. A method of approximation that was introduced for functional differential equations with a single state-dependent delay is now extended to handle multiple state-dependent delays. A discussion of the method of steps (MoS) is used to develop a version for the case of multiple state-dependent delays that aligns favorably with the method presented here. Some numerical results are presented to confirm the robustness of the implementation. Benefits of this method include producing a continuous function as the approximation, freedom from root finding techniques, and derivatives of the vector field not being required, although the delay needs to be continuously differentiable except at specific isolated points.
We derive variational integrators for stochastic Hamiltonian systems on Lie groups using a discrete version of the stochastic Hamiltonian phase space principle. The structure-preserving properties of the resulting scheme, such as symplecticity, preservation of the Lie-Poisson structure, preservation of the coadjoint orbits, and preservation of Casimir functions, are discussed, along with a discrete No ether theorem for subgroup symmetries. We also consider in detail the case of stochastic Hamiltonian systems with advected quantities, studying the associated structure-preserving properties in relation to semidirect product Lie groups. A full convergence proof for the scheme is provided for the case of the Lie group of rotations. Several numerical examples are presented, including simulations of the free rigid body and the heavy top.
. We investigate the benefits of higher-order time-integration for the quantized formulation of the Euler equations for incompressible fluid flow on the sphere. We show that higher-order isospectral symplectic Runge-Kutta (IsoSyRK) methods are a viable and efficient alternative to the widely used isospectral implicit midpoint method. When implemented using a block matrix approach, higher-order methods allow for effective parallelization and exhibit superior conservation of the Hamiltonian while reducing overall simulation time due to the larger time-step enabled by higher-order methods. We establish this with a series of numerical experiments that compare energy and enstrophy conservation, as well as execution time across integrators of different orders. A qualitative experiment illustrates further benefits of these integrators, showing that their enhanced conservation properties translate into more stable longtime behavior for a simulation starting from a sensitive initial condition.
