alongside with the well-known scheme for two-dimensional flows, a new original scheme for three-dimensional flows based on the concept of Lamb's vortex im- pulse, which provides a convenient vorticity measure, is presented. Chapter 5 contains particle-in-cell algorithms for the modeling of processes in collisionless plasma. Chapter 6 introduces statistical particle-in-cell methods used most- ly for numerical research of rarefied gas dynamics, disperse coagulate systems, and solid plasma. The subroutines of universal blocks used in "particle-in-cell" meth- ods written in Fortran 77, which are ready for use by the readers, are presented in the Supplement. The chapters devoted to concrete applications of particle-in-cell meth- ods begin with a brief statement of a mathematical problem. Then the corresponding algorithm is introduced on the basis of the general scheme and standard discretization. After this, the specific character of its im- plementation for a certain class of problems is considered. Examples of calculations give the readers an idea of the capabilities of particle-in-cell methods, their requirements to computers, and the degree of precision that can be achieved. Modifications of the presented schemes used in other supplements are briefly annotated. In selecting and presenting the material, we used our experience in mathematical modeling by particle methods as well as methodological approaches we developed in special courses of lectures for students of Novosibirsk State University. To make the book more compact, we tried to separate the general elements of the presented algorithms and to find an optimal combination between the rigor and heuristics in style. The contribution of the authors in the book is as follows. Chapters 1,2 and 3 were written by the authors jointly. Chapters 4,6 were written by Yu. N. Grigoryev, and Chapter 5 was written by V. A. Vshivkov and M. P. Fedoruk who also prepared the subroutines given in Supplement. The book is primarily intended for specialists in calculations who want to get a general idea of numerical particle-in-cell methods and the sphere of their application. As a methodological guide it will be of Preface iii interest to undergraduate and postgraduate students, mathematicians and physicists specializing in mathematical modeling. We hope that the book will also be of interest to specialists familiar with the methods. Authors Contents Introduction. Computational particle methods 5 General characteristics 5 Some applications 9 1. Particle-in-cell methods 15 1.1. Introduction 15 1.2. General scheme 16 1.3. Model particles and their properties 22 1.3.1. General properties of the particles 22 1.3.2. Widespread models of the particles 25 1.3.3. The improvement of the interpolation smoothness . 32 1.4. Errors of the particle-in-cell schemes 35 1.4.1. The self-force effect 35 1.4.2. The influence of the mesh periodicity 41 1.4.3. The errors of interpolation 43 1.5. The continuity equation in the particle method 48 2. Particle-in-cell methods on unstructured meshes 56 2.1. Introduction 56 2.2. Finite-elements bases 58 2.3. The Lagrangian step on unstructured meshes 62 2.3.1. Particle localization algorithms 63 2.3.2. The interpolation procedures 67 2.3.3. Numerical examples 72 2.3.4. The Lagrangian step with auxiliary Cartesian mesh 75 2.4. The Euler step. The finite-volume method 78 3. The particle methods in gas dynamics 85 3.1. Introduction 85 3.2. Basic equations 86 3.2.1. Gas dynamics equations in divergent form 86 3.2.2. The splitting scheme 87 3.3. The realization of the method 88 3.3.1. Harlow scheme 88 3.3.2. The "coarse" particle method and FLIC-method . 94 3.4. The combined particle method 97 3.5. The example of application 99 4. Vortex-in-cell methods 102 4.1. Introduction 102 4.2. Vorticity dynamics in two-dimensional flows 103 4.2.1. Basic equations 103 4.2.2. The conservation laws 104 2 Yu. Ν. Grigoryev, V. A. Vshivkov, Μ. P. Fedoruk 4.3. The vortex-in-cell method in two-dimensional case .... 104 4.3.1. Splitting on physical processes 104 4.3.2. The systems of discrete vortices and their properties 105 4.3.3. The computational cycle of the method 109 4.4. The dynamics of vortices in three-dimensional flows ... Ill 4.4.1. The vorticity evolution in the three-dimensional case 111 4.4.2. The equation for the Lamb impulse density .... 112 4.5. The vortex-in-cell scheme for three-dimensional flows . 114 4.5.1. The splitting scheme 114 4.5.2. The Lagrangian vortex mesh 114 4.5.3. Computational cycle of the three-dimensional scheme 116 4.6. The examples of applications 119 5. Particle-in-cell methods in collisionless plasma dynamicsl24 5.1. Introduction 124 5.2. Collisionless plasma basic equations 126 5.2.1. Kinetic equation with a self-consistent field .... 126 5.2.2. Particle motion equations 127 5.3. General scheme and computation cycle of the method . 129 5.3.1. Initial data 129 5.3.2. Computation cycle 131 5.3.3. Numerical schemes for dynamic equations of model particles 133 5.4. Conservation laws in model plasma 138 5.5. Examples of applications 143 5.5.1. The dynamics of ion-acoustic waves of the arbitrary amplitude in non-isothermal plasma 143 5.5.2. Shock waves in plasma with the magnetic field . 151 5.5.3. Numerical simulation of explosion phenomena in col- lisionless plasma at super-alfvenic speed 157 5.5.4. Interaction of the laser impulse with plasma ...