Subtheme: Geometric Integration (MAS1.3)

Of research group: Computational and Stochastic Dynamics

Coordinator of this project: Jason Frank
Startdate: 2001-01-01
Enddate: 2009-07-31

This subtheme addresses the numerical modelling of partial differential equations from application areas in geophysical fluids dynamics, including wave motion and transport, on long time scales. Important themes are conservation properties of discrete models, geometric integration, statistical accuracy measures for long-time simulations, and error control.

Members
Svetlana Dubinkina, Jason Frank, Bob Peeters, Jan Verwer.

Description of projects

• GEO - Geometric numerical methods for geophysical fluid dynamics
  Geometric numerical methods are designed to conserve, under discretization, numerous structural properties of the continuous model to which they are applied, for finite values of the discretization parameters. Consequently, the numerical solutions obtained with such methods may be more physically relevant than solutions obtained from general methods. Examples of structural properties in wave models are (multi-)symplectic structure, energy-momentum conservation laws, wave action, and properties of the dispersion relation. By preserving these features, we arrive at a discrete dynamical system which may be studied as an analog of the continuous one. Current activity in this project focuses on conservative and adaptive methods for Hamiltonian wave equations, with application to internal waves in the ocean.
• SIDiss - Hamiltonian-based numerical methods in forced-dissipative climate predictions
  Assessment of symplectic discretization of atmospheric dynamics in the presence of weak dissipation. Joint project (NWO/ALW Klimaatvariabiliteit) with Dr. O. Bokhove (P.I., U. Twente).
• IVP - Global error estimation and control for initial value problems
  Reliable and efficient global error estimation and control belongs to the main challenges in scientific computing. Since it requires incorporation of the conditioning of the problem and solution of additional equations for the errors it comes with genuine additional costs which seems to be an obstacle for further development. This project will focus on initial value problems for ordinary differential equations. Taking the backward error analysis point of view through a perturbed system approach, two techniques will be studied, viz. a novel technique based on the dual adjoint method combined with a small sample statistical initialization, and a classical technique based on the first variational equation with defect computations. The aim is to select the most efficient technique and to show that for truly ill-conditioned (unstable) initial-value problems this technique offers a substantial improvement compared to the standard error control technique solely based on local errors.
• SIAD - Symplectic Integration of Atmospheric Dynamics
  The mathematical equations modeling atmospheric flows are chaotic, and yet must be integrated over long time intervals for weather prediction and climate simulations. This calls for advanced numerical methods that produce statistically accurate solutions in the absence of traditional numerical accuracy. The goal of this project funded by the Climate Variability Program of NWO/ALW is to develop numerical approach for atmospheric simulations on climatic time-scales that is suitable for statistical ensemble simulations. The Hamiltonian Particle-Mesh (HPM) method was originally developed for planar shallow water flow, and subsequently extended to two-layer flows, spherical geometry, and 2-D vertical slice models. On the one hand, the HPM method will be further extended to more realistic 3D atmospheric models. At the same time, we are investigating effective measures of accuracy on long time-scale simulations. Joint work with S. Reich of Potsdam University. Articles on this work have appeared in SIAM J. Appl. Dyn. Systems, BIT and Atmospheric Science Letters.
• MSRK - Adaptive multisymplectic box schemes for Hamiltonian wave equations
  Adaptivity considerations for a class of geometric methods with remarkable conservation properties, applied to external and surface waves in oceanography.

Key publications
Publications of Geometric Integration

Project reports
Reports of Geometric Integration

Cooperation partners

• Onno Bokhove IMPACT, U. Twente
• Jens Lang U. Darmstadt.
• B. Leimkuhler, Mathematical Modelling Centre, University of Leicester.
• Leo Maas NIOZ, Texel
• Sebastian Reich Institute for Mathematics, Potsdam University.

Browse
See the web site of the Hamiltonian Particle-Mesh (HPM) method.