20:
1250:(FFT), transform the PDEs into an eigenvalue problem, but this time the basis functions are high order and defined globally over the whole domain. The domain itself is not discretized in this case, it remains continuous. Again, a trial solution is found by inserting the basis functions into the eigenvalue equation and then optimized to determine the best values of the initial trial parameters.
1190:
The global bound on energy error typically associated with symplectic algorithms still holds for the Boris algorithm, making it an effective algorithm for the multi-scale dynamics of plasmas. It has also been shown that one can improve on the relativistic Boris push to make it both volume preserving and have a constant-velocity solution in crossed E and B fields.
538:
1696:
which can be derived considering the harmonic oscillations of a one-dimensional unmagnetized plasma. The latter conditions is strictly required but practical considerations related to energy conservation suggest to use a much stricter constraint where the factor 2 is replaced by a number one order of
1490:
labels the grid point. To ensure that the forces acting on particles are self-consistently obtained, the way of calculating macro-quantities from particle positions on the grid points and interpolating fields from grid points to particle positions has to be consistent, too, since they both appear in
88:
For many types of problems, the classical PIC method invented by
Buneman, Dawson, Hockney, Birdsall, Morse and others is relatively intuitive and straightforward to implement. This probably accounts for much of its success, particularly for plasma simulation, for which the method typically includes
1189:
Because of its excellent long term accuracy, the Boris algorithm is the de facto standard for advancing a charged particle. It was realized that the excellent long term accuracy of nonrelativistic Boris algorithm is due to the fact it conserves phase space volume, even though it is not symplectic.
205:
The number of real particles corresponding to a super-particle must be chosen such that sufficient statistics can be collected on the particle motion. If there is a significant difference between the density of different species in the system (between ions and neutrals, for instance), separate real
1527:
In a real plasma, many other reactions may play a role, ranging from elastic collisions, such as collisions between charged and neutral particles, over inelastic collisions, such as electron-neutral ionization collision, to chemical reactions; each of them requiring separate treatment. Most of the
218:
The schemes used for the particle mover can be split into two categories, implicit and explicit solvers. While implicit solvers (e.g. implicit Euler scheme) calculate the particle velocity from the already updated fields, explicit solvers use only the old force from the previous time step, and are
214:
Even with super-particles, the number of simulated particles is usually very large (> 10), and often the particle mover is the most time consuming part of PIC, since it has to be done for each particle separately. Thus, the pusher is required to be of high accuracy and speed and much effort is
144:
to guarantee gauge invariant and conservation of charge, energy-momentum, and more importantly the infinitely dimensional symplectic structure of the particle-field system. These desired features are attributed to the fact that geometric PIC algorithms are built on the more fundamental
198:) is a computational particle that represents many real particles; it may be millions of electrons or ions in the case of a plasma simulation, or, for instance, a vortex element in a fluid simulation. It is allowed to rescale the number of particles, because the acceleration from the
1544:
As in every simulation method, also in PIC, the time step and the grid size must be well chosen, so that the time and length scale phenomena of interest are properly resolved in the problem. In addition, time step and grid size affect the speed and accuracy of the code.
1511:
As the field solver is required to be free of self-forces, inside a cell the field generated by a particle must decrease with decreasing distance from the particle, and hence inter-particle forces inside the cells are underestimated. This can be balanced with the aid of
337:
1270:). Particles can be situated anywhere on the continuous domain, but macro-quantities are calculated only on the mesh points, just as the fields are. To obtain the macro-quantities, one assumes that the particles have a given "shape" determined by the shape function
936:
1366:(CIC) scheme, which is a first order (linear) weighting scheme. Whatever the scheme is, the shape function has to satisfy the following conditions: space isotropy, charge conservation, and increasing accuracy (convergence) for higher-order terms.
1465:
331:
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848:
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153:
Inside the plasma research community, systems of different species (electrons, ions, neutrals, molecules, dust particles, etc.) are investigated. The set of equations associated with PIC codes are therefore the
1369:
The fields obtained from the field solver are determined only on the grid points and can't be used directly in the particle mover to calculate the force acting on particles, but have to be interpolated via the
533:{\displaystyle {\frac {\mathbf {v} _{k+1/2}-\mathbf {v} _{k-1/2}}{\Delta t}}={\frac {q}{m}}\left(\mathbf {E} _{k}+{\frac {\mathbf {v} _{k+1/2}+\mathbf {v} _{k-1/2}}{2}}\times \mathbf {B} _{k}\right),}
140:
Modern geometric PIC algorithms are based on a very different theoretical framework. These algorithms use tools of discrete manifold, interpolating differential forms, and canonical or non-canonical
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1124:
1748:
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1313:
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For an electrostatic plasma simulation using an explicit time integration scheme (e.g. leapfrog, which is most commonly used), two important conditions regarding the grid size
1499:. This can be achieved by choosing the same weighting scheme for particles and fields and by ensuring the appropriate space symmetry (i.e. no self-force and fulfilling the
80:
applications, the method amounts to following the trajectories of charged particles in self-consistent electromagnetic (or electrostatic) fields computed on a fixed mesh.
1819:
1380:
1360:
1338:
190:
The real systems studied are often extremely large in terms of the number of particles they contain. In order to make simulations efficient or at all possible, so-called
1592:
1569:
669:
2594:
Qin, H.; Liu, J.; Xiao, J.; et al. (2016). "Canonical symplectic particle-in-cell method for long-term large-scale simulations of the Vlasov-Maxwell system".
1232:
fields are calculated. Derivatives are then approximated with differences between neighboring grid-point values and thus PDEs are turned into algebraic equations.
240:
587:
2785:
Higuera, Adam V.; John R. Cary (2017). "Structure-preserving second-order integration of relativistic charged particle trajectories in electromagnetic fields".
1931:
1488:
561:
4187:
1941:
Within plasma physics, PIC simulation has been used successfully to study laser-plasma interactions, electron acceleration and ion heating in the auroral
680:
2647:
Xiao, J.; Qin, H.; Liu, J.; et al. (2015). "Explicit high-order non-canonical symplectic particle-in-cell algorithms for Vlasov-Maxwell systems".
3522:
3112:
2862:
771:
942:
2510:(1955). A Machine Calculation Method for Hydrodynamic Problems (Report). Los Alamos Scientific Laboratory of the University of California.
1524:, in which particles are grouped according to their cell, then these particles are paired randomly, and finally the pairs are collided.
1516:
between charged particles. Simulating the interaction for every pair of a big system would be computationally too expensive, so several
4652:
4278:
1825:
54:
4228:
130:
129:. This error is statistical in nature, and today it remains less-well understood than for traditional fixed-grid methods, such as
4667:
4204:
4178:
2911:
Birdsall, C.K. (1991). "Particle-in-cell charged-particle simulations, plus Monte Carlo collisions with neutral atoms, PIC-MCC".
1243:
that are localized in each element. The final solution is then obtained by optimization until the required accuracy is reached.
1236:
4657:
4502:
4272:
2846:
3126:
Shalaby, Mohamad; Broderick, Avery E.; Chang, Philip; Pfrommer, Christoph; Lamberts, Astrid; Puchwein, Ewald (23 May 2017).
4572:
4429:
4284:
1536:
scheme, which does not analyze all particles but uses the maximum collision probability for each charged species instead.
1069:
46:
1700:
202:
depends only on the charge-to-mass ratio, so a super-particle will follow the same trajectory as a real particle would.
2487:
1969:
2744:
1643:
1020:
4104:
4079:
3053:
2709:
145:
field-theoretical framework and are directly linked to the perfect form, i.e., the variational principle of physics.
4480:
3649:
Markidis, Stefano; Lapenta, Giovanni; Rizwan-uddin (17 Oct 2009). "Multi-scale simulations of plasma with iPIC3D".
1276:
4497:
1600:
68:
compilers were available. The method gained popularity for plasma simulation in the late 1950s and early 1960s by
4631:
4417:
2876:
Takizuka, Tomonor; Abe, Hirotada (1977). "A binary collision model for plasma simulation with a particle code".
4662:
4398:
4387:
4364:
4131:
Open source 3D Particle-In-Cell code for spacecraft plasma interactions (mandatory user registration required).
1880:
931:{\displaystyle \mathbf {u} '=\mathbf {u} +(\mathbf {u} +(\mathbf {u} \times \mathbf {h} ))\times \mathbf {s} ,}
42:
3069:
Byrne, F. N.; Ellison, M. A.; Reid, J. H. (1964). "The particle-in-cell computing method for fluid dynamics".
1968:
Hybrid models may use the PIC method for the kinetic treatment of some species, while other species (that are
1132:
4370:
1834:
1199:
4140:
2956:"A Monte Carlo collision model for the particle-in-cell method: applications to argon and oxygen discharges"
592:
4487:
4452:
3548:"PIConGPU - Particle-in-Cell Simulations for the Exascale Era - Helmholtz-Zentrum Dresden-Rossendorf, HZDR"
4492:
4171:
3796:
1235:
Using FEM, the continuous domain is divided into a discrete mesh of elements. The PDEs are treated as an
1362:
the observation point. Perhaps the easiest and most used choice for the shape function is the so-called
125:
Since the early days, it has been recognized that the PIC method is susceptible to error from so-called
4609:
4594:
4470:
4156:
2559:
Hideo Okuda (1972). "Nonphysical noises and instabilities in plasma simulation due to a spatial grid".
1756:
1500:
3772:
4256:
4236:
4218:
1460:{\displaystyle \mathbf {E} (\mathbf {x} )=\sum _{i}\mathbf {E} _{i}S(\mathbf {x} _{i}-\mathbf {x} ),}
53:, whereas moments of the distribution such as densities and currents are computed simultaneously on
4579:
4465:
4195:
3703:
1206:
58:
3463:"piccante: a spicy massively parallel fully-relativistic electromagnetic 3D particle-in-cell code"
1797:
1343:
1321:
4621:
4599:
4584:
4567:
4475:
4460:
4376:
4241:
4146:
4541:
4312:
4164:
4135:
3001:
Tskhakaya, D.; Matyash, K.; Schneider, R.; Taccogna, F. (2007). "The
Particle-In-Cell Method".
1492:
1247:
167:
134:
1574:
1551:
4589:
4435:
4351:
3106:
2725:
Boris, J.P. (November 1970). "Relativistic plasma simulation-optimization of a hybrid code".
1950:
1224:
With the FDM, the continuous domain is replaced with a discrete grid of points, on which the
1212:
326:{\displaystyle {\frac {\mathbf {x} _{k+1}-\mathbf {x} _{k}}{\Delta t}}=\mathbf {v} _{k+1/2},}
141:
3816:
3572:
4626:
4299:
3149:
3078:
3010:
2967:
2920:
2885:
2794:
2759:
2666:
2613:
2568:
2533:
1946:
1750:
is typical. Not surprisingly, the natural time scale in the plasma is given by the inverse
1532:
scheme, in which all particles carry information about their collision probability, or the
647:
4016:
106:
Models which include interactions of particles only through the average fields are called
8:
4393:
4307:
2069:
566:
3153:
3082:
3014:
2971:
2924:
2889:
2798:
2763:
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4557:
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3167:
3139:
3094:
3026:
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2603:
1916:
1824:
For an explicit electromagnetic plasma simulation, the time step must also satisfy the
1751:
1517:
1473:
546:
2113:
16:
Mathematical technique used to solve a certain class of partial differential equations
4246:
4100:
4075:
3752:
3171:
3098:
3049:
3030:
2983:
2979:
2936:
2897:
2842:
2705:
2580:
2507:
2135:
1513:
219:
therefore simpler and faster, but require a smaller time step. In PIC simulation the
3728:
2686:
2633:
4562:
4552:
4409:
3658:
3157:
3086:
3018:
2975:
2928:
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2205:
2197:
2175:
2153:
2131:
2109:
2087:
2065:
2043:
2021:
1198:
The most commonly used methods for solving
Maxwell's equations (or more generally,
760:{\displaystyle \mathbf {x} _{k+1}=\mathbf {x} _{k}+{\Delta t}\mathbf {v} _{k+1/2},}
674:
The equations of the Boris scheme which are substitute in the above equations are:
77:
4604:
4547:
4536:
3662:
2482:
2291:
1980:
1976:
1263:
1218:
231:
220:
4130:
1975:
PIC simulations have also been applied outside of plasma physics to problems in
1258:
The name "particle-in-cell" originates in the way that plasma macro-quantities (
4382:
4329:
3162:
3127:
2521:
2438:
2201:
2025:
1259:
1240:
1229:
1225:
175:
171:
73:
30:
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2545:
2394:
2313:
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2091:
1986:
4646:
4251:
3598:
3502:
2987:
2940:
2157:
199:
155:
69:
2833:. Lecture Notes in Physics 739. Vol. 739. Springer, Berlin Heidelberg.
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227:
is used which cancel out the magnetic field in the Newton-Lorentz equation.
19:
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4340:
4317:
3022:
2335:
1962:
1792:
4041:
3261:
4334:
4212:
3941:
3412:
2179:
2047:
50:
4186:
64:
PIC methods were already in use as early as 1955, even before the first
4126:
Particle-In-Cell and
Kinetic Simulation Software Center (PICKSC), UCLA.
3090:
2328:
Open source, but also has a private version with QED/radiative modules
1958:
1942:
114:(particle-particle). Models with both types of interactions are called
3623:
2932:
2771:
2678:
2372:
2209:
1953:, as well as ion-temperature-gradient and other microinstabilities in
1594:
should be fulfilled in order to ensure the stability of the solution:
843:{\displaystyle \mathbf {v} _{k+1/2}=\mathbf {u} '+q'\mathbf {E} _{k},}
4151:
3991:
3462:
3437:
3128:"SHARP: A Spatially Higher-order, Relativistic Particle-in-Cell Code"
3068:
2826:
2416:
1009:{\displaystyle \mathbf {u} =\mathbf {v} _{k-1/2}+q'\mathbf {E} _{k},}
2955:
1528:
collision models handling charged-neutral collisions use either the
110:(particle-mesh). Those which include direct binary interactions are
102:
Interpolation of the fields from the mesh to the particle locations.
4152:
open-pic - 3D Hybrid
Particle-In-Cell simulation of plasma dynamics
3211:"ALaDyn: A High-Accuracy PIC Code for the Maxwell-Vlasov Equations"
3210:
3185:
3144:
2661:
2608:
1496:
96:
Interpolation of charge and current source terms to the field mesh.
3312:"fbpic: Spectral, quasi-3D Particle-In-Cell code, for CPU and GPU"
2468:
Commercially available from Plasma Taiwan
Innovation Corporation.
4520:
3676:
3488:
3387:
3311:
3000:
1954:
644:), and velocities are calculated in-between the usual time steps
65:
45:. In this method, individual particles (or fluid elements) in a
148:
41:) method refers to a technique used to solve a certain class of
4359:
3866:
3841:
3801:
3757:
3507:
2829:. In Fehske, Holger; Schneider, Ralf; WeiĂźe, Alexander (eds.).
4125:
3966:
3362:
3236:
3125:
3337:
2784:
1495:. Above all, the field interpolation scheme should conserve
4514:
4508:
4323:
3648:
3547:
3286:
3046:
Smoothed
Particle Hydrodynamics: A Meshfree Particle Method
1987:
Electromagnetic particle-in-cell computational applications
3891:
4121:
Beam, Plasma & Accelerator
Simulation Toolkit (BLAST)
1520:
have been developed instead. A widely used method is the
4120:
4090:
4069:
3599:"Smilei — A Particle-In-Cell code for plasma simulation"
2699:
1266:, etc.) are assigned to simulation particles (i.e., the
1202:(PDE)) belong to one of the following three categories:
563:
refers to "old" quantities from the previous time step,
3916:
2646:
2593:
1919:
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1837:
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1703:
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1603:
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1072:
1023:
945:
862:
774:
683:
650:
595:
569:
549:
340:
243:
4188:
Numerical methods for partial differential equations
2350:
1539:
1119:{\displaystyle \mathbf {s} =2\mathbf {h} /(1+h^{2})}
589:
to updated quantities from the next time step (i.e.
3573:"ComputationalRadiationPhysics / PIConGPU — GitHub"
1239:and initially a trial solution is calculated using
223:is used, a second-order explicit method. Also the
158:as the equation of motion, solved in the so-called
4092:
3413:"osiris-code/osiris: OSIRIS Particle-In-Cell code"
2732:. Naval Res. Lab., Washington, D.C. pp. 3–67.
1925:
1905:
1866:
1813:
1783:
1743:{\displaystyle \Delta t\leq 0.1\omega _{pe}^{-1},}
1742:
1685:
1631:
1586:
1563:
1482:
1459:
1354:
1332:
1307:
1178:
1118:
1057:
1008:
930:
842:
759:
663:
636:
581:
555:
532:
325:
3287:"FBPIC documentation — FBPIC 0.6.0 documentation"
2729:4th Conference on Numerical Simulation of Plasmas
4644:
1686:{\displaystyle \Delta t\leq 2\omega _{pe}^{-1},}
1058:{\displaystyle \mathbf {h} =q'\mathbf {B} _{k},}
4070:Birdsall, Charles K.; A. Bruce Langdon (1985).
2700:Birdsall, Charles K.; A. Bruce Langdon (1985).
1253:
206:to super-particle ratios can be used for them.
2953:
2910:
4172:
4141:Plasma Theory and Simulation Group (Berkeley)
4091:Hockney, Roger W.; James W. Eastwood (1988).
1308:{\displaystyle S(\mathbf {x} -\mathbf {X} ),}
149:Basics of the PIC plasma simulation technique
4143:Contains links to freely available software.
2520:
1632:{\displaystyle \Delta x<3.4\lambda _{D},}
3111:: CS1 maint: DOI inactive as of May 2024 (
3043:
2875:
2861:: CS1 maint: DOI inactive as of May 2024 (
2558:
215:spent on optimizing the different schemes.
4179:
4165:
4147:Introduction to PIC codes (Univ. of Texas)
2524:(1983). "Particle simulation of plasmas".
76:, Hockney, Birdsall, Morse and others. In
4017:"Educational Particle-In-Cell code suite"
3161:
3143:
2824:
2820:
2818:
2816:
2660:
2607:
1906:{\displaystyle \Delta x\sim \lambda _{D}}
99:Computation of the fields on mesh points.
2827:"Chapter 6: The Particle-in-Cell Method"
1179:{\displaystyle q'=\Delta t\times (q/2m)}
18:
3651:Mathematics and Computers in Simulation
2742:
1867:{\displaystyle \Delta t<\Delta x/c,}
1503:) of the field solver at the same time
93:Integration of the equations of motion.
4645:
4136:Simple Particle-In-Cell code in MATLAB
4072:Plasma Physics via Computer Simulation
2813:
2702:Plasma Physics via Computer Simulation
2506:
1340:is the coordinate of the particle and
637:{\displaystyle t_{k+1}=t_{k}+\Delta t}
4160:
2724:
209:
4430:Moving particle semi-implicit method
4341:Weighted essentially non-oscillatory
2298:The Virtual Laser Plasma Lab (VLPL)
1972:) are simulated with a fluid model.
83:
4095:Computer Simulation Using Particles
2913:IEEE Transactions on Plasma Science
2831:Computational Many-Particle Physics
1246:Also spectral methods, such as the
1193:
13:
4279:Finite-difference frequency-domain
3701:
3119:
2488:Multiphase particle-in-cell method
2387:Available from Tech-X Corporation
1884:
1847:
1838:
1704:
1647:
1604:
1578:
1555:
1147:
721:
628:
401:
282:
185:
14:
4679:
4114:
4042:"ricardo-fonseca / ZPIC — GitHub"
2954:Vahedi, V.; Surendra, M. (1995).
2745:"Why is Boris algorithm so good?"
1784:{\displaystyle \omega _{pe}^{-1}}
1540:Accuracy and stability conditions
4653:Numerical differential equations
3942:"berkeleylab / Warp — Bitbucket"
3523:"Fraunhofer IST Team Simulation"
2878:Journal of Computational Physics
2561:Journal of Computational Physics
1447:
1433:
1415:
1393:
1385:
1348:
1326:
1295:
1287:
1085:
1074:
1042:
1025:
993:
956:
947:
921:
907:
899:
888:
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865:
827:
806:
777:
730:
707:
686:
512:
477:
448:
430:
375:
346:
296:
270:
249:
49:frame are tracked in continuous
4632:Method of fundamental solutions
4418:Smoothed-particle hydrodynamics
4063:
4034:
4009:
3984:
3959:
3934:
3909:
3884:
3859:
3834:
3809:
3789:
3765:
3753:"Tristan v2 public github page"
3745:
3721:
3695:
3669:
3642:
3616:
3591:
3565:
3540:
3515:
3495:
3481:
3455:
3430:
3405:
3380:
3355:
3330:
3304:
3279:
3254:
3229:
3203:
3178:
3062:
3037:
3003:Contributions to Plasma Physics
2994:
2960:Computer Physics Communications
2947:
2904:
2869:
1936:
194:are used. A super-particle (or
4668:Computational electromagnetics
4273:Alternating direction-implicit
3729:"Tristan v2 wiki | Tristan v2"
2778:
2736:
2718:
2693:
2640:
2587:
2552:
2514:
2500:
2225:Available from Fraunhofer IST
2070:10.1088/0741-3335/57/11/113001
1697:magnitude smaller. The use of
1451:
1428:
1397:
1389:
1299:
1283:
1200:partial differential equations
1173:
1156:
1113:
1094:
914:
911:
895:
884:
43:partial differential equations
1:
4285:Finite-difference time-domain
3777:princetonuniversity.github.io
3733:princetonuniversity.github.io
3624:"SmileiPIC / Smilei — GitHub"
3388:"OSIRIS open-source - OSIRIS"
2743:Qin, H.; et al. (2013).
2626:10.1088/0029-5515/56/1/014001
2493:
2114:10.1016/S0168-9002(01)00024-9
1506:
230:For plasma applications, the
4658:Computational fluid dynamics
4324:Advection upstream-splitting
3992:"ECP-WarpX / WarpX — GitHub"
3663:10.1016/j.matcom.2009.08.038
3044:Liu, G.R.; M.B. Liu (2003).
2980:10.1016/0010-4655(94)00171-W
2898:10.1016/0021-9991(77)90099-7
2581:10.1016/0021-9991(72)90048-4
2292:10.1016/j.matcom.2009.08.038
2136:10.1016/0010-4655(95)00010-D
1814:{\displaystyle \lambda _{D}}
1355:{\displaystyle \mathbf {X} }
1333:{\displaystyle \mathbf {x} }
1254:Particle and field weighting
7:
4335:Essentially non-oscillatory
4318:Monotonic upstream-centered
3704:"Relativistic Laser Plasma"
2476:
10:
4684:
4595:Infinite difference method
4213:Forward-time central-space
2439:10.1016/j.nima.2018.01.035
2202:10.1016/j.crme.2014.07.005
1994:Computational application
234:takes the following form:
178:fields, calculated in the
89:the following procedures:
4529:
4498:Poincaré–Steklov operator
4451:
4408:
4350:
4298:
4265:
4257:Method of characteristics
4227:
4203:
4194:
3797:"Tristan v2: Citation.md"
3773:"QED Module | Tristan v2"
3503:"piclas-framework/piclas"
3132:The Astrophysical Journal
2839:10.1007/978-3-540-74686-7
2825:Tskhakaya, David (2008).
2807:10.1016/j.jcp.2003.11.004
2546:10.1103/RevModPhys.55.403
2526:Reviews of Modern Physics
2395:10.1016/j.jcp.2003.11.004
2314:10.1017/S0022377899007515
2270:10.1016/j.cpc.2017.09.024
2092:10.1016/j.cpc.2016.02.007
1207:Finite difference methods
4515:Tearing and interconnect
4509:Balancing by constraints
3163:10.3847/1538-4357/aa6d13
2158:10.1007/3-540-47789-6_36
2026:10.3847/1538-4357/aa6d13
1791:and length scale by the
1587:{\displaystyle \Delta t}
1564:{\displaystyle \Delta x}
4622:Computer-assisted proof
4600:Infinite element method
4388:Gradient discretisation
3603:Maisondelasimulation.fr
3093:(inactive 2024-05-03).
2841:(inactive 2024-05-03).
2248:10.1145/2503210.2504564
1933:is the speed of light.
127:discrete particle noise
4610:Petrov–Galerkin method
4371:Discontinuous Galerkin
3867:"LANL / VPIC — GitHub"
3023:10.1002/ctpp.200710072
2336:10.5281/zenodo.7566725
1927:
1907:
1868:
1815:
1785:
1744:
1687:
1633:
1588:
1565:
1522:binary collision model
1484:
1461:
1356:
1334:
1309:
1248:fast Fourier transform
1213:Finite element methods
1180:
1120:
1059:
1010:
932:
844:
761:
665:
638:
583:
557:
534:
327:
142:symplectic integrators
26:
4663:Mathematical modeling
4590:Isogeometric analysis
4436:Material point method
3967:"WarpX Documentation"
3392:osiris-code.github.io
1951:magnetic reconnection
1928:
1908:
1869:
1816:
1786:
1745:
1688:
1634:
1589:
1566:
1485:
1462:
1357:
1335:
1310:
1181:
1121:
1060:
1011:
933:
845:
762:
666:
664:{\displaystyle t_{k}}
639:
584:
558:
535:
328:
24:Ebola Virus Particles
22:
4627:Integrable algorithm
4453:Domain decomposition
4021:picksc.idre.ucla.edu
3071:Methods Comput. Phys
3048:. World Scientific.
2180:10.5281/zenodo.48703
2048:10.5281/zenodo.49553
2006:Canonical Reference
1947:magnetohydrodynamics
1917:
1881:
1835:
1798:
1757:
1701:
1644:
1601:
1575:
1552:
1474:
1470:where the subscript
1381:
1344:
1322:
1277:
1133:
1070:
1021:
943:
860:
772:
681:
648:
593:
567:
547:
543:where the subscript
338:
241:
4471:Schwarz alternating
4394:Loubignac iteration
3971:ecp-warpx.github.io
3154:2017ApJ...841...52S
3083:1964SSRv....3..319B
3015:2007CoPP...47..563T
2972:1995CoPhC..87..179V
2925:1991ITPS...19...65B
2890:1977JCoPh..25..205T
2799:2004JCoPh.196..448N
2764:2013PhPl...20h4503Q
2727:Proceedings of the
2671:2015PhPl...22k2504X
2618:2016NucFu..56a4001Q
2573:1972JCoPh..10..475O
2538:1983RvMP...55..403D
2281:Apache License 2.0
2128:Available from ATK
2106:Available from ATK
1780:
1736:
1679:
1518:Monte Carlo methods
1501:action-reaction law
1493:Maxwell's equations
582:{\displaystyle k+1}
168:Maxwell's equations
4617:Validated numerics
3702:Dreher, Matthias.
3579:. 28 November 2017
3469:. 14 November 2017
3217:. 18 November 2017
3091:10.1007/BF00230516
2787:Physics of Plasmas
2752:Physics of Plasmas
2649:Physics of Plasmas
2508:Harlow, Francis H.
2428:3-Clause-BSD-LBNL
2406:3-Clause-BSD-LBNL
2081:3-Clause-BSD-LBNL
1923:
1903:
1864:
1811:
1781:
1760:
1740:
1716:
1683:
1659:
1629:
1584:
1571:and the time step
1561:
1530:direct Monte-Carlo
1514:Coulomb collisions
1480:
1457:
1412:
1352:
1330:
1305:
1268:particle weighting
1237:eigenvalue problem
1176:
1116:
1055:
1006:
928:
840:
757:
661:
634:
579:
553:
530:
323:
210:The particle mover
27:
4640:
4639:
4580:Immersed boundary
4573:Method of moments
4488:Neumann–Dirichlet
4481:abstract additive
4466:Fictitious domain
4410:Meshless/Meshfree
4294:
4293:
4196:Finite difference
3683:. 31 January 2020
3677:"iPic3D — GitHub"
3630:. 29 October 2019
3527:ist.fraunhofer.de
3318:. 8 November 2017
2933:10.1109/27.106800
2848:978-3-540-74685-0
2772:10.1063/1.4818428
2679:10.1063/1.4935904
2474:
2473:
2373:10.1063/1.2840133
2210:10.1063/1.5097638
1959:vacuum discharges
1926:{\displaystyle c}
1483:{\displaystyle i}
1403:
556:{\displaystyle k}
505:
421:
408:
289:
166:of the code, and
84:Technical aspects
4675:
4585:Analytic element
4568:Boundary element
4461:Schur complement
4442:Particle-in-cell
4377:Spectral element
4201:
4200:
4181:
4174:
4167:
4158:
4157:
4110:
4098:
4085:
4057:
4056:
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4027:
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3863:
3857:
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3838:
3832:
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3829:
3827:
3813:
3807:
3806:
3793:
3787:
3786:
3784:
3783:
3769:
3763:
3762:
3749:
3743:
3742:
3740:
3739:
3725:
3719:
3718:
3716:
3714:
3699:
3693:
3692:
3690:
3688:
3673:
3667:
3666:
3646:
3640:
3639:
3637:
3635:
3620:
3614:
3613:
3611:
3609:
3595:
3589:
3588:
3586:
3584:
3569:
3563:
3562:
3560:
3558:
3552:picongpu.hzdr.de
3544:
3538:
3537:
3535:
3533:
3519:
3513:
3512:
3499:
3493:
3492:
3485:
3479:
3478:
3476:
3474:
3459:
3453:
3452:
3450:
3448:
3442:Aladyn.github.io
3434:
3428:
3427:
3425:
3423:
3409:
3403:
3402:
3400:
3398:
3384:
3378:
3377:
3375:
3373:
3359:
3353:
3352:
3350:
3348:
3334:
3328:
3327:
3325:
3323:
3308:
3302:
3301:
3299:
3297:
3283:
3277:
3276:
3274:
3272:
3258:
3252:
3251:
3249:
3247:
3233:
3227:
3226:
3224:
3222:
3207:
3201:
3200:
3198:
3196:
3182:
3176:
3175:
3165:
3147:
3123:
3117:
3116:
3110:
3102:
3066:
3060:
3059:
3041:
3035:
3034:
3009:(8–9): 563–594.
2998:
2992:
2991:
2966:(1–2): 179–198.
2951:
2945:
2944:
2908:
2902:
2901:
2873:
2867:
2866:
2860:
2852:
2822:
2811:
2810:
2782:
2776:
2775:
2749:
2740:
2734:
2733:
2722:
2716:
2715:
2697:
2691:
2690:
2664:
2644:
2638:
2637:
2611:
2591:
2585:
2584:
2556:
2550:
2549:
2518:
2512:
2511:
2504:
2417:10.1063/1.860024
1991:
1990:
1932:
1930:
1929:
1924:
1912:
1910:
1909:
1904:
1902:
1901:
1873:
1871:
1870:
1865:
1857:
1820:
1818:
1817:
1812:
1810:
1809:
1790:
1788:
1787:
1782:
1779:
1771:
1752:plasma frequency
1749:
1747:
1746:
1741:
1735:
1727:
1692:
1690:
1689:
1684:
1678:
1670:
1638:
1636:
1635:
1630:
1625:
1624:
1593:
1591:
1590:
1585:
1570:
1568:
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1562:
1489:
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1436:
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1411:
1396:
1388:
1361:
1359:
1358:
1353:
1351:
1339:
1337:
1336:
1331:
1329:
1314:
1312:
1311:
1306:
1298:
1290:
1219:Spectral methods
1194:The field solver
1185:
1183:
1182:
1177:
1166:
1143:
1125:
1123:
1122:
1117:
1112:
1111:
1093:
1088:
1077:
1064:
1062:
1061:
1056:
1051:
1050:
1045:
1039:
1028:
1015:
1013:
1012:
1007:
1002:
1001:
996:
990:
979:
978:
974:
959:
950:
937:
935:
934:
929:
924:
910:
902:
891:
880:
872:
868:
849:
847:
846:
841:
836:
835:
830:
824:
813:
809:
800:
799:
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780:
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748:
733:
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710:
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689:
670:
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643:
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624:
623:
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588:
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531:
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378:
369:
368:
364:
349:
342:
332:
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329:
324:
319:
318:
314:
299:
290:
288:
280:
279:
278:
273:
264:
263:
252:
245:
170:determining the
35:particle-in-cell
4683:
4682:
4678:
4677:
4676:
4674:
4673:
4672:
4643:
4642:
4641:
4636:
4605:Galerkin method
4548:Method of lines
4525:
4493:Neumann–Neumann
4447:
4404:
4346:
4313:High-resolution
4290:
4261:
4223:
4190:
4185:
4117:
4107:
4082:
4074:. McGraw-Hill.
4066:
4061:
4060:
4050:
4048:
4040:
4039:
4035:
4025:
4023:
4015:
4014:
4010:
4000:
3998:
3990:
3989:
3985:
3975:
3973:
3965:
3964:
3960:
3950:
3948:
3940:
3939:
3935:
3925:
3923:
3915:
3914:
3910:
3900:
3898:
3892:"Tech-X - VSim"
3890:
3889:
3885:
3875:
3873:
3865:
3864:
3860:
3850:
3848:
3840:
3839:
3835:
3825:
3823:
3815:
3814:
3810:
3795:
3794:
3790:
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3746:
3737:
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3727:
3726:
3722:
3712:
3710:
3700:
3696:
3686:
3684:
3675:
3674:
3670:
3647:
3643:
3633:
3631:
3622:
3621:
3617:
3607:
3605:
3597:
3596:
3592:
3582:
3580:
3571:
3570:
3566:
3556:
3554:
3546:
3545:
3541:
3531:
3529:
3521:
3520:
3516:
3501:
3500:
3496:
3487:
3486:
3482:
3472:
3470:
3461:
3460:
3456:
3446:
3444:
3436:
3435:
3431:
3421:
3419:
3411:
3410:
3406:
3396:
3394:
3386:
3385:
3381:
3371:
3369:
3361:
3360:
3356:
3346:
3344:
3336:
3335:
3331:
3321:
3319:
3310:
3309:
3305:
3295:
3293:
3291:fbpic.github.io
3285:
3284:
3280:
3270:
3268:
3260:
3259:
3255:
3245:
3243:
3235:
3234:
3230:
3220:
3218:
3209:
3208:
3204:
3194:
3192:
3184:
3183:
3179:
3124:
3120:
3104:
3103:
3067:
3063:
3056:
3042:
3038:
2999:
2995:
2952:
2948:
2909:
2905:
2874:
2870:
2854:
2853:
2849:
2823:
2814:
2783:
2779:
2747:
2741:
2737:
2723:
2719:
2712:
2704:. McGraw-Hill.
2698:
2694:
2645:
2641:
2592:
2588:
2557:
2553:
2519:
2515:
2505:
2501:
2496:
2483:Plasma modeling
2479:
1989:
1981:fluid mechanics
1939:
1918:
1915:
1914:
1897:
1893:
1882:
1879:
1878:
1853:
1836:
1833:
1832:
1805:
1801:
1799:
1796:
1795:
1772:
1764:
1758:
1755:
1754:
1728:
1720:
1702:
1699:
1698:
1671:
1663:
1645:
1642:
1641:
1620:
1616:
1602:
1599:
1598:
1576:
1573:
1572:
1553:
1550:
1549:
1542:
1509:
1475:
1472:
1471:
1446:
1437:
1432:
1431:
1419:
1414:
1413:
1407:
1392:
1384:
1382:
1379:
1378:
1372:field weighting
1347:
1345:
1342:
1341:
1325:
1323:
1320:
1319:
1294:
1286:
1278:
1275:
1274:
1264:current density
1256:
1241:basis functions
1196:
1162:
1136:
1134:
1131:
1130:
1107:
1103:
1089:
1084:
1073:
1071:
1068:
1067:
1046:
1041:
1040:
1032:
1024:
1022:
1019:
1018:
997:
992:
991:
983:
970:
960:
955:
954:
946:
944:
941:
940:
920:
906:
898:
887:
876:
864:
863:
861:
858:
857:
831:
826:
825:
817:
805:
804:
791:
781:
776:
775:
773:
770:
769:
744:
734:
729:
728:
720:
711:
706:
705:
690:
685:
684:
682:
679:
678:
655:
651:
649:
646:
645:
619:
615:
600:
596:
594:
591:
590:
568:
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564:
548:
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516:
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510:
491:
481:
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462:
452:
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429:
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427:
423:
413:
400:
389:
379:
374:
373:
360:
350:
345:
344:
343:
341:
339:
336:
335:
310:
300:
295:
294:
281:
274:
269:
268:
253:
248:
247:
246:
244:
242:
239:
238:
232:leapfrog method
225:Boris algorithm
221:leapfrog method
212:
192:super-particles
188:
186:Super-particles
151:
135:semi-Lagrangian
86:
17:
12:
11:
5:
4681:
4671:
4670:
4665:
4660:
4655:
4638:
4637:
4635:
4634:
4629:
4624:
4619:
4614:
4613:
4612:
4602:
4597:
4592:
4587:
4582:
4577:
4576:
4575:
4565:
4560:
4555:
4550:
4545:
4542:Pseudospectral
4539:
4533:
4531:
4527:
4526:
4524:
4523:
4518:
4512:
4506:
4500:
4495:
4490:
4485:
4484:
4483:
4478:
4468:
4463:
4457:
4455:
4449:
4448:
4446:
4445:
4439:
4433:
4427:
4421:
4414:
4412:
4406:
4405:
4403:
4402:
4396:
4391:
4385:
4380:
4374:
4368:
4362:
4356:
4354:
4352:Finite element
4348:
4347:
4345:
4344:
4338:
4332:
4330:Riemann solver
4327:
4321:
4315:
4310:
4304:
4302:
4296:
4295:
4292:
4291:
4289:
4288:
4282:
4276:
4269:
4267:
4263:
4262:
4260:
4259:
4254:
4249:
4244:
4239:
4237:Lax–Friedrichs
4233:
4231:
4225:
4224:
4222:
4221:
4219:Crank–Nicolson
4216:
4209:
4207:
4198:
4192:
4191:
4184:
4183:
4176:
4169:
4161:
4155:
4154:
4149:
4144:
4138:
4133:
4128:
4123:
4116:
4115:External links
4113:
4112:
4111:
4105:
4087:
4086:
4080:
4065:
4062:
4059:
4058:
4033:
4008:
3983:
3958:
3933:
3908:
3883:
3858:
3833:
3808:
3788:
3764:
3744:
3720:
3694:
3668:
3641:
3615:
3590:
3564:
3539:
3514:
3494:
3480:
3454:
3429:
3404:
3379:
3354:
3329:
3303:
3278:
3253:
3228:
3202:
3177:
3118:
3077:(3): 319–343.
3061:
3054:
3036:
2993:
2946:
2903:
2884:(3): 205–219.
2868:
2847:
2812:
2777:
2735:
2717:
2710:
2692:
2639:
2596:Nuclear Fusion
2586:
2567:(3): 475–486.
2551:
2532:(2): 403–447.
2513:
2498:
2497:
2495:
2492:
2491:
2490:
2485:
2478:
2475:
2472:
2471:
2469:
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2419:
2410:
2407:
2404:
2402:
2398:
2397:
2388:
2385:
2382:
2380:
2379:VSim (Vorpal)
2376:
2375:
2366:
2363:
2360:
2358:
2354:
2353:
2351:
2348:
2345:
2343:
2339:
2338:
2329:
2326:
2323:
2321:
2317:
2316:
2307:
2304:
2301:
2299:
2295:
2294:
2285:
2282:
2279:
2277:
2273:
2272:
2263:
2260:
2257:
2255:
2251:
2250:
2241:
2238:
2235:
2233:
2229:
2228:
2226:
2223:
2220:
2218:
2214:
2213:
2195:
2192:
2189:
2187:
2183:
2182:
2173:
2170:
2167:
2165:
2161:
2160:
2151:
2148:
2145:
2143:
2139:
2138:
2129:
2126:
2123:
2121:
2117:
2116:
2107:
2104:
2101:
2099:
2095:
2094:
2085:
2082:
2079:
2077:
2073:
2072:
2063:
2060:
2057:
2055:
2051:
2050:
2041:
2038:
2035:
2033:
2029:
2028:
2019:
2017:
2014:
2012:
2008:
2007:
2004:
2001:
1998:
1995:
1988:
1985:
1957:, furthermore
1938:
1935:
1922:
1900:
1896:
1892:
1889:
1886:
1875:
1874:
1863:
1860:
1856:
1852:
1849:
1846:
1843:
1840:
1808:
1804:
1778:
1775:
1770:
1767:
1763:
1739:
1734:
1731:
1726:
1723:
1719:
1715:
1712:
1709:
1706:
1694:
1693:
1682:
1677:
1674:
1669:
1666:
1662:
1658:
1655:
1652:
1649:
1639:
1628:
1623:
1619:
1615:
1612:
1609:
1606:
1583:
1580:
1560:
1557:
1541:
1538:
1534:null-collision
1508:
1505:
1479:
1468:
1467:
1456:
1453:
1449:
1445:
1440:
1435:
1430:
1427:
1422:
1417:
1410:
1406:
1402:
1399:
1395:
1391:
1387:
1350:
1328:
1316:
1315:
1304:
1301:
1297:
1293:
1289:
1285:
1282:
1260:number density
1255:
1252:
1222:
1221:
1216:
1210:
1195:
1192:
1175:
1172:
1169:
1165:
1161:
1158:
1155:
1152:
1149:
1146:
1142:
1139:
1127:
1126:
1115:
1110:
1106:
1102:
1099:
1096:
1092:
1087:
1083:
1080:
1076:
1065:
1054:
1049:
1044:
1038:
1035:
1031:
1027:
1016:
1005:
1000:
995:
989:
986:
982:
977:
973:
969:
966:
963:
958:
953:
949:
938:
927:
923:
919:
916:
913:
909:
905:
901:
897:
894:
890:
886:
883:
879:
875:
871:
867:
851:
850:
839:
834:
829:
823:
820:
816:
812:
808:
803:
798:
794:
790:
787:
784:
779:
767:
756:
751:
747:
743:
740:
737:
732:
726:
723:
719:
714:
709:
704:
699:
696:
693:
688:
658:
654:
633:
630:
627:
622:
618:
614:
609:
606:
603:
599:
578:
575:
572:
552:
541:
540:
529:
525:
519:
514:
509:
504:
498:
494:
490:
487:
484:
479:
474:
469:
465:
461:
458:
455:
450:
442:
437:
432:
426:
420:
417:
412:
406:
403:
396:
392:
388:
385:
382:
377:
372:
367:
363:
359:
356:
353:
348:
333:
322:
317:
313:
309:
306:
303:
298:
293:
287:
284:
277:
272:
267:
262:
259:
256:
251:
211:
208:
187:
184:
180:(field) solver
164:particle mover
150:
147:
104:
103:
100:
97:
94:
85:
82:
78:plasma physics
31:plasma physics
15:
9:
6:
4:
3:
2:
4680:
4669:
4666:
4664:
4661:
4659:
4656:
4654:
4651:
4650:
4648:
4633:
4630:
4628:
4625:
4623:
4620:
4618:
4615:
4611:
4608:
4607:
4606:
4603:
4601:
4598:
4596:
4593:
4591:
4588:
4586:
4583:
4581:
4578:
4574:
4571:
4570:
4569:
4566:
4564:
4561:
4559:
4556:
4554:
4551:
4549:
4546:
4543:
4540:
4538:
4535:
4534:
4532:
4528:
4522:
4519:
4516:
4513:
4510:
4507:
4504:
4501:
4499:
4496:
4494:
4491:
4489:
4486:
4482:
4479:
4477:
4474:
4473:
4472:
4469:
4467:
4464:
4462:
4459:
4458:
4456:
4454:
4450:
4443:
4440:
4437:
4434:
4431:
4428:
4425:
4422:
4419:
4416:
4415:
4413:
4411:
4407:
4400:
4397:
4395:
4392:
4389:
4386:
4384:
4381:
4378:
4375:
4372:
4369:
4366:
4363:
4361:
4358:
4357:
4355:
4353:
4349:
4342:
4339:
4336:
4333:
4331:
4328:
4325:
4322:
4319:
4316:
4314:
4311:
4309:
4306:
4305:
4303:
4301:
4300:Finite volume
4297:
4286:
4283:
4280:
4277:
4274:
4271:
4270:
4268:
4264:
4258:
4255:
4253:
4250:
4248:
4245:
4243:
4240:
4238:
4235:
4234:
4232:
4230:
4226:
4220:
4217:
4214:
4211:
4210:
4208:
4206:
4202:
4199:
4197:
4193:
4189:
4182:
4177:
4175:
4170:
4168:
4163:
4162:
4159:
4153:
4150:
4148:
4145:
4142:
4139:
4137:
4134:
4132:
4129:
4127:
4124:
4122:
4119:
4118:
4108:
4106:0-85274-392-0
4102:
4099:. CRC Press.
4097:
4096:
4089:
4088:
4083:
4081:0-07-005371-5
4077:
4073:
4068:
4067:
4047:
4043:
4037:
4022:
4018:
4012:
3997:
3993:
3987:
3972:
3968:
3962:
3947:
3946:bitbucket.org
3943:
3937:
3922:
3918:
3912:
3897:
3893:
3887:
3872:
3868:
3862:
3847:
3843:
3837:
3822:
3821:esgeetech.com
3818:
3812:
3804:
3803:
3798:
3792:
3778:
3774:
3768:
3760:
3759:
3754:
3748:
3734:
3730:
3724:
3709:
3705:
3698:
3682:
3678:
3672:
3664:
3660:
3656:
3652:
3645:
3629:
3625:
3619:
3604:
3600:
3594:
3578:
3574:
3568:
3553:
3549:
3543:
3528:
3524:
3518:
3510:
3509:
3504:
3498:
3490:
3484:
3468:
3464:
3458:
3443:
3439:
3433:
3418:
3414:
3408:
3393:
3389:
3383:
3368:
3364:
3363:"Orbital ATK"
3358:
3343:
3339:
3338:"Orbital ATK"
3333:
3317:
3313:
3307:
3292:
3288:
3282:
3267:
3263:
3257:
3242:
3238:
3232:
3216:
3212:
3206:
3191:
3187:
3181:
3173:
3169:
3164:
3159:
3155:
3151:
3146:
3141:
3137:
3133:
3129:
3122:
3114:
3108:
3100:
3096:
3092:
3088:
3084:
3080:
3076:
3072:
3065:
3057:
3055:981-238-456-1
3051:
3047:
3040:
3032:
3028:
3024:
3020:
3016:
3012:
3008:
3004:
2997:
2989:
2985:
2981:
2977:
2973:
2969:
2965:
2961:
2957:
2950:
2942:
2938:
2934:
2930:
2926:
2922:
2918:
2914:
2907:
2899:
2895:
2891:
2887:
2883:
2879:
2872:
2864:
2858:
2850:
2844:
2840:
2836:
2832:
2828:
2821:
2819:
2817:
2808:
2804:
2800:
2796:
2793:(5): 052104.
2792:
2788:
2781:
2773:
2769:
2765:
2761:
2758:(5): 084503.
2757:
2753:
2746:
2739:
2731:
2728:
2721:
2713:
2711:0-07-005371-5
2707:
2703:
2696:
2688:
2684:
2680:
2676:
2672:
2668:
2663:
2658:
2655:(11): 12504.
2654:
2650:
2643:
2635:
2631:
2627:
2623:
2619:
2615:
2610:
2605:
2602:(1): 014001.
2601:
2597:
2590:
2582:
2578:
2574:
2570:
2566:
2562:
2555:
2547:
2543:
2539:
2535:
2531:
2527:
2523:
2517:
2509:
2503:
2499:
2489:
2486:
2484:
2481:
2480:
2470:
2467:
2464:
2462:
2459:
2458:
2455:
2452:
2449:
2447:
2444:
2443:
2440:
2436:
2433:
2430:
2427:
2425:
2422:
2421:
2418:
2414:
2411:
2408:
2405:
2403:
2400:
2399:
2396:
2392:
2389:
2386:
2383:
2381:
2378:
2377:
2374:
2370:
2367:
2364:
2362:3-Clause-BSD
2361:
2359:
2356:
2355:
2352:
2349:
2346:
2344:
2341:
2340:
2337:
2333:
2330:
2327:
2325:3-Clause-BSD
2324:
2322:
2319:
2318:
2315:
2311:
2308:
2305:
2302:
2300:
2297:
2296:
2293:
2289:
2286:
2283:
2280:
2278:
2275:
2274:
2271:
2267:
2264:
2261:
2258:
2256:
2253:
2252:
2249:
2245:
2242:
2239:
2236:
2234:
2231:
2230:
2227:
2224:
2221:
2219:
2216:
2215:
2212:
2211:
2207:
2203:
2199:
2196:
2193:
2190:
2188:
2185:
2184:
2181:
2177:
2174:
2171:
2168:
2166:
2163:
2162:
2159:
2155:
2152:
2149:
2146:
2144:
2141:
2140:
2137:
2133:
2130:
2127:
2124:
2122:
2119:
2118:
2115:
2111:
2108:
2105:
2102:
2100:
2097:
2096:
2093:
2089:
2086:
2083:
2080:
2078:
2075:
2074:
2071:
2067:
2064:
2061:
2058:
2056:
2053:
2052:
2049:
2045:
2042:
2039:
2036:
2034:
2031:
2030:
2027:
2023:
2020:
2018:
2015:
2013:
2010:
2009:
2005:
2003:Availability
2002:
1999:
1996:
1993:
1992:
1984:
1982:
1978:
1973:
1971:
1966:
1964:
1963:dusty plasmas
1960:
1956:
1952:
1948:
1944:
1934:
1920:
1898:
1894:
1890:
1887:
1861:
1858:
1854:
1850:
1844:
1841:
1831:
1830:
1829:
1827:
1826:CFL condition
1822:
1806:
1802:
1794:
1776:
1773:
1768:
1765:
1761:
1753:
1737:
1732:
1729:
1724:
1721:
1717:
1713:
1710:
1707:
1680:
1675:
1672:
1667:
1664:
1660:
1656:
1653:
1650:
1640:
1626:
1621:
1617:
1613:
1610:
1607:
1597:
1596:
1595:
1581:
1558:
1546:
1537:
1535:
1531:
1525:
1523:
1519:
1515:
1504:
1502:
1498:
1494:
1477:
1454:
1443:
1438:
1425:
1420:
1408:
1404:
1400:
1377:
1376:
1375:
1373:
1367:
1365:
1364:cloud-in-cell
1302:
1291:
1280:
1273:
1272:
1271:
1269:
1265:
1261:
1251:
1249:
1244:
1242:
1238:
1233:
1231:
1227:
1220:
1217:
1214:
1211:
1208:
1205:
1204:
1203:
1201:
1191:
1187:
1170:
1167:
1163:
1159:
1153:
1150:
1144:
1140:
1137:
1108:
1104:
1100:
1097:
1090:
1081:
1078:
1066:
1052:
1047:
1036:
1033:
1029:
1017:
1003:
998:
987:
984:
980:
975:
971:
967:
964:
961:
951:
939:
925:
917:
903:
892:
881:
873:
869:
856:
855:
854:
837:
832:
821:
818:
814:
810:
801:
796:
792:
788:
785:
782:
768:
754:
749:
745:
741:
738:
735:
724:
717:
712:
702:
697:
694:
691:
677:
676:
675:
672:
656:
652:
631:
625:
620:
616:
612:
607:
604:
601:
597:
576:
573:
570:
550:
527:
523:
517:
507:
502:
496:
492:
488:
485:
482:
472:
467:
463:
459:
456:
453:
440:
435:
424:
418:
415:
410:
404:
394:
390:
386:
383:
380:
370:
365:
361:
357:
354:
351:
334:
320:
315:
311:
307:
304:
301:
291:
285:
275:
265:
260:
257:
254:
237:
236:
235:
233:
228:
226:
222:
216:
207:
203:
201:
200:Lorentz force
197:
196:macroparticle
193:
183:
181:
177:
173:
169:
165:
161:
157:
156:Lorentz force
146:
143:
138:
136:
132:
128:
123:
121:
117:
113:
109:
101:
98:
95:
92:
91:
90:
81:
79:
75:
71:
67:
62:
60:
57:(stationary)
56:
52:
48:
44:
40:
36:
32:
25:
21:
4441:
4424:Peridynamics
4242:Lax–Wendroff
4094:
4071:
4064:Bibliography
4049:. Retrieved
4045:
4036:
4024:. Retrieved
4020:
4011:
3999:. Retrieved
3995:
3986:
3974:. Retrieved
3970:
3961:
3949:. Retrieved
3945:
3936:
3924:. Retrieved
3921:warp.lbl.gov
3920:
3911:
3899:. Retrieved
3895:
3886:
3874:. Retrieved
3870:
3861:
3849:. Retrieved
3845:
3836:
3824:. Retrieved
3820:
3811:
3800:
3791:
3780:. Retrieved
3776:
3767:
3756:
3747:
3736:. Retrieved
3732:
3723:
3711:. Retrieved
3708:2.mpq.mpg.de
3707:
3697:
3685:. Retrieved
3680:
3671:
3654:
3650:
3644:
3632:. Retrieved
3627:
3618:
3606:. Retrieved
3602:
3593:
3581:. Retrieved
3576:
3567:
3555:. Retrieved
3551:
3542:
3530:. Retrieved
3526:
3517:
3506:
3497:
3483:
3471:. Retrieved
3466:
3457:
3445:. Retrieved
3441:
3432:
3420:. Retrieved
3416:
3407:
3395:. Retrieved
3391:
3382:
3370:. Retrieved
3366:
3357:
3345:. Retrieved
3341:
3332:
3320:. Retrieved
3315:
3306:
3294:. Retrieved
3290:
3281:
3269:. Retrieved
3265:
3256:
3244:. Retrieved
3240:
3231:
3219:. Retrieved
3214:
3205:
3193:. Retrieved
3189:
3180:
3135:
3131:
3121:
3107:cite journal
3074:
3070:
3064:
3045:
3039:
3006:
3002:
2996:
2963:
2959:
2949:
2919:(2): 65–85.
2916:
2912:
2906:
2881:
2877:
2871:
2830:
2790:
2786:
2780:
2755:
2751:
2738:
2730:
2726:
2720:
2701:
2695:
2652:
2648:
2642:
2599:
2595:
2589:
2564:
2560:
2554:
2529:
2525:
2522:Dawson, J.M.
2516:
2502:
2465:Proprietary
2384:Proprietary
2347:Proprietary
2303:Proprietary
2222:Proprietary
2204:
2125:Proprietary
2103:Proprietary
2016:Proprietary
1974:
1967:
1940:
1937:Applications
1876:
1823:
1793:Debye length
1695:
1547:
1543:
1533:
1529:
1526:
1521:
1510:
1469:
1371:
1368:
1363:
1317:
1267:
1257:
1245:
1234:
1223:
1197:
1188:
1128:
852:
673:
542:
229:
224:
217:
213:
204:
195:
191:
189:
179:
163:
159:
152:
139:
126:
124:
119:
115:
111:
107:
105:
87:
63:
38:
34:
28:
23:
4558:Collocation
3657:(7): 1509.
3422:13 December
3397:13 December
2453:Open Repo:
2431:Open Repo:
2409:Open Repo:
2365:Open Repo:
2320:Tristan v2
2284:Open Repo:
2262:Open Repo:
2240:Open Repo:
2194:Open Repo:
2172:Open Repo:
2150:Open Repo
2084:Open Repo:
2062:Open Repo:
2040:Open Repo:
51:phase space
4647:Categories
4247:MacCormack
4229:Hyperbolic
4051:29 October
4046:GitHub.org
4026:29 October
4001:29 October
3996:GitHub.org
3976:29 October
3951:1 December
3926:1 December
3901:1 December
3896:Txcorp.com
3876:29 October
3871:github.com
3846:github.com
3826:1 December
3817:"VizGrain"
3782:2022-12-15
3738:2022-12-15
3713:1 December
3687:31 January
3681:GitHub.com
3634:29 October
3628:GitHub.com
3608:1 December
3583:1 December
3577:GitHub.com
3557:1 December
3473:1 December
3467:GitHub.com
3447:1 December
3438:"Piccante"
3417:GitHub.com
3372:1 December
3367:Mrcwdc.com
3347:1 December
3342:Mrcwdc.com
3322:1 December
3316:GitHub.com
3296:1 December
3266:GitHub.com
3221:1 December
3215:GitHub.com
3195:1 December
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2662:1510.06972
2609:1503.08334
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1943:ionosphere
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4503:Balancing
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2164:PICCANTE
2147:GNU AGPL
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4399:Smoothed
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3532:7 August
3489:"PICLas"
3271:14 March
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3186:"ALaDyn"
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4521:FETI-DP
4401:(S-FEM)
4320:(MUSCL)
4308:Godunov
3262:"EPOCH"
3237:"EPOCH"
3150:Bibcode
3079:Bibcode
3011:Bibcode
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2886:Bibcode
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4420:(SPH)
4390:(GDM)
4379:(SEM)
4337:(ENO)
4275:(ADI)
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2748:(PDF)
2683:S2CID
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