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Dive into the research topics where Jason Frank is active.

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Featured researches published by Jason Frank.


SIAM Journal on Scientific Computing | 2001

On the Construction of Deflation-Based Preconditioners

Jason Frank; C. Vuik

In this article we introduce new bounds on the effective condition number of deflated and preconditioned-deflated symmetric positive definite linear systems. For the case of a subdomain deflation such as that of Nicolaides [SIAM J. Numer. Anal., 24 (1987), pp. 355--365], these theorems can provide direction in choosing a proper decomposition into subdomains. If grid refinement is performed, keeping the subdomain grid resolution fixed, the condition number is insensitive to the grid size. Subdomain deflation is very easy to implement and has been parallelized on a distributed memory system with only a small amount of additional communication. Numerical experiments for a steady-state convection-diffusion problem are included.


Applied Numerical Mathematics | 1997

On the stability of implicit-explicit linear multistep methods

Jason Frank; Willem Hundsdorfer; Jan Verwer

In many applications, large systems of ordinary differential equations (ODEs) have to be solved numerically that have both stiff and nonstiff parts. A popular approach in such cases is to integrate the stiff parts implicitly and the nonstiff parts explicitly. In this paper we study a class of implicit-explicit (IMEX) linear multistep methods intended for such applications. The paper focuses on the linear stability of popular second order methods like extrapolated BDF, Crank-Nicolson Leap-Frog and a particular class of Adams methods. We present results for problems with decoupled eigenvalues and comment on some specific CFL restrictions associated with advection terms.


SIAM Journal on Scientific Computing | 2006

Linear PDEs and Numerical Methods That Preserve a Multisymplectic Conservation Law

Jason Frank; Brian E. Moore; Sebastian Reich

Multisymplectic 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 multisymplectic methods is still largely missing. In this paper, we study linear multisymplectic PDEs and their discretization by means of numerical dispersion relations. It is found that multisymplectic methods in the sense of Bridges and Reich Phys. Lett. A, 284 (2001), pp. 184-193] and Reich J. Comput. Phys., 157 (2000), pp. 473-499], such as Gauss-Legendre Runge-Kutta methods, possess a number of desirable properties such as nonexistence of spurious roots and conservation of the sign of the group velocity. A certain CFL-type restriction on


computational science and engineering | 2003

A Hamiltonian Particle-Mesh Method for the Rotating Shallow-Water Equations

Jason Frank; Georg A. Gottwald; Sebastian Reich

\Delta t/\Delta x


Journal of Computational Physics | 2007

Statistical mechanics of Arakawa's discretizations

Svetlana Dubinkina; Jason Frank

might be required for methods higher than second order in time. It is also demonstrated by means of the explicit midpoint method that multistep methods may exhibit spurious roots in the numerical dispersion relation for any value of


International Journal of Computer Mathematics | 2007

On the multisymplecticity of partitioned Runge-Kutta and splitting methods

Brett N. Ryland; Robert I. McLachlan; Jason Frank

\Delta t/\Delta x


Quarterly Journal of the Royal Meteorological Society | 2007

The remapped particle-mesh semi-Lagrangian advection scheme

Colin J. Cotter; Jason Frank; Sebastian Reich

despite being multisymplectic in the sense of discrete variational mechanics [J. E. Marsden, G. P. Patrick, and S. Shkoller, Commun. Math. Phys., 199 (1999), pp. 351-395].


Applied Numerical Mathematics | 1999

Parallel implementation of a multiblock method with approximate subdomain solution

Jason Frank; C. Vuik

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 α-Euler models.


Journal of Computational Physics | 2010

Statistical relevance of vorticity conservation in the Hamiltonian particle-mesh method

Svetlana Dubinkina; Jason Frank

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 used. This is illustrated for quasi-geostrophic flow with topographic forcing, for which a well established statistical mechanics exists. Statistical mechanical theories are constructed for the discrete dynamical systems arising from three discretizations due to Arakawa [Arakawa, Computational design for long-term numerical integration of the equations of fluid motion: two-dimensional incompressible flow. Part I. J. Comput. Phys. 1 (1966) 119-143] which conserve energy, enstrophy or both. Numerical experiments with conservative and projected time integrators show that the statistical theories accurately explain the differences observed in statistics derived from the discretizations.


Journal of Physics A | 2006

Conservation of wave action under multisymplectic discretizations

Jason Frank

Although Runge–Kutta and partitioned Runge–Kutta methods are known to formally satisfy discrete multisymplectic conservation laws when applied to multi-Hamiltonian PDEs, they do not always lead to well-defined numerical methods. We consider the case study of the nonlinear Schrödinger equation in detail, for which the previously known multisymplectic integrators are fully implicit and based on the (second order) box scheme, and construct well-defined, explicit integrators, of various orders, with local discrete multisymplectic conservation laws, based on partitioned Runge–Kutta methods. We also show that two popular explicit splitting methods are multisymplectic.

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Svetlana Dubinkina

Université catholique de Louvain

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C. Vuik

Delft University of Technology

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Barry Koren

Eindhoven University of Technology

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