arXiv subjects

Jan L. Cieśliński

Publications and source records attributed to Jan L. Cieśliński.

25 records · Page 2Linked to original sources

arXiv2012

Modified van der Pauw method based on formulas solvable by the Banach fixed point method

We propose a modification of the standard van der Pauw method for determining the resistivity and Hall coefficient of flat thin samples of arbitrary shape. Considering a different choice of resistance measurements we derive a new formula which can be numerically solved (with respect to sheet resistance) by the Banach fixed point method for any values of experimental data. The convergence is especially fast in the case of almost symmetric van der Pauw configurations (e.g., clover shaped samples).

physics.class-ph↗

arXiv2011

Locally exact modifications of numerical integrators

We present a new class of exponential integrators for ordinary differential equations. They are locally exact, i.e., they preserve the linearization of the original system at every point. Their construction consists in modifying existing numerical schemes in order to make them locally exact. The resulting schemes preserve all fixed points and are A-stable. The most promising results concern the discrete gradient method (modified implicit midpoint rule) where we succeeded to preserve essential geometric properties and the final results have a relatively simple form. In the case of one-dimensional Hamiltonian systems numerical experiments show that our modifications can increase the accuracy by several orders of magnitude. The main result of this paper is the construction of energy-preserving locally exact discrete gradient schemes for arbitrary multidimensional Hamiltonian systems in canonical coordinates.

math.NA↗

arXiv2010

Energy-preserving numerical schemes of high accuracy for one-dimensional Hamiltonian systems

We present a class of non-standard numerical schemes which are modifications of the discrete gradient method. They preserve the energy integral exactly (up to the round-off error). The considered class contains locally exact discrete gradient schemes and integrators of arbitrary high order. In numerical experiments we compare our integrators with some other numerical schemes, including the standard discrete gradient method, the leap-frog scheme and a symplectic scheme of 4th order. We study the error accumulation for very long time and the conservation of the energy integral.

math.NA↗

arXiv2010

Some implications of a new definition of the exponential function on time scales

We present a new approach to exponential functions on time scales and to timescale analogues of ordinary differential equations. We describe in detail the Cayley-exponential function and associated trigonometric and hyperbolic functions. We show that the Cayley-exponential is related to implicit midpoint and trapezoidal rules, similarly as delta and nabla exponential functions are related to Euler numerical schemes. Extending these results on any Padé approximants, we obtain Padé-exponential functions. Moreover, the exact exponential function on time scales is defined. Finally, we present applications of the Cayley-exponential function in the q-calculus and suggest a general approach to dynamic systems on Lie groups.

math.CA↗

arXiv2010

Discrete gradient algorithms of high order for one-dimensional systems

We show how to increase the order of one-dimensional discrete gradient numerical integrator without losing its advantages, such as exceptional stability, exact conservation of the energy integral and exact preservation of the trajectories in the phase space. The accuracy of our integrators is higher by several orders of magnitude as compared with the standard discrete gradient scheme (modified midpoint rule) and, what is more, our schemes have very high accuracy even for large time steps.

physics.comp-ph↗

arXiv2010

Improved q-exponential and q-trigonometric functions

We propose a new definition of the q-exponential function. Our q-exponential function maps the imaginary axis into the unit circle and the resulting q-trigonometric functions are bounded and satisfy the Pythagorean identity.

math.CA↗

arXiv2004

Why the phase shifts for solitons on a vortex filament are so large?

The phase ``jumps'' for solitons interacting on a vortex filament, observed in experiments, have been unaccounted for since more than twenty years. Using explicit formulas describing the interaction of two solitons on a thin vortex filament in the Localized Induction Approximation we show that an appropriate choice of the parameters of the solitons leads to large phase shifts. This result does not depend on the axial flow along the filament.

physics.flu-dyn↗