With small 8, these cases show recurrent behavior limit cycles. The energy ratio is independent of 6, so that even a very small nonlinearity can produce a significant error. One usually associates instability with an increase in amplitude, but a decrease can be equally as damaging a divergence from correct behavior. Hint: the series converges rapidly enough say, two terms to allow use of a pocket calculator. With suitable definitions of K and S, la holds for multipole algorithms as well Chapter For a Maxwellian, the contribution of the terms for which kpv,At 2 1 in the sum over p is approximately since the s 2 2 terms are much smaller.
Thus if v,At 2 A x , the sum over p converges in only a few terms. This is not surprising in electrostatic codes, since field grid points are at the same positions relative to the plasma at every time step. Thus a limited form of Galilean invariance is restored. More complicated examples of combined A x and At analysis appear in Chen, Langdon, and Birdsall , and Chapter There is a difference worth emphasizing between the finite spatial gridding and finite time stepping.
As the particle dynamics places the particles essentially at all x, and interpolation within a cell is used to obtain charge density and force, spatial information exists for all x and for all k.
However, in time, information is generated only at 0, At, 2At, 3 A t ,. For propagation across the magnetic field at wavelengths approximating the Larmor radius, there are waves near harmonics of the cyclotron frequency. As in Section on the unmagnetized case, we ignore the effect of the spatial grid used for the fields and concentrate on the time integration. We consider small perturbations of a uniform plasma in a uniform magnetic field parallel to the z axis.
An external magnetic field can be incorporated into the particle equations in such a way that the zero-order orbits in constant fields are the exact helices plus E x B drift with the correct gyrofrequency 0,. The difference equation for z, is the same as used in Section for unmagnetized plasmas. The existence of the constants of motion v1 and v, is important in applications and in the following analysis.
Apart from this, choosing other difference equations alters our results only in uninteresting ways Problem e. We consider a wave propagating in the x-z plane. This E" is substituted into 1 written for perturbed quantities. With boundary conditions x!
We assume that the unperturbed particles are distributed with uniform density no, uniformly in angle J,, and with velocity distribution fo vl, v;. With the time-reversible difference schemes used here, such a distribution of particles is constant in the absence of perturbing fields. For a Maxwellian f o the integral can be done analytically Problem c.
In constructing the difference scheme the design criteria were to make the particle respond accurately to forces at frequencies low compared to w, Hockney ; Buneman, ; Hockney and Eastwood, , Eq. Then it is surprising to find that the behavior of e near the nrh harmonic is very accurate even if n o , A t is not small. The result is a polynomial of degree N - 1 or N - 2, whose roots are easy to calculate we used a variant of Muller's method in order to avoid the awkward calculation of the polynomial coefficients and attendant loss of accuracy.
Thus the finite-difference scheme introduces no new modes, and combines harmonic modes when their corresponding cyclotron harmonics are aliases of one another. The additional terms and modes, introduced by increasing nmaxexcessively, only slightly affect the roots corresponding to smaller n terms, especially if one thinks of the effect of collisions, finite kz, slightly nonuniform magnetic field, etc. This usually requires an implicit scheme, in which the calculation of the positions x, requires knowledge of the fields at the same time Problem a , rather than the preceding time, as in the explicit methods discussed so far.
An approximate solution must be quite accurate if stability is to be retained. Recently there has been experimentation with schemes for accurate prediction of the fields at the next time step Mason, ; Friedman, Langdon, and Cohen, ; Denavit, ; Brackbill and Forslund, ; Langdon, Cohen, and Friedman, ; Barnes et al. Once found, the particles can be advanced one at a time. If their new charge density is not consistent with the predicted electric field, then convergent iteration is possible. In this way, the number of coupled equations to be solved is the order of the number of cells, rather than the number of particles.
We will describe these developments in Chapters 14 and Parallel work on design of implicit time differencing schemes applies the analysis of this chapter next section, and Cohen, Langdon, and Friedman, b. There is a fundamental limitation in using particle electrons Langdon, b; Langdon. Cohen, and Friedman, Thus when kv,At 2 1 we are unable to reproduce even Debye shielding correctly! A Vlasov equation model for the electrons may be more practical than a particle model when kv,At cannot be kept small.
A limitation on electric field gradient is noted by Denavit and elaborated upon by Langdon, Cohen, and Friedman There has been considerable experience accumulated in the implicit time integration of the equations of fluid flow, diffusion, chemical kinetics, magnetohydrodynamics, and many other fields. Now particle simulation is also beginning to profit from the use of implicit methods. In an orbit-averaged magneto-inductive algorithm Cohen et al.
An explicit solution for the electromagnetic fields, dropping radiation and electrostatics, is obtained using a current density that is accumulated from the particle data at each small step and temporally averaged over the fast, orbital time-scale.
The calculations required per particle are not reduced; the real gain is the great reduction in the number of particles needed. The averaged contributions from each particle can substitute for those from many particles in a conventional code. This increased efficiency 'allows use of realistic parameters, such as the ratio of the ion cyclotron frequency to the rate of ion slowing due to collisions with electrons, in simulations of "magnetic mirror" experiments.
In order to apply orbit-averaging to a model including electrostatic fields and extend the simulation to large w , A t , an implicit field solution must be performed Cohen, Freis, and Thomas, Show that such a scheme becomes unstable, as At--, when applied to a single particle simple harmonic oscillator, if the a, term is omitted. Including a, makes the scheme implicit Cohen, Lungdon, and Friedman, In this section we see how to derive properties of an integration scheme in a direct and reliable manner.
A different approach often seen uses a finite-difference analog of the Vlasov equation Lindman, ; Godfiey, ; Hockney and Eastwood, When v is defined at half-steps, as in the leap frog scheme, one can define a velocity v, see, e. This method fails when applied as stated to other equations of motion which are not "measure-preserving" Problem b , such as the damped schemes described later in this section.
For example, it is clear in the case of damped oscillations in a potential well that measure decreases and f increases. While it is possible to repair this derivation Problem c , we find it far simpler to use 1.
In the rest of this section, we consider two classes of time integration schemes with error of the same or better order as the leap-frog method. Included are implicit schemes which were designed and analyzed using the methods of this chapter. The order of this equation is k because there is a span of k 1 times involved , and it is implicit if co is nonzero.
At long wavelengths the particles undergo simple harmonic oscillations at the plasma frequency. As shown later, the two roots near foo have an error in their real part which is second or higher order in A t , and an error in their imaginary part which is third or higher order in At. The dispersion function has already been given, However, for the parameters used in Feix , w,At Also it may sometimes be an advantage that x and v are given at the same times.
Instead, we remain explicit at the expense of introducing an O A t 3 damping. The difference scheme may now be written by identifying powers of z wth time levels e.
The damping arises because time centering is lost in this term. This scheme seems preferable to the preceding two, if one is willing to save a for use in the next time step. Its impulse response decays rather than vanishing after several steps. Hint: J xn?
For the Euler scheme, show that in which the f o term arises because J f 1. Derive the dispersion function and show that it agrees with the final result in Problem b. Show that D , is equivalent to Improved energy conservation is an ostensible benefit. It was equally inevitable that mathematical elegance would obscure the practical properties.
In this chapter, we derive the algorithms and explore their properties using the methods of Chapter 8. This chapter draws heavily on Lungdon We begin by showing that the momentum-conserving algorithm cannot conserve energy, then show how it can be adjusted so that it does.
The algorithm derived from the variational principle follows this prescription and in addition provides the Poisson algorithm. The loss of momentum conservation and the overall accuracy of the variational procedure are discussed in the remaining sections. Although the energy-conserving algorithms have not demonstrated superiority in practice, they have many interesting properties. Although we do not go beyond electrostatic fields, the electromagnetic case is developed by Lewis b, and applied by Denavit To see why the sum of energies is not exactly constant, but is often very nearly so, we express the rates of change in terms of the particle current density J and particle force F, in one dimension: whereas 4 - which is S d x E J for a real plasma.
Although we see that energy is not conserved microscopically, in many momentum conserving simulations, the observed macroscopic "total energy" changes by amounts small compared to other energies of importance, e.
When this is so, our results suggest that most of the exchange of energy between fields and particles has taken place at long wavelengths. Since this is where the model most accurately simulates the plasma, a good energy check gives credibility to the simulation. In particular, 1 applies when the Coulomb force is smoothed at short range. The charge density is defined at the grid points as usual: If we obtain the force on the irh particle from Fix--a axi 3 and the electric potential 4 is obtained from p by some procedure yet to determined, then, assuming accurate time integration, total energy is conserved trivially.
Since this term therefore contains contributions from all cells, its evaluation would be far too expensive. It is helpful to rewrite 4 as We show later in this section that the second sum is generally zero. The particle force is then obtained from the first sum in 51, as The gradient of S is performed analytically and is therefore exact. In a code, 4j is calculated from p j in one step, and F i is calculated in a later step using that 4j, which is now fixed.
The prescription given by Lewis a in his Eq. Note that the same interpolation function S is used in 2 and 6. Lewis gave examples using first-order linear interpolation in one and two dimensions. However, in Lewis a, b and Langdon b, , there was no restriction to these weights. Zero-order interpolation NGP in which S is a discontinuous function is not suitable because the gradient in 6 does not exist. We said the second sum in 5 vanishes; this is true if Problem a j j where 1 and 2 refer to two different density distributions and their corresponding potentials.
The reciprocity result holds when 4 is the solution of a difference equation of the form with Aj, Amj 10 Problem b. The symmetry of A j , usually arises naturally in the formulation of Poisson's equation in a general curvilinear coordinate system. In a neutral plasma with periodic boundary conditions and a Poisson equation which is symmetric to reflection in the lattice planes, this symmetry of Ajmis ensured. The first equality follows from reflection; the second, from translation by the amount j m.
A is symmetric in Lewis' prescription for the Poisson difference equation 6 since the integral is invariant under interchange of subscripts j and j'. This latter property is very much less restrictive than 6 , and therefore, the energyconserving property is shared by a much wider class of algorithms than that derived by Lewis. These conditions affect the sign of the field energy Problem d.
If this ratio is positive, then the self-potential energy is nonnegative. The conclusions are: when there is reciprocity, as per 81, the force used in an energy-conserving code is identical to the negative gradient of the total field energy. The discussion of 3 shows also that reciprocity is required.
This means that there are jumps as a particle moves through a cell boundary, leading to enhanced noise and self-heating, just as with NGP in momentum-conserving programs. Generalize this to two dimensions. This result applies to the energy in most field solution methods, while Problem Sa is specific to Lagrangian formulations.
Since the energy-conserving property applies exactly only when the time integration is exact for the particle equations of motions, we assume time is continuous. We therefore write the potential as where gj,,, is the Green's function for the difference Poisson's equation.
Any fixed charge density can be either included in p and regarded as due to infinitely massive particles or regarded as a contributor to 4 j , e x t. By analogy with real electrostatic field theory, we expect that the potential energy of the system due to the fields of the particles is see Jackson, , p. Let us see when this is true. To obtain an energy-conserving system, therefore, we want the second sum on the right-hand side to vanish. An alternate proof of the sufficiency of reciprocity for energy conservation was given in Section The relevance of symmetry is indicated by Problem b.
Show that the particle accelerates in its own field, gaining kinetic energy while the field energy is constant zero. This representation has a finite set of variables, and the usual variational principle provides equations governing these variables. To avoid notational complexity, we retain vector coordinates instead of "generalized coordinates and specialize to electrostatic fields. Lewis, , treats formally the case of generalized coordinates and the full electromagnetic field.
In rationalized cgs Heaviside-Lorentz units Punofsky and Phillips, , p. With the representation 21, we find Equation 5 is the usual equation of motion with the same force term as in 6. It is the gradient of the interpolated potential, rather than the interpolated first difference of the potential. The connection of this feature to the existence of a conserved energy is shown in Section Equation 6 is the same, in our notation, as eq.
In one dimension and with linear S, we recover from 6 the simplest difference approximation to the Poisson equation. However, in two or three dimensions the resulting Poisson difference equation is not familiar. We find in Section that 6 produces the correct cold-plasma oscillation frequency at any wavelength! It automatically compensates for the increase in smoothing as one goes to higher-order splines. It is tempting to think that these simulation algorithms ought to be optimal in some sense.
To answer that, one must decide what properties of the simulation are to be accurate, and then go outside the variational principle to the methods of Chapter 8 to analyze and adjust the algorithm. For example, we see in Section that for warm plasmas, 6 does not give the most accurate oscillation frequencies.
This is because a warm plasma responds less to short-wavelength noise than to errors at long and medium wavelengths. The variational principle cannot "know" this. In systems where such analysis is difficult or the better algorithms are difficult to implement e. This proof can be adapted to general coordinate systems. The nonnegative property and symmetry of the Poisson operator facilitate numerical solution of 6. Compare 3 and 4. Comparing to 7 and 89 15 , we see that 1 is formally the same, differing only in the definition of K.
Note K 2 2 0, so the field energy is nonnegative. To see this, we use the results of the last section to relate the transforms of the particle density and force field - iq2kS k CS kp n kp P Suppose an interpolation is used which is free of aliasing.
If S k is not constant in the first zone, so that there are errors in the interpolation, Lewis' Poisson algorithm makes compensating errors in calculating 4 to yield good overall accuracy. This would be important to the practical realization of high-accuracy algorithms, since if S k is constant in the first zone, then S x drops off very slowly with increasing x and also does not remain positive.
However, band-limited interpolation is not very practical in plasma simulation, and if it were, the momentum-conserving algorithm would also conserve energy and could be made as accurate. Lindman private communication observed that small oscillations of a cold, nondrifting plasma in a linear-weighting Lewis model occur at exactly the correct frequency except for time-integration errors.
This interesting observation is true for any weighting function S , and in one, two, or three dimensions, as shown in Section Here is an instance in which the variational principle does as well as can be done. However, this turns out to be an exceptional case, as seen in Section As a measure of accuracy in realistic cases, we examine the "averaged force" Fo k , defined in Section Imagine holding the particles fixed while displacing not rotating the grid.
Then Fo x is the average of F x over all such displacements. We make use of it in Sections and in discussing two examples. This is done both with and without use of Fourier transforms. We are left with simply independent of k. This is the correct result, having no error due to finite Ax. Although this derivation is very short [once 1 has been derived] it is instructive to repeat the derivation ab initio without using Fourier transforms. The meaning of linearization and the fluid limit is clarified, as is the nature of the oscillations.
For brevity the discussion is kept to one dimension; the generalization is trivial. We assume that the unperturbed particle positions xi0 are equally spaced, and there is an integer number of particles per cell.
They are neutralized by a fixed background. After perturbing the particle positions by xil, This Taylor expansion of 2 is where linearization first enters. We now differentiate this twice in time. The acceleration x i l is given by 6 , but evaluated at x-i, linearization again.
Rearranging the sum, we have Assuming that the number of particles per cell, noAx, is large, the particle sum may be replaced by an integral. Then, comparing with 6 , we have simply Each p , oscillates at the plasma frequency and independently of the others. Alternatively the 4 j can oscillate independently. This can be understood by working through the linear case with only one pi or 4 j oscillating. The oscillations of particles in the same cell are not independent.
If the kinetic energy is evaluated as in ESl Section in a linearweighting model, placing the particles so that grid points fall between them, and keeping oscillation amplitudes low enough that particles do not cross grid points, then total energy is conserved to within roundoff error Problem e.
However, this is a very special situation! The derivation breaks down if the particles have any drift motion. In this case the dispersion relation 1 may be used. Some features of this case are discussed in Section and in Langdon b. We now show how this failure is associated with aliasing. Both are manifestations of the nonuniformity of the system dynamics.
When the Lagrangian is not invariant under displacement, momentum is not conserved in general. Thus the total particle force is zero and momentum is conserved. This is essentially because the particles can no longer sense the positions of the grid points; the nonuniformity of the grid is removed from the dynamics. We examine this self-force later, but let us assume for the present that we are not interested in such a force error on the microscopic level perhaps because it averages to zero unless there is some macroscopic manifestation.
We now give two examples in which a large change may take place in the total momentum. A dramatic failure of momentum conservation is illustrated in Figure a and Figure b. Instability is predicted and observed in a cold plasma drifting through the grid with a fixed neutralizing background; see Section and Problem a.
There need not be two or more plasma components drifting relative to each other. Clearly this instability is not physically valid; its origin is in aliasing errors. Let us divide the energy into three nonnegative parts: kinetic energy associated with the mean motion, kinetic energy of motion relative to the mean, and field energy.
The sum of these can be made to remain as nearly constant as desired, by decreasing the time step. As the instability develops, the latter two contributions to the total energy both increase, so the first contribution decreases. Therefore, the mean velocity and momentum must be decreasing. The force errors produce a drag on the mean motion.
A cold beam passing through a fixed uniform neutralizing background is made unstable by the grid via aliasing. Later behavior is more affected by the small number of particles used The range of variation in the total energy is 0.
From Langdon, Collisions also produce a drag leading to decreasing momentum, as predicted for a warm, uniform, stable plasma drifting through the grid Section The loss of energy of mean motion is compensated for by an increase in temperature. We do not claim to have shown that the lack of momentum conservation is necessarily damaging in practice, but only that it can have macroscopically visible consequences.
The initial drift velocity was 0. The two are coupled by the grid and grow quickly together a. The distribution after saturation shows little structure b. This has been observed in numerical solutions of the exact including aliasing dispersion relations Section In this section we show that the aliasing errors can be fourth or fifth order in A x , in the linear-weighting case.
Therefore, the dispersion errors at long wavelengths can be dominated by second-order errors in FO if the variational principle is used, as discussed at the end of Section The effects of finite grid spacing are treated exactly. For simplicity, we work in one dimension.
We now examine the difference between E and eo, due to the aliasing terms. On the other hand, with YO f 0 these terms can become destabilizing. First, we note that F is discontinuous. In one dimension, F is a step function Figure a.
Thus this case may be expected to be noisier than the common linear-weighting algorithms, but the overall computational time is shorter because the expression for F is much simpler. It is also difficult to integrate in time accurately enough to realize an improvement in energy conservation. In empirical studies Lewis et al. Since a particle has many neighbors, it is more significant that the two-particle interaction force is nonconservative.
However, the single particle case is simple and of some interest. Consider a single particle in a large one-dimensional system, using linear interpolation. Let us attempt to assess its importance in a single-species plasma. The well-depth energy Wself may be compared to the thermal energy: where m is the particle mass, v, is the rms thermal velocity, and A D is the Debye length. Similar results are anticipated in two and three dimensions Problem la.
Furthermore, since other particles in the cell contribute forces comparable to a particle's self-force, the latter becomes relatively small compared to normal many-body interactions. We have so far ignored time-integration errors, in effect, assuming the time step A t is kept negligibly small. If A t is held constant while A x is decreased then, when uselfAt 2 2, the self-force oscillation becomes unstable.
A similar argument may be made in two and three dimensions. Thus the time-integration errors appear to increase the significance of the self-force, in this limit, but not disastrously. One might try to restore momentum conservation by adding a new force to each particle which cancels its self-force. The unmodified algorithm correctly calculates no forces. To see that no such approach succeeds, note that in the Vlasov limit, the self-acceleration vanishes and the system is the same as without the selfforce cancellation.
The discussion of Section shows that momentum is not conserved in the Vlasov limit. Nor does any smoothing of p or 4 restore momentum conservation. As an important example, we now consider oscillations of a warm plasma, using the accuracy of the period and rate of decay or growth as a measure of the accuracy of the fields. Therefore, a lower-order error in FO reduces the order of overall accuracy. One way to do this is to solve the same Poisson equation but with as the source density. One is then left with an 0 Ax4 error in FO and from aliasing terms, resulting in a fourth-order error in o k.
These remarks hold also in two or three dimensions. The Poisson algorithms obtained from the variational principle are not optimal from the present point of view. Whether they are optimal in some other situation apart from the singular case of cold-plasma oscillations remains to be shown.
There should be no surprise that problems arise with the variational principle when the basis functions are an incomplete set. Many other such examples are known, e. The frequency uselfis only an approximate indication of the average force gradient, since a single-particle oscillation is no longer simple harmonic. This is all that is needed, if Fourier transform methods are to be used in the simulation.
The new factor in 1 is added by the variational derivation in order to compensate for the low-pass filtering smoothing effect of S2 as compared to SI. These emerge naturally from Eq. The self-force in this example scales as in 1 and 2 , but with smaller coefficients.
In general, the deficiencies due to aliasing the instability, momentum nonconservation, grid noise are reduced. Again one achieves best accuracy in the complex frequency o k with a Poisson algorithm different than that specified by the variational principle. PROBLEM a Equation 2 leads to a matrix equation in which the nonzero elements are all in a band, five elements across, down the diagonal, plus a few in the other corners.
However, 1 shows how to solve two tridiagonal systems instead, each corresponding to factors of K2. Show that the corresponding difference equations can be written as Is such a factorization possible in two dimensions? Nevins and A. We begin with a little history and an outline of this chapter.
The first method considered by Dawson and co-workers for twodimensional simulation employed a truncated Fourier series expansion of the field, evaluated at each particle position. In applications requiring much spatial detail Le. Instead, the field and its derivatives are evaluated at a number of spatial grid points many fewer than the number of particles , and the field at a particle is evaluated as a truncated Taylor series expansion about the nearest grid point. They interpret this expansion in terms of multipole moments.
In order to reduce storage requirements, the "subtracted multipole" method evaluates only the field from the Fourier sum; its derivatives are evaluated using finite differences. This algorithm is easily expressed in the formulations of Chapter 8; doing so facilitates comparison with other methods.
The multipole method is derived in Section in its original form, and in Section in the "subtracted form. In Section we show how to derive the standard one-dimensional linear weighting as a dipole scheme, and derive a new two-dimensional dipole scheme. Adding the quadrupole xy moment yields the familiar bilinear or area weighting!
Compared to published dipole algorithms, those derived here give smoother spatial variation for comparable resolution. These are used in Section to examine overall accuracy. The predecessor of the multipole method is a field algorithm in which the particle is considered to be a finite-size cloud and the field is represented by a truncated Fourier series Dawson, , p.
This sum must be evaluated for each particle. This method provides smooth variation of the fields but the computational effort per particle increases as spatial resolution and therefore the number of Fourier modes increases. To speed up the force calculation, a spatial grid is introduced. The particle force is evaluated by Taylor expansion of the exponential in I : where Xj is the location of the grid point nearest the particle x i.
The force on each particle is evaluated as the sum 4. The whole scheme is illustrated in Figure a. These require a total of 2 M 1 real transforms, normally done as pairs of complex transforms Appendix A. When the particle coordinates reside in secondary storage, such as a rotating magnetic disk, one prefers to collect the new densities as the particles are advanced to their new positions, rather than going through the particle list twice per time step.
In this case, both the old force and the new density must be in fast memory, so we must store 2 M 1 quantities per grid point. In one dimension this is much easier than in two, where F and k become vectors. In practice, multipole codes have often been used in the monopole i. Let us now examine the spatial variation of the force field using the dipole approximation, as compared to a standard linear-interpolation method.
In the latter case, the force appears as in Figure ll-2b, continuous and piecewise linear. In the multipole expansion method, the force is given by a truncated Taylor series 4 , expanding about the nearest grid point. Accuracy is good near the grid point and degrades rapidly toward the midpoint between cells. Furthermore, after the particle crosses the midpoint, the expansion is about a new grid point. Hence the force jumps discontinuously at the midpoint, as in Figure c.
A deleterious effect of this discontinuity is increased aliasing errors, as we see in Sections and In practice, the magnitude olf this jump Problem c is decreased by choosing the parameter CI in S to be Ax or larger.
For a given Fourier mode, this does not decrease the size of the jump relative to the force itself. Rather, both are suppressed at short wavelenghths, at some cost in resolution.
In this approach, the derivatives of the force at the grid points are formed by using a finite-difference operator on the grid. Hence, one need only calculate and store the force at each grid point. The force may be obtained by finitedifference operation on the potential. The multipole densities are combined into a charge density by difference operators symmetric to those used for the force.
Of the multipole schemes, a subtracted dipole expansion SDPE has been most used. The dipole moment is now Aq 2Ax , and must equal q x, - X,. Find S X,,l - x, by comparing to 3 rewritten asp,. The answers agree with 5. Which is more efficient? Next we derive an improved twodimensional dipole algorithm by making an expansion about the nearest cell center. Then we show that area weighting is a dipole scheme which also includes the xy quadrupoIe moment.
If the multipole interpretation is valued, one may use standard linear and area weighting with a clear conscience. We conclude with some opinions on the design of multipole algorithms which optimize accuracy and computational demands. Let us construct a monopole plus dipole with charges at two grid points instead of three. For a charge q at position x between grid points j and j 1, the monopole is constructed by placing half the charge at each grid point. This is as smooth as the improved subtracted dipole scheme and can be evaluated just as quickly Problem c.
In two dimensions, form the monopole and two dipole moments as indicated in Figure ll-4a, centered in the middle of the cell. The smaller discontinuities in this weighting function, compared to that of Kruer et af. Underlying these observations are the relations given in 3 and Problem b, which follow from 4 and 8 - 5 6 ].
Any linear weighting can be regarded as a dipole method, in one, two, or three dimensions. Bilinear weighting includes, in addition, one quadrupole moment Problem b. Similarly, show that the particle force is a linear combination of q and qx in each cell, in both schemes.
Therefore, the computational expense of the SDPE and the standard linear scheme are the same, when both are optimally calculated and use equal storage. In Section , k is understood without saying so to be confined to the first Brillouin zone, i. This result is in the same form as 8 - 7 0 4 with 9 , with K given by 1 and the addition of the spatial smoothing Hence, it is possible to draw on other results in Chapter 8, such as the discussion of momentum conservation Problem 1 1 -5a.
If we could keep all moments, then Problem ll-5b s. The results of this section readily generalize to two and three dimensions Problem ll-5e.
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