Fer Expansion Research Articles

Geometric integration methods for general nonlinear dynamic equation based on Magnus and Fer expansions

Based on Magnus or Fer expansion for solving linear differential equation and operator semi-group theory, Lie group integration methods for general nonlinear dynamic equation are studied. Approximate schemes of Magnus type of 4th, 6th and 8th order are constructed which involve only 1, 4 and 10 different commutators, and the time symmetry properties of the schemes are proved. In the meantime, the integration methods based on Fer expansion are presented. Then by connecting the Fer expansion methods with Magnus expansion methods some techniques are given to simplify the construction of Fer expansion methods. Furthermore time-symmetric integrators of Fer type are constructed. These methods belong to the category of geometric integration methods and can preserve many qualitative properties of the original dynamic system. Supported by the National Natural Science Foundation of China (Grant Nos. 10372084, 10472059), Fok Ying Tung Youth Teacher Foundation (Grant No. 71005), the Doctoral Program Foundation of Edu...

Complexity Theory for Lie-Group Solvers

Commencing with a brief survey of Lie-group theory and differential equations evolving on Lie groups, we describe a number of numerical algorithms designed to respect Lie-group structure: Runge–Kutta–Munthe-Kaas schemes, Fer and Magnus expansions. This is followed by derivation of the computational cost of Fer and Magnus expansions, whose conclusion is that for order four, six, and eight an appropriately discretized Magnus method is always cheaper than a Fer method of the same order. Each Lie-group method of the kind surveyed in this paper requires the computation of a matrix exponential. Classical methods, e.g., Krylov-subspace and rational approximants, may fail to map elements in a Lie algebra to a Lie group. Therefore we survey a number of approximants based on the splitting approach and demonstrate that their cost is compatible (and often superior) to classical methods.

The Fer expansion and time-symmetry: A Strang-type approach

We introduce a symmetric version of the Fer expansion for the solution of the ODE y′= γ( t, y) y. We show that the schemes are time-symmetric for linear problems. Furthermore, symmetry implies order 2 p+2 whereby the classical Fer expansion would display order 2 p+1 only. This reduces significantly the computational cost of the classical Fer expansion. We prove the convergence of the proposed symmetric schemes and obtain an estimate of the radius of convergence. Numerical experiments for two methods in this class confirm that time-symmetry is preserved to machine accuracy for linear problem, while, for nonlinear problems is not preserved exactly but to a higher order of accuracy.

10.14322/publons.r864509

10.14322/publons.r864509

Magnus and Fer expansions for matrix differential equations: the convergence problem

Approximate solutions of matrix linear differential equations by matrix exponentials are considered. In particular, the convergence issue of Magnus and Fer expansions is treated. Upper bounds for the convergence radius in terms of the norm of the defining matrix of the system are obtained. The very few previously published bounds are improved. Bounds to the error of approximate solutions are also reported. All results are based just on algebraic manipulations of the recursive relation of the expansion generators.

Fer's factorization as a symplectic integrator

In this paper we analyze the main features of the so-called Fer expansion as a new method for integrating the ordinary differential equations derived from explicitly time-dependent Hamiltonian dynamical systems. This method is based on a factorization of the evolution operator as an infinite product of exponentials of Lie operators and thus exactly preserves the Poincare integral invariants. Third and fourth-order expansions are considered and numerical results are presented for a quadratic Hamiltonian with various time-dependent frequencies. Comparison is done with other numerical integration schemes.

Lie algebraic approach to Fer's expansion for classical Hamiltonian systems

The so-called Fer's expansion is proposed as a solution for the time-evolution operator of classical time-dependent Hamiltonian systems. The quadratic Hamiltonians treated as examples show that, under very different regimes, the second-order approximation already gives extremely good results.