Table Of ContentNumerical "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
/ / / V S P / //
Utrecht · Boston, 2002
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First published in 2002
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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
