PAC2013 - List of Authors (Erdelyi, B.)

Paper	Title	Page
MOPBA13	Optimization of the Multipole to Local Translation Operator in the Adaptive Fast Multipole Method	201
	S. Abeyratne, B. Erdelyi Northern Illinois University, DeKalb, Illinois, USA B. Erdelyi ANL, Argonne, USA
	The Fast Multipole Method (FMM) is an accurate and fast way to calculate potentials/fields created by a very large number of particles. The run time of the FMM is significantly less than that of the pairwise calculation if the particle number, N is sufficiently large. Two major parts in the FMM are the upward pass and the downward pass. The upward pass calculates multipole expansions and then performs multipole- to-multipole translations. The downward pass calculates multipole-to- local expansions and local-to local expansions. The multipole-to-local translation in the downward pass is the most time consuming translation in the FMM. In order to make the FMM more efficient, we sought to minimize the time taken by the multipole-to-local translation. The promising and practical strategy to minimize the multipole-to-local translation time is to replace the 3D multipole-to-local translation with a 1D multipole-to-local translation in conjunction with rotations of the coordinate axes. In this paper we show how to perform the 1D multipole-to-local translation and the time comparisons between the two FMM variants.

MOPBA15	Study and Comparison of the Method of Moments and the Single Level Fast Multipole Method for 2D Space Charge Tracking	207
	A.J. Gee, B. Erdelyi Northern Illinois University, DeKalb, Illinois, USA B. Erdelyi ANL, Argonne, USA
	Funding: Partially funded by Department of Energy, Office of High Energy Physics under Contract No. DE-FG02-08ER41532. Strong space charge is a significant impediment in charged particle beam physics, particularly at the high intensity frontier. For future applications, where particles must occupy the smallest region possible, quickly and accurately and efficient modeling space charge modeling is essential, for instance, to minimize the space charge contribution to beam dispersion. In this paper, we study and compare the performance for the method of moments (MoM) and the single-level fast multipole method (SLFMM) in 2D. The method of moments has been widely used to solve computational electromagnetic problems but assumes a series-expandable smooth distribution function, limiting its reliability in some cases. The fast multipole method was more recently developed and shows remarkable accuracy with difficult beam distributions. We demonstrate these methods using a simplified version of the University of Maryland electron ring (UMER). We present some multi-particle tracking results obtained using these methods. Future work will study the space charge inclusive transfer maps calculated from these methods.

MOPBA14	Numerical Integrator for Coulomb Collisions	204
	A.A. Al Marzouk, B. Erdelyi Northern Illinois University, DeKalb, Illinois, USA
	The trajectories of protons interacting through Coulomb forces were computed using a numerical integrator based on Picard's iteration method. This is a variable order, adaptive integrator with dense output. We show different cases by varying some parameters such as the impact parameter, the relative velocity of the protons and the order of the differential algebraic (DA) vector. The accuracy of the trajectories was tested by changing the order of the DA vector while fixing the other parameters. The impact parameter between the protons and the velocity of the incident proton has the most impact on the trajectories. The maximum time step is determined by the radius of convergence of the expansions, while a fixed accuracy is attained by varying the order.

MOPBA15	Study and Comparison of the Method of Moments and the Single Level Fast Multipole Method for 2D Space Charge Tracking	207
	A.J. Gee, B. Erdelyi Northern Illinois University, DeKalb, Illinois, USA B. Erdelyi ANL, Argonne, USA
	Funding: Partially funded by Department of Energy, Office of High Energy Physics under Contract No. DE-FG02-08ER41532. Strong space charge is a significant impediment in charged particle beam physics, particularly at the high intensity frontier. For future applications, where particles must occupy the smallest region possible, quickly and accurately and efficient modeling space charge modeling is essential, for instance, to minimize the space charge contribution to beam dispersion. In this paper, we study and compare the performance for the method of moments (MoM) and the single-level fast multipole method (SLFMM) in 2D. The method of moments has been widely used to solve computational electromagnetic problems but assumes a series-expandable smooth distribution function, limiting its reliability in some cases. The fast multipole method was more recently developed and shows remarkable accuracy with difficult beam distributions. We demonstrate these methods using a simplified version of the University of Maryland electron ring (UMER). We present some multi-particle tracking results obtained using these methods. Future work will study the space charge inclusive transfer maps calculated from these methods.

MOPBA16	A Picard Iteration Based Integrator	210
	H.D. Schaumburg, B. Erdelyi Northern Illinois University, DeKalb, Illinois, USA
	Picard iteration is mainly used as a theoretical tool to establish the existence and uniqueness of a solution to an initial value problem. We have developed a method based on Picard iteration that computes the exact Taylor polynomial of the solution to arbitrary order. The method has been implemented in COSY infinity to numerically solve Coulomb interactions.

Paper

Title

Page

Optimization of the Multipole to Local Translation Operator in the Adaptive Fast Multipole Method

201

S. Abeyratne, B. Erdelyi
Northern Illinois University, DeKalb, Illinois, USA
B. Erdelyi
ANL, Argonne, USA

The Fast Multipole Method (FMM) is an accurate and fast way to calculate potentials/fields created by a very large number of particles. The run time of the FMM is significantly less than that of the pairwise calculation if the particle number, N is sufficiently large. Two major parts in the FMM are the upward pass and the downward pass. The upward pass calculates multipole expansions and then performs multipole- to-multipole translations. The downward pass calculates multipole-to- local expansions and local-to local expansions. The multipole-to-local translation in the downward pass is the most time consuming translation in the FMM. In order to make the FMM more efficient, we sought to minimize the time taken by the multipole-to-local translation. The promising and practical strategy to minimize the multipole-to-local translation time is to replace the 3D multipole-to-local translation with a 1D multipole-to-local translation in conjunction with rotations of the coordinate axes. In this paper we show how to perform the 1D multipole-to-local translation and the time comparisons between the two FMM variants.

MOPBA15

Study and Comparison of the Method of Moments and the Single Level Fast Multipole Method for 2D Space Charge Tracking

207

A.J. Gee, B. Erdelyi
Northern Illinois University, DeKalb, Illinois, USA
B. Erdelyi
ANL, Argonne, USA

Funding: Partially funded by Department of Energy, Office of High Energy Physics under Contract No. DE-FG02-08ER41532.
Strong space charge is a significant impediment in charged particle beam physics, particularly at the high intensity frontier. For future applications, where particles must occupy the smallest region possible, quickly and accurately and efficient modeling space charge modeling is essential, for instance, to minimize the space charge contribution to beam dispersion. In this paper, we study and compare the performance for the method of moments (MoM) and the single-level fast multipole method (SLFMM) in 2D. The method of moments has been widely used to solve computational electromagnetic problems but assumes a series-expandable smooth distribution function, limiting its reliability in some cases. The fast multipole method was more recently developed and shows remarkable accuracy with difficult beam distributions. We demonstrate these methods using a simplified version of the University of Maryland electron ring (UMER). We present some multi-particle tracking results obtained using these methods. Future work will study the space charge inclusive transfer maps calculated from these methods.

MOPBA14

Numerical Integrator for Coulomb Collisions

204

A.A. Al Marzouk, B. Erdelyi
Northern Illinois University, DeKalb, Illinois, USA

The trajectories of protons interacting through Coulomb forces were computed using a numerical integrator based on Picard's iteration method. This is a variable order, adaptive integrator with dense output. We show different cases by varying some parameters such as the impact parameter, the relative velocity of the protons and the order of the differential algebraic (DA) vector. The accuracy of the trajectories was tested by changing the order of the DA vector while fixing the other parameters. The impact parameter between the protons and the velocity of the incident proton has the most impact on the trajectories. The maximum time step is determined by the radius of convergence of the expansions, while a fixed accuracy is attained by varying the order.

MOPBA15

Study and Comparison of the Method of Moments and the Single Level Fast Multipole Method for 2D Space Charge Tracking

207

A.J. Gee, B. Erdelyi
Northern Illinois University, DeKalb, Illinois, USA
B. Erdelyi
ANL, Argonne, USA

Funding: Partially funded by Department of Energy, Office of High Energy Physics under Contract No. DE-FG02-08ER41532.
Strong space charge is a significant impediment in charged particle beam physics, particularly at the high intensity frontier. For future applications, where particles must occupy the smallest region possible, quickly and accurately and efficient modeling space charge modeling is essential, for instance, to minimize the space charge contribution to beam dispersion. In this paper, we study and compare the performance for the method of moments (MoM) and the single-level fast multipole method (SLFMM) in 2D. The method of moments has been widely used to solve computational electromagnetic problems but assumes a series-expandable smooth distribution function, limiting its reliability in some cases. The fast multipole method was more recently developed and shows remarkable accuracy with difficult beam distributions. We demonstrate these methods using a simplified version of the University of Maryland electron ring (UMER). We present some multi-particle tracking results obtained using these methods. Future work will study the space charge inclusive transfer maps calculated from these methods.

MOPBA16

A Picard Iteration Based Integrator

210

H.D. Schaumburg, B. Erdelyi
Northern Illinois University, DeKalb, Illinois, USA

Picard iteration is mainly used as a theoretical tool to establish the existence and uniqueness of a solution to an initial value problem. We have developed a method based on Picard iteration that computes the exact Taylor polynomial of the solution to arbitrary order. The method has been implemented in COSY infinity to numerically solve Coulomb interactions.