This note, written for the NIC Winter School ”Quantum Simulations of Complex Many-Body Systems: From Theory to Algorithms”, discusses and compares – in theoretical respects – various old and new approaches to numerical time integration for quantum dynamics: implicit vs. exponential midpoint rule; splitting, Chebyshev and Lanczos approximations to the exponential; Magnus integrators; integrators...

It is the aim of this paper to investigate a suitable approach to compute solutions of the powerful Michaelis-Menten enzyme reaction equation with less computational effort. We obtain analytical-numerical solutions using piecewise finite series by means of the differential transformation method, DTM. In addition, we compute a global analytical solution by a modal series expansion. The Michaelis...

In this work we have considered the Taylor series expansion of the dynamic scaling relation of the magnetization with respect to small initial magnetization values in order to study the dynamic scaling behaviour of 2and 3-dimensional Ising models. We have used the literature values of the critical exponents and of the new dynamic exponent x0 to observe the dynamic finite-size scaling behaviour ...

We analyze Monte Carlo simulation and series-expansion data for the susceptibility of the three-state Potts model in the critical region. The amplitudes of the susceptibility on the highand the lowtemperature sides of the critical point as extracted from the Monte Carlo data are in good agreement with those obtained from the series expansions and their (universal) ratio compares quite well with...

Query expansion is a well-known method for improving average effectiveness in information retrieval. However, the most effective query expansion methods rely on costly retrieval and processing of feedback documents. We explore alternative methods for reducing queryevaluation costs, and propose a new method based on keeping a brief summary of each document in memory. This method allows query exp...

The class of commutator-free Magnus integrators is known to provide a favourable alternative to standard Magnus integrators, in particular for large-scale applications arising in the time integration of non-autonomous linear evolution equations. A high-order commutator-free Magnus integrator is given by a composition of several exponentials that comprise certain linear combinations of the value...

We study the problem of efficient integration of variational equations in multi-dimensional Hamiltonian systems. For this purpose, we consider a Runge-Kutta-type integrator, a Taylor series expansion method and the so-called ‘Tangent Map’ (TM) technique based on symplectic integration schemes, and apply them to the Fermi-Pasta-Ulam β (FPU-β) lattice of N nonlinearly coupled oscillators, with N ...

Backward error analysis has become an important tool for understanding the long time behavior of numerical integration methods. This is true in particular for the integration of Hamiltonian systems where backward error analysis can be used to show that a symplectic method will conserve energy over exponentially long periods of time. Such results are typically based on two aspects of backward er...

The Taylor series expansionand least-square-based Lattice Boltzmann method (TLLBM) is a flexible Lattice Boltzmann approach capable of simulating incompressible viscous flows with arbitrary geometry. The method is based on the standard Lattice Boltzmann equation (LBE), Taylor series expansion and the least square optimisation. The final formulation is an algebraic form and essentially has no li...

In this paper, we propose a wavelet-Taylor–Galerkin method for solving the twodimensional Navier–Stokes equations. The discretization in time is performed before the spatial discretization by introducing second-order generalization of the standard time stepping schemes with the help of Taylor series expansion in time step. WaveletTaylor–Galerkin schemes taking advantage of the wavelet bases cap...