1,424 research outputs found
High order time integrators for the simulation of charged particle motion in magnetic quadrupoles
Magnetic quadrupoles are essential components of particle accelerators like
the Large Hadron Collider. In order to study numerically the stability of the
particle beam crossing a quadrupole, a large number of particle revolutions in
the accelerator must be simulated, thus leading to the necessity to preserve
numerically invariants of motion over a long time interval and to a substantial
computational cost, mostly related to the repeated evaluation of the magnetic
vector potential. In this paper, in order to reduce this cost, we first
consider a specific gauge transformation that allows to reduce significantly
the number of vector potential evaluations. We then analyze the sensitivity of
the numerical solution to the interpolation procedure required to compute
magnetic vector potential data from gridded precomputed values at the locations
required by high order time integration methods. Finally, we compare several
high order integration techniques, in order to assess their accuracy and
efficiency for these long term simulations. Explicit high order Lie methods are
considered, along with implicit high order symplectic integrators and
conventional explicit Runge Kutta methods. Among symplectic methods, high order
Lie integrators yield optimal results in terms of cost/accuracy ratios, but non
symplectic Runge Kutta methods perform remarkably well even in very long term
simulations. Furthermore, the accuracy of the field reconstruction and
interpolation techniques are shown to be limiting factors for the accuracy of
the particle tracking procedures.Comment: 39 pages, 18 figure
On the Construction of Splitting Methods by Stabilizing Corrections with Runge-Kutta Pairs
In this technical note a general procedure is described to construct
internally consistent splitting methods for the numerical solution of
differential equations, starting from matching pairs of explicit and diagonally
implicit Runge-Kutta methods. The procedure will be applied to suitable
second-order pairs, and we will consider methods with or without a mass
conserving finishing stage. For these splitting methods, the linear stability
properties are studied and numerical test results are presented.Comment: 18 page
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