The computational complexity of differential and integral equations: an information-based approach
The computational complexity of differential and integral equations: an information-based approach
Low-order polynomial approximation of propagators for the time-dependent Schro¨dinger equation
Journal of Computational Physics
Efficient solution of parabolic equations by Krylov approximation methods
SIAM Journal on Scientific and Statistical Computing
On the numerical integration of ordinary differential equations by symmetric composition methods
SIAM Journal on Scientific Computing
On Krylov Subspace Approximations to the Matrix Exponential Operator
SIAM Journal on Numerical Analysis
Numerical solution of isospectral flows
Mathematics of Computation
Exponential Integrators for Large Systems of Differential Equations
SIAM Journal on Scientific Computing
High order Runge-Kutta methods on manifolds
proceedings of the on Numerical analysis of hamiltonian differential equations
Approximating the exponential from a Lie algebra to a Lie group
Mathematics of Computation
Collocation and Relaxed Collocation for the Fer and the Magnus Expansions
SIAM Journal on Numerical Analysis
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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.