7,232 research outputs found
Guidance, flight mechanics and trajectory optimization. Volume 6 - The N-body problem and special perturbation techniques
Analytical formulations and numerical integration methods for many body problem and special perturbative technique
Estimating numerical integration errors
Algorithm for use in estimating accumulated numerical integration error
Bayesian Analysis of ODE's: solver optimal accuracy and Bayes factors
In most relevant cases in the Bayesian analysis of ODE inverse problems, a
numerical solver needs to be used. Therefore, we cannot work with the exact
theoretical posterior distribution but only with an approximate posterior
deriving from the error in the numerical solver. To compare a numerical and the
theoretical posterior distributions we propose to use Bayes Factors (BF),
considering both of them as models for the data at hand. We prove that the
theoretical vs a numerical posterior BF tends to 1, in the same order (of the
step size used) as the numerical forward map solver does. For higher order
solvers (eg. Runge-Kutta) the Bayes Factor is already nearly 1 for step sizes
that would take far less computational effort. Considerable CPU time may be
saved by using coarser solvers that nevertheless produce practically error free
posteriors. Two examples are presented where nearly 90% CPU time is saved while
all inference results are identical to using a solver with a much finer time
step.Comment: 28 pages, 6 figure
Fourth Order Algorithms for Solving the Multivariable Langevin Equation and the Kramers Equation
We develop a fourth order simulation algorithm for solving the stochastic
Langevin equation. The method consists of identifying solvable operators in the
Fokker-Planck equation, factorizing the evolution operator for small time steps
to fourth order and implementing the factorization process numerically. A key
contribution of this work is to show how certain double commutators in the
factorization process can be simulated in practice. The method is general,
applicable to the multivariable case, and systematic, with known procedures for
doing fourth order factorizations. The fourth order convergence of the
resulting algorithm allowed very large time steps to be used. In simulating the
Brownian dynamics of 121 Yukawa particles in two dimensions, the converged
result of a first order algorithm can be obtained by using time steps 50 times
as large. To further demostrate the versatility of our method, we derive two
new classes of fourth order algorithms for solving the simpler Kramers equation
without requiring the derivative of the force. The convergence of many fourth
order algorithms for solving this equation are compared.Comment: 19 pages, 2 figure
An intelligent processing environment for real-time simulation
The development of a highly efficient and thus truly intelligent processing environment for real-time general purpose simulation of continuous systems is described. Such an environment can be created by mapping the simulation process directly onto the University of Alamba's OPERA architecture. To facilitate this effort, the field of continuous simulation is explored, highlighting areas in which efficiency can be improved. Areas in which parallel processing can be applied are also identified, and several general OPERA type hardware configurations that support improved simulation are investigated. Three direct execution parallel processing environments are introduced, each of which greatly improves efficiency by exploiting distinct areas of the simulation process. These suggested environments are candidate architectures around which a highly intelligent real-time simulation configuration can be developed
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
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