Adaptive symplectic and reversible integrators

The so-called structure-preserving methods which reproduce the fundamental properties like symplecticness, time reversibility, volume and energy preservation of the original model of the underlying physical problem became very important in recent years. It has been shown theoretically and experimentally, that these methods are superior to the standard integrators, especially in long term computation. In the paper the adaptivity issues are discussed for symplectic and reversible methods designed for integration of Hamiltonian systems. Molecular dynamics models and N-body problems, as Hamiltonian systems, are challenging mathematical models in many aspects; the wide range of time scales, very large number of differential equations, chaotic nature of trajectories, restriction to very small step sizes in time, etc. Recent results on variable step size integrators, multiple time stepping methods, regularization techniques with applications to classical and quantum molecular dynamics, to N- body atomic problems and planetary motion will be presented.


Noncommutative nonlinear sigma models and integrability
Kürkcüoğlu, Seçkin (American Physical Society (APS), 2008-09-01)
We first review the result that the noncommutative principal chiral model has an infinite tower of conserved currents and discuss the special case of the noncommutative CP1 model in some detail. Next, we focus our attention to a submodel of the CP1 model in the noncommutative spacetime A(theta)(R2+1). By extending a generalized zero-curvature representation to A(theta)(R2+1) we discuss its integrability and construct its infinitely many conserved currents. A supersymmetric principal chiral model with and wi...
Yücel, Hamdullah; Benner, Peter (2018-01-01)
In this paper, we investigate a posteriori error estimates of a control-constrained optimal control problem governed by a time-dependent convection diffusion equation. The control constraints are handled by using the primal-dual active set algorithm as a semi-smooth Newton method and by adding a Moreau-Yosida-type penalty function to the cost functional. Residual-based error estimators are proposed for both approaches. The derived error estimators are used as error indicators to guide the mesh refinements. ...
Energy preserving model order reduction of the nonlinear Schrodinger equation
Karasözen, Bülent (2018-12-01)
An energy preserving reduced order model is developed for two dimensional nonlinear Schrodinger equation (NLSE) with plane wave solutions and with an external potential. The NLSE is discretized in space by the symmetric interior penalty discontinuous Galerkin (SIPG) method. The resulting system of Hamiltonian ordinary differential equations are integrated in time by the energy preserving average vector field (AVF) method. The mass and energy preserving reduced order model (ROM) is constructed by proper orth...
Adaptive Harmonic Balance Methods, A Comparison
Sert, Onur; Ciğeroğlu, Ender (2016-01-28)
Harmonic balance method (HBM) is one of the most popular and powerful methods, which is used to obtain response of nonlinear vibratory systems in frequency domain. The main idea of the method is to express the response of the system in Fourier series and converting the nonlinear differential equations of motion into a set of nonlinear algebraic equations. System response can be obtained by solving this nonlinear equation set in terms of the unknown Fourier coefficients. The accuracy of the solution is great...
Discontinuous Galerkin finite element methods with shock-capturing for nonlinear convection dominated models
Yücel, Hamdullah; BENNER, Peter (2013-11-11)
In this paper, convection-diffusion-reaction models with nonlinear reaction mechanisms, which are typical problems of chemical systems, are studied by using the upwind symmetric interior penalty Galerkin (SIPG) method. The local spurious oscillations are minimized by adding an artificial viscosity diffusion term to the original equations. A discontinuity sensor is used to detect the layers where unphysical oscillations occur. Finally, the proposed method is tested on various single- and multi-component prob...
Citation Formats
B. Karasözen, “Adaptive symplectic and reversible integrators,” 1998, vol. 536, Accessed: 00, 2020. [Online]. Available: