Nuclear Instruments and Methods in Physics Research A

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1 Nuclear Instruments and Methods in Physics Research A 645 (20) Contents lists available at ScienceDirect Nuclear Instruments and Methods in Physics Research A journal homeage: The fast multiole method in the differential algebra framework He Zhang, Martin Berz Deartment of Physics and Astronomy, Michigan State University, East Lansing, MI , USA article info Available online 2 January 20 Keywords: Sace charge effect Fast multiole method Differential algebra abstract A method is resented that allows the comutation of sace charge effects of arbitrary and large distributions of articles in an efficient and accurate way based on a variant of the Fast Multiole Method (FMM). It relies on an automatic multigrid-based decomosition of charges in near and far regions and the use of high-order differential algebra methods to obtain decomositions of far fields that lead to an error that scales with a high ower of the order. Given an ensemble of N articles, the method allows the comutation of the self-fields of all articles on each other with a comutational exense that scales as O(N). Using remainder-enhanced DA methods, it is also ossible to obtain rigorous estimates of the errors of the methods. Furthermore, the method allows the comutation of all high-order multioles of the sace charge fields that are necessary for the comutation of high-order transfer mas and all resulting aberrations. & 20 Elsevier B.V. All rights reserved.. Introduction The Coulomb interaction between the articles inside a bunch is one of the most imortant collective effects in the study of beam dynamics. This effect becomes more significant when the article beam has lower energy such as in contemlated time resolved electric microscoes, or has higher density such as in modern high brilliance article accelerators or free electron laser devices [,2]. Numerical simulation is usually necessary to study this effect. If the Coulomb interaction is calculated directly via airwise interaction, the comutational time scales with the square of the article number N, which makes it ractically difficult to simulate a large number of articles. More efficient methods, such as the article article interaction (PPI) method, the article in cell (PIC) method, the tree code and the fast multiole method (FMM), have been ut forward and widely used in many simulation codes. The PPI method uses macroarticles, each of which reresents a grou of real articles. The field on each macroarticle is calculated according to the charge distribution of the whole bunch. The macroarticle reresentation entails an inevitable modeling error. Furthermore, a secific distribution is assumed, for examle an ellitical Gaussian distribution [3], since otherwise one has to use some comlicated algorithm to fit the real charge distribution; see for examle [4 6]. On the other hand, the PIC method reassigns the charge density onto a reviously chosen mesh, solves the Poisson equation to calculate the field on the mesh oints, from which the field on Corresonding author. address: zhanghe@msu.edu (H. Zhang). each article is calculated by the interolation; see for examle [7 9]. The PIC method naturally suffers from comutational error due to statistics, and it is very difficult to calculate the derivatives of the field by this method, which are required in the comutation of high-order mas by differential algebraic methods. However, significant breakthroughs have haened outside the beam hysics regarding the efficiency of the direct field comutation via Coulomb s law. The year 986 brought the ublication of the tree code algorithm, in which the otentials of articles far away from the observer are reresented by multiole exansions in owers of /r, covering larger and larger boxes further and further away from the oint of interest. It could be shown that the cost of the method scales with O(N log N) [0,]. In 987, the FMM was ublished, in which the multiole exansions are converted into the local exansions in the near region of the observer, and the efficiency is further increased to O(N) [2,3]. In the original FMM, the three dimensional otential of a oint charge is usually exanded in terms of sherical harmonic functions or exonential functions in sherical coordinates around suitable nearby exansion oints [4]. In 2006, an accelerated Cartesian exansions FMM algorithm for any otential of the form r v with ositive v were resented [5]. In the following we will resent a formulation of the FMM algorithm in the differential algebra (DA) framework. We also use the Cartesian exansions, and we will see that the DA formalism considerably simlifies the necessary algorithms, also for otentials differing from /r, and in articular those associated with common macroarticle aroaches for the avoidance of collisions. Furthermore, the method allows the comutation of all derivatives of the otential in a natural way u to a redetermined /$ - see front matter & 20 Elsevier B.V. All rights reserved. doi:0.06/j.nima

2 H. Zhang, M. Berz / Nuclear Instruments and Methods in Physics Research A 645 (20) order, and thus allows the comutation of high-order transfer mas and aberrations. 2. The algorithm of the FMM The key idea of the FMM is to reresent the otential of suitable grous of source articles that are far away from the observer in terms of exansions involving owers of /r, which we refer to as a far multiole exansion, and utilize that far enough away, higher owers of /r are less and less significant. This makes it ossible to comute the action of source articles on observer articles in a more efficient manner. Furthermore, far multiole exansions corresonding to certain grous can be translated and combined. And finally, far multioles can also be locally exanded involving owers of r, which we call a local exansion. This local exansion thus allows the treatment of grous of nearby observer articles in a combined manner. In ractice, one first encloses all the articles in a grou of cube boxes, the zero level boxes. Then each of these cube boxes is cut into eight equal small cube boxes, which we call the first level boxes. Then each first level box is cut in the same way into eight smaller boxes, leading to the second level boxes. This rocess is continued until a re-secified level is reached. In ractice, the number of levels is determined such that the average number of articles in the finest boxes is near a re-secified value, the size of which will affect the efficiency of the method. Fig.. Sequential cutting of an original box. Fig. 2. The roagation of the far multiole exansions. We call boxes of the same level neighbors if they touch. For a given box A, we denote the region made of all same level neighbors and A itself as the near region of A, and everything else as the far region A. For a box A, a next higher level box B containing A is called a arent box of A, and A is called one of the child boxes of B. Aarently each child box has only one arent box, and each arent box has eight child boxes. A 2D examle about how to cut the boxes is shown in Fig.. In the first level, the square box is cut into four square boxes, and in the second level, 6 boxes are cut out, and so on. Furthermore, boxes 2 through 9 are all neighbors of box, and together they describe the near region of. All the unnumbered boxes are in the far region of the box. There is no difference between the 2D case and the 3D case in rincile, excet that each box is cut into four child boxes in the 2D case rather than eight in the 3D case. The FMM algorithm consists of two arts, the first of which we now describe:. Cut the box of interest into the desired levels. 2. In each box of the finest level, calculate the far multiole exansion of the articles inside that box around the center of the box. 3. From the finest level to the second finest level, translate the multiole exansions from the center of the children boxes into the center of their arent box, and then add them u to get the far multiole exansion of the arent box, as Fig. 2 shows. 4. Reeat this at all levels. After this art is comleted, we now have a far multiole exansion for each box in each level. We now roceed to the second art of the FMM. First, for any box A having a arent B, we define the interaction list to be the collection of those boxes of the same level that belong to the far region of A, but to the near region of its arent B. These boxes of medium distance to A will be imortant in the subsequent algorithm because they require secial treatment. To illustrate the concet of the interaction list, consider Fig. 3, which shows a box of interest in black, its near region in white, and its interaction list in light gray, for three different levels. The second art of the FMM algorithm now begins at the first level at which boxes have interaction lists.. For each box with an interaction list, comute a local exansion around its center oint from the far exansions of the boxes making u the interaction list. 2. At the next finer level, again comute a local exansion around the center of each box from the far exansion of the boxes making u the interaction list. To this, add the local exansion Fig. 3. The behavior of the interaction list of boxes at three levels.

3 340 H. Zhang, M. Berz / Nuclear Instruments and Methods in Physics Research A 645 (20) of the arent s interaction list by re-exanding the arent s local exansion to its own center, as shown in Fig Proceed until the lowest level is reached. After the comletion of the second ste, each box at the finest level now ossesses a local exansion of all contributions to the otential from outside its near region. The gradient of this local exansion consequently reresents the field contributions from everything in the far region. The third ste of the method now consists of determining the field on each article of interest.. For each article, determine the finest level box to which the article belongs, and evaluate the local exansion of the field of that box at the article s coordinates. This results in an aroximation of the field at the article due to all other articles outside the near region of the box in which the article lies. 2. To the far field, exlicitly add the Coulomb field of all other articles in the near region. We note that the accuracy of this aroximation deends on the orders used for the far exansions and the local exansions that occur, and in rincile can be made as high as desired. In the remainder of the aer we will discuss how to erform all the relevant far and local exansions using the differential algebraic methods. 3. A brief review of DA and COSY Before we continue to resent our work on the FMM in the DA framework, we want to give a brief introduction of the DA method. The basic concets and the deductions will be resented directly without roof, lease refer to Ref. [6] for details. Consider the vector sace of the infinitely differentiable functions C ðr v Þ, in which we can define an equivalence relation ¼ n between two functions a,bac ðr v Þ via a¼ n b if a(0)¼b(0) and if all the artial derivatives of a and b at 0 agree u to the order n. Note that the oint 0 is selected for convenience, and any other oint could be chosen as well. The set of all b that satisfies b¼ n a is called the equivalence class of a, which is denoted by [a] n.we denote all the equivalence classes with resect to ¼ n on C ðr v Þ as n D v. The addition, scalar multilication and multilication on n D v can be defined as Eq. (). ½aŠ n þ½bš n :¼½aþbŠ n c ½aŠ n :¼½c aš n Fig. 4. The translation of the local exansions. ½aŠ n ½bŠ n :¼½abŠ n ðþ where a,ba n D v and c is a scalar, so that n D v is an algebra. We can also define the derivation v as Eq. v ½aŠ n a v n where x v is the vth variable of the function a. The v v ð½aš½bšþ ¼ ½aŠð@ v ½bŠÞþð@ v ½aŠÞ ½bŠ: ð3þ An algebra with a derivation is called a differential algebra. There are v secial classes d v ¼[x v ], whose elements are all infinitely small. According to the fixed oint theorem [6], the inverse and the roots of any element that is not infinitely small in n D v exist and can be calculated easily. Further more, all real ower series can be extend to the DA within their radius of convergence. If a function a in n D v has all the derivatives c ¼ þ þ Jv Jv v, then [a] can be written as ½aŠ¼ X c J,...,Jv dj... d Jv v : ð4þ Thus d J... d Jv v is a basis of the vector sace of n D v. The Eq. (4) reminds us of the Taylor exansion of a function. Actually if we have a function f in C ðr v Þ and f T is its Taylor exansion u to order n, obviously we have f¼ n f T in n D v. In ractice this means we can exress f by its Taylor exansion u to an arbitrary order n as an element in n D v, and we can calculate the derivative classes of f and any other function that can be derived by alying the elemental oerations, divisions, roots, and ower series on f [7]. A beam otical system can be described in terms of the transfer ma method, which relates the final ositions and velocities of the articles to the initial conditions, so that it makes the tracking more efficient than solving the dynamic equations element by element. The ower of the DA make it ossible to calculate the ma u to any arbitrary order [8]. COSY Infinity s a rogram develoed for high erformance modern scientific comuting, which is used in beam otical system design. COSY suorts various advanced data tyes, such as DA, TM and VE [9]. In our following work on the FMM, we will use the DA data tye. By evaluating a function f in DA data tye in COSY, one can get its Taylor exansion f T u to an arbitrary redetermined order n automatically, because a DA vector carries all its coefficients c, which are exactly the coefficients of f J,...,Jv T.If we consider two functions f,g A n D v which can also be viewed as two mas M f and M g, the function g(f) or the comosition of the two mas M f 3M g can be easily calculated by the command POLVAL in COSY. If f is a linear function or a linear ma, a more efficient command DATRN can be used [9]. Furthermore, the TM data tye can be used for the rigorous calculation. In the future we will use the TM in our algorithm to get rigorous error bounds [20]. COSY also suorts arallel calculation. Our current code can be easily revised for the arallel calculation and run in a cluster machine. 4. FMM in the DA framework 4.. The far multiole exansion Given a unit oint charge located at osition ~r i, its otential at the location ~r of an observer is given by f ¼ j~r ~r i j : ð5þ The key idea of the FFM method lies in grouing together articles that are sufficiently near and treat them with a combined exansion. As it turns out, it is not immediately obvious what the best choice of variables is to achieve this end. It is known that using sherical coordinates, f can be exressed as series of the sherical harmonic functions, which are comosed of roducts of cos y, e ic and /r [4]. The desire to oerate directly in Cartesian coordinates ultimately leads to the insiration of exanding the function with resect to /r, x/r 2, y/r 2, and z/r 2. As it turns out this

4 H. Zhang, M. Berz / Nuclear Instruments and Methods in Physics Research A 645 (20) is one variable more than necessary and contains redundancy because (x,y,z) already determine r. But the use of four variables leads to a articularly transarent algorithm. Back to the FMM, we consider a cube box centered at the origin, whose side length is a, and there are n articles inside the box whose coordinates are (x i,y i,z i ). At any oint (x,y,z), whose distance to the origin r ba, the otential can be exressed as f ¼ Xn i ¼ Xn i ¼ Xn i qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx i xþ 2 þðy i yþ 2 þðz i zþ 2 = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 2 þy 2 þz sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ x2 i þy2 i þz2 i x 2 þy 2 þz 2x ix 2 x 2 þy 2 þz 2y iy 2 x 2 þy 2 þz 2z iz 2 x 2 þy 2 þz 2 d qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þðx 2 i þy2 i þz2 i Þd2 2x id 2 2y i d 3 2z i d 4 with d ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ x 2 þy 2 þz 2 r, d x 2 ¼ x 2 þy 2 þz ¼ x 2 r 2 d 3 ¼ y x 2 þy 2 þz 2 ¼ y r 2, d 4 ¼ z x 2 þy 2 þz ¼ z 2 r : 2 If we consider oints (x,y,z) whose distance to the origin r ba, the exression above can aarently exanded in owers of the quantities d, d 2, d 3 and d 4. Since these quantities to go zero as r becomes larger and larger, the resulting exansion mathematically reresent a Laurent series, and in common hysical terminology a multiole exansion. To avoid confusion with the common definition of multiole exansions in article otics, we call this exansion the far multiole exansion if it is not clear from the context. Using the variables d, d 2, d 3 and d 4 as the exansion variables in the DA algorithm, COSY can now readily comute the far multiole exansions to any order automatically. Furthermore, quite imortantly, the same method can be alied to other tyes of the otential, for examle the Coulomb otential in other dimensions, the common van der Waals otentials, or the otentials of certain macroarticle charge distributions, and in the DA formalism, without much additional difficulty far multiole exansions can be calculated by mere evaluation in DA arithmetic. We now use the formula to calculate the multiole exansions in the finest level boxes. To assess the erformance of the method, we erformed some test calculations of the far multiole exansion. One grou of results are resented in Table. We ut 50 electrons into each of the two cube boxes, whose centers are (0,0,0) and (4,0,0) and whose side length are both 2. In terms of the terminology introduced above, this means the second box lies in the far region of the first and is hence subject to the far multiole exansion; but it reresents the closes such box, where thus the convergence of the method is articularly difficult to achieve. Table The relative errors of the otential. DA order Error ð6þ The otential on the electrons in the second box is calculated and reeated for 000 times with different random numbers, so that we have 50,000 data oints together. The relative error is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P calculated by n i ¼ ðdf=fþ2 with n¼50,000. We can see that as the DA order increases, the relative error decreases, which shows the convergence of this multiole exansion Translation of a multiole exansion If we have a multiole exansion in the cube box centered at the origin (0,0,0), we can translate it into another frame whose origin is at ðxu o,yu o,zu o Þ. The distance from the observer to the origin of the new frame should be much larger than the side length of the box centered at ðxu o,yu o,zu o Þ, which assures that the new DA variables du...du 4 in Eq. (7) are small enough to still have convergence. Secifically, we have du ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðx xu o Þ 2 þðy yu o Þ 2 þðz zu o Þ 2 x u2 þy u2 þz u2 ru du 2 ¼ x xu o du 3 ¼ y yu o du 4 ¼ z zu o ¼ xu ¼ yu ¼ zu In Eq. (7), ðxu,yu,zuþ is the osition of the observer in the new frame of ðxu o,yu o zu o Þ, and ru is the distance from the observer to the new origin ðxu o,yu o,zu o Þ. The new DA variables and the old DA variables in Eq. (6) are related via d ¼ r ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx xu o þxu o Þ 2 þðy yu o þyu o Þ 2 þðz zu o þzu o Þ 2 ¼ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u t ðx xu o Þ 2 þðy yu o Þ 2 þðz z o Þ 2 þx o u 2 þy o u 2 þz o u 2 þ2ðx xu o Þxu o þ2ðy yu o Þyu o þ2ðz zu o Þzu o ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi =d u 2 þx o u 2 þy o u 2 þz o u 2 þ2xu o du 2 =d u 2 þ2yu o du 3 =d u 2 þ2zu o du 4 =d u 2 du ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þðx o u 2 þy o u 2 þz o u 2 Þd u2 þ2xu o du 2 þ2yu o du 3 þ2zu o du 4 ffiffiffi ¼ du R, d 2 ¼ xd 2 ¼ðx xu o þxu o Þd 2 ¼ðdu 2 þxu o d u2 ÞR d 3 ¼ yd 2 ¼ðy yu o þyu o Þd 2 ¼ðdu 3 þyu o d u2 ÞR d 4 ¼ zd 2 ¼ðz zu o þzu o Þd 2 ¼ðdu 4 þzu o d u2 ÞR with R ¼ : þðx o u 2 þy o u 2 þz o u 2 Þd u 2 þ2xu o du 2 þ2yu o du 3 þ2zu o du 4 If we substitute Eq. (8) into Eq. (6), we obtain the multiole exansion in the new frame. If one considers Eq. (6) as a ma M c2m (d,d 2,d 3,d 4 ) and Eq. (8) as a ma M ðdu,du 2,du 3,du 4 Þ, the substitution is actually merely the comosition of the two mas: M m2m ¼ M c2m 3M ð9þ which is a common oeration in DA methods, and which here yields the exression of f in the new frame with the new DA variables. With Eqs. (8) and (9), to calculate the multiole exansion in a box not in the finest level, we can translate all ð7þ ð8þ

5 342 H. Zhang, M. Berz / Nuclear Instruments and Methods in Physics Research A 645 (20) the multiole exansions of all its children boxes to its center, then take the summation Conversion of a far multiole exansion into a local exansion The otential of a far multiole exansion at the frame at (0,0,0) on an observer (x,y,z) can be converted into a local exansion in the near region of the observer. Since the new frame has its origin ðxu o,yu o,zu o Þ close to the observer, it is natural to choose the coordinates of the observer ðxu,yu,zuþ in the new frame as the new DA variables, as Eq. (2) shows: du ¼ xu du 2 ¼ yu du 3 ¼ zu: ð0þ So, in contrast to the far multiole exansion, which corresonds to a Laurent exansion, the local exansion corresonds to a conventional Taylor exansion. The old DA variables in Eq. (6) and the new DA variables in Eq. (2) have the relation as shown in Eq. (): d ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ffiffiffi R x 2 þy 2 þz 2 ðxu o þdu Þ 2 þðyu o þdu 2 Þ 2 þðzu o þdu 3 Þ 2 d 2 ¼ d 3 ¼ d 4 ¼ with x x 2 þy 2 þz 2 ¼ xu o þdu ðxu o þdu Þ 2 þðyu o þdu 2 Þ 2 þðzu o þdu 3 Þ 2 ¼ðxu o þdu ÞR y x 2 þy 2 þz 2 ¼ yu o þdu 2 ðxu o þdu Þ 2 þðyu o þdu 2 Þ 2 þðzu o þdu 3 Þ 2 ¼ðyu o þdu 2 ÞR z x 2 þy 2 þz 2 ¼ zu o þdu 3 ðxu o þdu Þ 2 þðyu o þdu 2 Þ 2 þðzu o þdu 3 Þ 2 ¼ðzu o þdu 3 ÞR: ðþ R ¼ ðxu o þdu Þ 2 þðyu o þdu 2 Þ 2 þðzu o þdu 3 Þ : 2 If we call Eq. () the ma M 2 ðdu, du 2, du 3 Þ, the local exansion can again be written as a comosition of M c2m and M 2 as M m2l ¼ M c2m 3M 2 : ð2þ Having Eqs. () and (2), we can convert the multioles in the interaction list of each box into the local exansion inside the box itself. The result of a test calculation to show the convergence of the local exansion is resented in Table 2. The set u of the boxes and electrons, the number of the data oints, and the definition the relative error are the same with those of Table. We can see that the relative error decreases as we increase the DA order Translation of a local exansion Consider two local frames whose origins are both close to the observer. If we know the local exansion of the frame at (0,0,0) is Table 2 The relative errors of the otential. DA order Error as Eq. (2) shows, we can translate it into the other local frame at ðxu o,yu o,zu o Þ. Since the new frame s origin is also close to the observer, we can still choose the coordinates of the observer in the new frame as the new DA variables du, du 2, and du 3. Obviously the old DA variables d, d 2, and d 3, which are the coordinates of the observer in the local frame at (0,0,0), have the simle relation shown in Eq. (3) with the new DA variables: d ¼ xu o þdu d 2 ¼ yu o þdu 2 d 3 ¼ zu o þdu 3 : ð3þ We call Eq. (3) the ma M 3, then the new local exansion can be calculated by comosing M m2l with M 3 as M l2l ¼ M m2l 3M 3 : ð4þ We can translate the local exansion of a box into its children boxes using Eqs. (3) and (4) Reresentation of otential and the field as olynomials In order to calculate the high-order transfer ma of a beam otical system, one needs not only the field on the reference orbit but also the derivatives of the field [8]. If we have the field of the Coulomb interaction and its derivatives, we can include the Coulomb interaction into the ma. Eq. (4) rovides the exansion of the otential f with resect to the osition of the observer u to the order n. This exansion converges inside the corresonding finest box. The constant art of the exansion is the value of the otential at the center of the box, and the coefficient of each monomial is the value of the corresonding artial derivative of the otential. Table 3 shows an examle of the local exansion. In a large cube box, we ut 0,000 electrons. The cube box is cut into the third level. And the third level box size is 2, which means if we ut a frame at the center of one finest box, we have x,y,za½,š. The local exansion of each third level box is calculated u to the 0th order, and the coefficients till the third order of one third level box are resented in Table 3. The forth column and the eighth column show the exonents of each DA variables in each monomial. For examle, the index trile 2 0 denotes the monomial x 2 y z 0 ¼x 2 y. The second and sixth columns show the coefficient of each monomial. From the table, we observe the decreasing trend of the coefficients according to the increase of the order. To obtain sufficiently meaningful statistics, we reeat the above rocess for 40 times with different grous of electrons. Each time we find the maximum absolute value of the coefficients for each order and calculate the average absolute value for the coefficients for each order. Then we take the average over the 40 rocesses and resent the result in Fig. 5. The green dots show the maximum value of the coefficients, and the red dots show the average value of the coefficients. It is clear that the coefficients decrease as the order increase, which suggests the convergence of the local exansion. If we want to calculate the exansion of the Coulomb interaction field on a reference oint, we only need to determine which finest box the oint belongs to, then by Eqs. (3) and (4) we can translate the local exansion of the otential from the center of the finest box to the reference oint. Taking the derivative of the osition (x,y,z), we get the exansion of the field (E x,e y,e z )uto the order n. The exansion of the field only includes the contribution of the electrons in the far region. As to the electrons in the near region, we obtain the extension of their field by reresenting each of them by Gaussian distribution, which will be discussed in the following.

6 H. Zhang, M. Berz / Nuclear Instruments and Methods in Physics Research A 645 (20) Table 3 A local exansion. I Coefficient n Ex. I Coefficient n Ex Avg. Coef. Max. Coef Coefficient 0 0. Time (s) order Fig. 5. The coefficients of the different orders of the local exansion. (For interretation of the references to color in this figure legend, the reader is referred to the web version of this article.) 4.6. Gaussian macroarticles instead of oint charges Sometimes it is imortant to use a finite number of articles to reresent a continuous distribution. There are two imortant cases where this arises; the first is when the number of actual articles is substantially larger than the number than what can ractically be handled in a simulation. The second case is the desire to comute a transfer ma and aberrations of the system at hand, which requires the knowledge of high-order field exansions near the reference oint, which cannot be done reliably if oint charges are near. To avoid this, one can use 3D sherical Gaussian distributions instead of oint charges as x2 y2 z2 f ðx,y,zþ¼ ffiffiffiffiffiffi ex ¼ ffiffiffiffiffiffi ð 2 sþ 3 s2 s2 s 2 ð 2 sþ 3 ex r2 : ð5þ s 2 It is easy to see from Gauss s law that the field along the radius can be exressed as ffiffiffiffiffiffi r E r ¼ ffiffiffiffiffiffi 2 erf ffiffiffi s 3 2rs 2 ex r2 ð6þ r 2 2 s 3 2 s 2s 2 where erfðr= ffiffiffi 2 sþ is the error function. By setting a roer value for s, we can use the Gaussian distribution in Eq. (5) to reresent a oint charge. When r bs, E r is equal to the field of a oint charge. When r-0, E r -0. To calculate the exansion of the error function, we need to find all the derivatives of it. Considering the definition of the error function Z x erfðxþ¼ 2 ffiffiffi e t2 dt 0 it is easily to determine its derivative derfðxþ dx N (0 6 ) ¼ Fig. 6. Comutation time for different numbers of electrons. ð7þ 2 ffiffiffi e x2 : ð8þ For any given oint x, we calculate erf(x) by the rational Chebyshev aroximation [2] and d erf(x)/dx by Eq. (8), which is infinitely differentiable and allows the comutation of all higher derivatives Examle calculations exhibiting the linear scaling roerty To show the erformance of the method, for reasons of sace we limit ourselves here to the most imortant characteristic, namely the scaling of the comutation time to the number of articles. We erform a series of simulations with exansion order 5 and with varying numbers of electrons N. Fig. 6 shows the required comutation time for a simulation with varying article numbers N. We observe the nearly linear behavior, with a slight deviation around N ¼ 0:4 0 6, at which oint the number of articles crosses a threshold for transitions from a three-level scheme to a four-level scheme, which leads to slightly imroved erformance after the transition. Overall, the exected linearity is well achieved. 5. Conclusions and future work We have shown how we aly the FMM in the DA framework to calculate the electrostatic otential. Future work will see the

7 344 H. Zhang, M. Berz / Nuclear Instruments and Methods in Physics Research A 645 (20) alication to other otentials, the derivation of exlicit highorder time steing schemes, and the direct connection to codes erforming ma integration. Furthermore, we will develo arallel versions of the algorithms, making use of COSY s suort for MPI-tye arallelization at the machine level. Finally, we will use the Taylor model data tye in COSY to obtain fully rigorous error estimates of the methods. Acknowledgments For many fruitful discussions we are indebted to He Huang, Kyoko Makino and Alexander Wittig. Financial suort was areciated from the US Deartment of Energy under contract number DE-FG02-08ER4546. References [] G.H. Jansen, Coulomb Interactions in Particle Beams (Advances in Electronics and Electron Physics Sulement 2), Academic Press, 990. [2] P. Kruit, G. Jansen, Sace Charge and Statistical Coulomb Effects, second ed., Handbook of Charged Particle Otics, 2009, [3] M. Martini, M. Prome, Particle Accelerators 2 (97) 289. [4] R.W. Garnett, T.P. Wangler, Sace-charge calculation for bunched beams with 3-D ellisoidal symmetry, in: Proceedings of the 99 IEEE Particle Accelerator Conference (APS Beam Physics), 99, [5] P.M. Laostolle, A.M. Lombardi, S. Nath, E. Tanke, S. Valero, T.P. Wangler, A new aroach to sace charge for linac beam dynamics codes, in: Proceedings of the 8th International Linear Accelerator Conference, 996, [6] P. Laostolle, A.M. Lombardi, E. Tanke, S. Valero, R.W. Garnett, T.P. Wangler, Nuclear Instruments and Methods A 379 () (996) 2. [7] N. Pichoff, J.M. Lagniel, S. Nath, Simulation results with an alternate 3D sace charge routine, PICNIC, in: Proceedings of the XIX International Linac Conference, vol. 4, 998, [8] G. Polau, U. van Rienen, B. van der Geer, M. de Loos, IEEE Transactions on Magnetics 40 (2 Part 2) (2004) 74. [9] Y.K. Batygin, Nuclear Instruments and Methods A 539 (3) (2005) 455. [0] J. Barnes, P. Hut, Nature 324 (986) 4. [] J.E. Barnes, Journal of Comutational Physics 87 () (990) 6. [2] L. Greengard, V. Rokhlin, Journal of Comutational Physics 73 (2) (987) 325. [3] J. Carrier, L. Greengard, V. Rokhlin, SIAM Journal on Scientific and Statistical Comuting 9 (988) 669. [4] R.K. Beatson, L. Greengard, Wavelets, Multilevel Methods and Ellitic PDEs (997). [5] B. Shanker, H. Huang, Journal of Comutational Physics 226 () (2007) 732. [6] M. Berz, Modern Ma Methods in Particle Beam Physics, Academic Press, San Diego, 999 Also available at /htt://bt.a.msu.edu/ubs. [7] M. Berz, Nuclear Instruments and Methods A 298 (990) 426. [8] M. Berz, Nuclear Instruments and Methods 363 (995) 00. [9] M. Berz, K. Makino, COSY INFINITY Version 9.0 rogrammer s manual, Technical Reort MSUHEP , Deartment of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, see also /htt://cosyinfi nity.orgs, [20] M. Berz, AIP Conference Proceedings 405 (997). [2] W.J. Cody, Mathematics of Comutation 23 (07) (969) 63.