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- Jason Frank, Cornelis Vuik
- SIAM J. Scientific Computing
- 2001

In this article we introduce new bounds on the e ective condition number of de ated and preconditioned-de ated symmetric positive de nite linear systems. For the case of a subdomain de ation such as that of Nicolaides (1987), these theorems can provide direction in choosing a proper decomposition into subdomains. If grid re nement is done keeping the… (More)

The Hamiltonian particle-mesh (HPM) method is generalized to the spherical shallow water equations, utilizing constrained particle dynamics on the sphere and smoothing with Merilees’ double-periodic FFT formulation of O(J log J) in the latitudinal gridsize. The time step for the explicit, symplectic integrator depends only on the uniform smoothing length.… (More)

- Svetlana Dubinkina, Jason Frank
- J. Comput. Physics
- 2007

The results of statistical analysis of simulation data obtained from long time integrations of geophysical fluid models greatly depend on the conservation properties of the numerical discretization chosen. This is illustrated for quasi-geostrophic flow with topographic forcing, for which a well established statistical mechanics exists. Statistical… (More)

- Jason Frank, Brian E. Moore, Sebastian Reich
- SIAM J. Scientific Computing
- 2006

Multi-symplectic methods have recently been proposed as a generalization of symplectic ODE methods to the case of Hamiltonian PDEs. Their excellent long time behavior for a variety of Hamiltonian wave equations has been demonstrated in a number of numerical studies. A theoretical investigation and justification of multi-symplectic methods is still largely… (More)

This paper addresses nonphysical reflections encountered in the discretization of wave equations on nonuniform grids. Such nonphysical solutions are commonly attributed to spurious modes in the numerical dispersion relation. We provide an example of a discretization in which a (nonspurious) physical mode is spuriously energized at a grid nonuniformity. It… (More)

A new particle-mesh method is proposed for the rotating shallow-water equations. The spatially truncated equations are Hamiltonian and satisfy a Kelvin circulation theorem. The generation of non-smooth components in the layer-depth is avoided by applying a smoothing operator similar to what has recently been discussed in the context of -Euler models.

In this note we study the asymptotic limit of large variance in a stochastically perturbed thermostat model, the Nosé-Hoover-Langevin device. We show that in this limit, the model reduces to a Langevin equation with one-dimensional Wiener process, and that the perturbation is in the direction of the conjugate momentum vector. Numerical experiments with a… (More)

- Jason Frank, Sergiy Zhuk
- CDC
- 2014

The paper presents symplectic Möbius integrators for Riccati equations. All proposed methods preserve symmetry, positivity and quadratic invariants for the Riccati equations, and non-stationary Lyapunov functions. In addition, an efficient and numerically stable discretization procedure based on reinitialization for the associated linear Hamiltonian system… (More)

- Wander Wadman, Daan Crommelin, Jason Frank
- 2013 Winter Simulations Conference (WSC)
- 2013

As intermittent renewable energy penetrates electrical power grids more and more, assessing grid reliability is of increasing concern for grid operators. Monte Carlo simulation is a robust and popular technique to estimate indices for grid reliability, but the involved computational intensity may be too high for typical reliability analyses. We show that… (More)

- Colin J. Cotter, Jason Frank, Sebastian Reich
- SIAM J. Applied Dynamical Systems
- 2004

We develop a particle-mesh method for two-layer shallow-water equations subject to the rigid-lid approximation. The method is based on the recently proposed Hamiltonian particle-mesh (HPM) method and the interpretation of the rigid-lid approximation as a set of holonomic constraints. The suggested spatial discretization leads to a constrained Hamiltonian… (More)