Structure-preserving reduced basis methods for Poisson systems

Energy-preserving methods for Poisson systems

We present and analyze energy-conserving methods for the numerical inte-gration of IVPs of Poisson type systems that are able to preserve some Casimirs.Their derivation and analysis is done within the framework following the ideasof Hamiltonian BVMs (HBVMs) (see [1] and references therein). The pro-posed methods turn out to be equivalent to those recently derived in [2], giv...

Energy-preserving integrators for stochastic Poisson systems

A new class of energy-preserving numerical schemes for stochastic Hamiltonian systems with non-canonical structure matrix (in the Stratonovich sense) is proposed. These numerical integrators are of mean-square order one and also preserve quadratic Casimir functions. In the deterministic setting, our schemes reduce to methods proposed in [9] and [6].

Linear energy-preserving integrators for Poisson systems

For Hamiltonian systems with non-canonical structure matrix a new class of numerical integrators is proposed. The methods exactly preserve energy, are invariant with respect to linear transformations, and have arbitrarily high order. Those of optimal order also preserve quadratic Casimir functions. The discussion of the order is based on an interpretation as partitioned Runge–Kutta method with ...

Stochastic Reduced Basis Methods

Numerical Methods for Stochastic Systems Preserving Symplectic Structure

Stochastic Hamiltonian systems with multiplicative noise, phase flows of which preserve symplectic structure, are considered. To construct symplectic methods for such systems, sufficiently general fully implicit schemes, i.e., schemes with implicitness both in deterministic and stochastic terms, are needed. A new class of fully implicit methods for stochastic systems is proposed. Increments of ...