A direct implicit particle-in-cell (PIC) simulation model with full electromagnetic (EM) effects has been implemented in 2-D Cartesian geometry. The model, implemented with the D/sub 1/ time differencing scheme, was first implemented in a 1-D electrostatic (ES) version to gain some experience with spatial differencing in forms suitable for extension to the full EM field in two dimensions. The implicit EM field solve is considerably different from the implicit ES code. The EM field calculation requires an inductive part as well as the electrostatic and the B field must be self-consistently advanced

Hewett, D.W.

Langdon, A.B.

Hewett, D. W.

Langdon, A. B.

UNT Digital Library

T ~ \ x~ RECEIVES BY rit l A U B 0 9 Wi UCRL-92041 PREPRINT / SOLUTION OF THE FIELD EQUATIONS FOR 2-D ELECTROMAGNETIC DIRECT IMPLICIT PLASMA SIMULATION D. W. HEUETT A. B. LANGDON UCRL—92041 DE85 016073 This paper was prepared fo r the Eleventh In te r ­nat ional Conference on Numerical Simulat ion of Plasmas held in Montreal , Canada, June 25-27, 1985. January 2, 1985 This is a preprint of a paper intended for publication in a journal or proceedings, ! changes may be made before publication, this preprint is made available with the understanding thai it will not be cited or reproduced without the permission of the author. DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thr-ieof, nor any of their employees, makes any warranty, express or imp'ied, or assumes any legal liability or responsi­bility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights, refer­ence herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recom­mendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those or the United States Government or any agency thereof. MASTER DISTRIBUTION OF THIS DOCUMENT IS UNLIMITED Solution of the Field Equations for 2-D Electromagnetic Direct Implicit Plasma Simulation D. W. Hewett, A. B. Longdon X-Division, LLNL, Livermore CA 94550 '* A direct implicit par t icl e-i n-ce I I (PIC) simulation model with full electromagnetic (EM) effects has been implemented in 2—D Cartesian geometry. The model, implemented with the D, time differencing scheme[l], was first implemented in a 1—D electrostatic (ES) version to gain some experience with spatial differencing in forms suitable for extension to the full EM field in two dimensions[2]. The implicit EM field solve is considerably different from the implicit ES code. The EM £ field calculation requires an inductive part as well as the electrostatic and the B field must be seIf-consistently advanced. The field equations, with D, temporal differencing, can be e'xpressed by the following equations. £n+1 " En " c A t * * B n + i + «**l i„ +J (la) B n + i - e n_j = -cAt VxE n (1b) In = i( En + 1 +In-1 ) O O V - E n + 1 = 4 n p n + 1 (1d) The particle source terms p and Ĵ  are obtained in two steps: first we advance the particles to the ~ level using everything BUT the advanced ,E_ field required by the implicit method. We now accumulate charge and current densities by summing over particles to get the ~ level source terms for the field solve. In addition the two tensors ^ and £ are so constructed such that the total charge and current densities, p n + 1 ond jJ n +±. can be recovered by 4 P„ + , ( ^ ) = P n + 1 ( X j - V.*.E n +1 (2a) A U ^ i Q ^ ) = Atf^jQ^) + X'En + 1 - VX£.E n +1 (2b) D e t a i l s o f t h i s p r o c e d u r e can be found in Langdon and B a r n e s [ 3 ] . The f i e l d e q u a t i o n s are s o l v e d by s u b s t i t u t i n g these exp ress i ons f o r pn+« ond j i n+1 ' n * n e f ' e ' d e q u a t i o n s i n e q s . 1 . R e a r r a n g i n g , we have This work v/as performed under the auspices of the U. S. Department of Energy by the Lawrence Livermore National Laboratory under Controct No. W-7405-Eng-48 H e w e t t . Langdon 2 £ - A[V2£ - X(X-£) ] + 4fr(x-E + £><£•£) = ft fi = £ n - 4rrAt 2 + cAt Vxg^j - A XX£x£n_, A = c z A t 2 / 2 Z - U + 4t r jc ) -£ n + 1 = 4 r r p n + 1 I n - } <!„_, + £) £ n + i = 6 n 4 - cAt Vx| n (3o) (3b) (3c ) (3d) where t he £ w i t h o u t s u b s c r i p t i s £ n + 1 . The e q u a t i o n s f o r £ gove rn the t ime e v o l u t i o n of b o t h the e l e c t r o s t a t i c and i n d u c t i v e p a r t s o f the e l e c t r i c f i e l d — p r o v i d e d we guaran tee the cons i s tency r e l a t i o n e q . (1d) i s s a t i s f i e d . The r e d u c t i o n o f the f i e l d a l g o r i t h m to a f i n i t e d i f f e r e n c e p r o c e d u r e ' o n a mesh has severa l p i t f a l l s which can erode the u t i J i t y and r o b u s t n e s s o f the c o d e . The r e l a t i v e p o s i t i o n s on the mesh a t wh ich e v a l u a t i o n s are made can s i g n i f i c a n t l y e f f e c t the numer ica l d i f f u s i o n t h a t t h e r e s u l t i n g a l g o r i t h m e x h i b i t s . Almost as i m p o r t a n t , p o s i t i o n i n g o f t h e f i e l d and sou rce components can determine the e f f o r t r e q u i r e d to imp lement and m o d i f y v a r i o u s c h o i c e s f o r boundary c o n d i t i o n s . P r e v i o u s ID ES s t u d i e s [ 2 ] p r o v i d e gu idance fo r the s e l e c t i o n of l o c a t i o n s fo r q u a n t i t i e s l i k e l y to dominate when EM e f f e c t s are less i m p o r t a n t . We hove p r e s e n t l y chosen the s to rage l o c a t i o n s j+ i W J i i+i where the • implies all components and i+1 Hewett, Langdon 3 y k = Vmin + k (y„.ax-ymin)/ N y • Quantities not specified are known at cell corners—integer j,k values. The solution of eq. 3a is the most interesting part of the field solve. The other field equations, 3b-3d. can simply be evaluated point by point once £ n + 1 is known. Eq. 3a involves a coupling of E_ components and plasma source terms that has proven difficult to solve if not properly arranged. 1-D tests, for example, quickly revealed the advantage of solving all components simultaneously with a linear system solver. Experimentation with Picard iteration—solving for each component in turn using the most recently obtained values for the other components—not only exhibited generally slow convergence but also gave additional restrictive constraints on At. For w c e At > 2, simple Picard iteration does not converge. The 1—D results suggest that a simultaneous solution is probably essential in 2-D as well. Since a direct solve would require far too much storage, an iterative solution has been implememted. We have achieved good results with an alternating direction method which solves all three component equations simultaneously in each direction pass-. As in 1-D no new constraints on At emerge and the convergence is remarkably rapid in those cases in which the j£ coefficients dominate. Even if the plasma terms are small, convergence is still more than acceptable if the terms with A coefficients are comparable to other coefficients in eqn 3a. For example, typical problems on a 32X32 mesh can easily be iterated to residue-to-field ratios of 1 0 - 4 for less than 10% of the particle pushing cost assuming 15 particles/cell. The desired boundary conditions must be built into the coefficients sent to the linear system solver. In general, we allow field values needed at each boundary point to be unknowns and define boundary value equations for each such that boundary point and nearest interior neighbor are coupled to the desired boundary condition. For example. 01 E y(j=0,k-4) + a2 E y(j=1,k+1) = a3(k+i) (6) provides a Dirichlet condition on E„(x_- ) for a1=1, a2=0 and given r Z * m 1 n ' 3 a3(y) as the desired boundary value as a function of y at x mj Other choices allow easy imposition of Neumann conditions and both Dirichlet and Neumann conditions can be implemented for either integer or half-integer mesh storage. (Periodic boundary conditions do not fit into this scheme. A separate linear system solver has been constructed ?,o provide this capability.) Hewett, Langdon 4 It is frequently useful to have the capability to run with EM wove boundary conditions. If we wish to consider a TM wave incident on the simulated plasma from the region to the left of x m } for example, we need a relation to connect E and B z at x m i n at the advanced time (because the field calculation is also implicit). The desired relation is 2V Ey j=0 n+1 + E y j=0 J = { B 2 d n + i - [ i(B z j =_i n + i + B z j 4 n + i ) - B 2 b n + i ] } (7) right-traveling left-traveling reflected rf speci f ied i nput where Bz j = _ j n + £ , B z j =^ n +^ and E y . 0 n + 1 have not yet been determined. B l d (t) and B z b (t) are the specified rf input and bias field level as functions of time respectively. To use eq. 7 os a boundary condition for the E field calculation eq. 3a, we need to eliminate these unknown B z quantities with expressions for known — quantities and earlier field quantities. Using Ampere's law eq. 1a to eliminate B z j__i n + i and Faraday's law eq. 1b to eliminate B •_i +i and assuming no plasma currents at x ; , we have [(1 - i/"1 - v) E y j = 0 n + 1 + v E y j = 1 n + 1] = 2 B z d - (1 + "" 1) Ey J-O n + 2 Bzb - * (E y j = 1 „_, " E y j = 0 „-i) (8) where v = c dt/dx. This relation is in the form of eq. 6 as our algorithm requires. Examples of the use of this relation will be presented. References 1. A.B. Langdon, "Electromagnetic Direct Implicit PIC Simulation", UCRL-88979, March 1983. 2. A.B. Langdon, D.W. Hewett, and A. Friedman, LLNL Laser Fusion Annual Report 1983. 3. A.B. Langdon ond D.C. Barnes. "Direct Implicit Plasma Simulation", a chapter in the volume "Multiple Time Scales" in the series Computational Techniques, Academic Press, 1984. 

Solution of the field equations for 2-D electromagnetic direct implicit plasma simulation

Abstract

Similar works

Full text

Available Versions

UNT Digital Library