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Geometry, compatibility and structure preservation in computational differential equations

Participation in INI programmes is by invitation only. Anyone wishing to apply to participate in the associated workshop(s) should use the relevant workshop application form.

3rd July 2019 to 19th December 2019
Elena Celledoni
Douglas Arnold
Franco Brezzi
Arieh Iserles
Elizabeth Mansfield
Reinout Quispel

Programme Theme

Computations of differential equations are a fundamental activity in applied mathematics. While historically the main quest was to derive all-purpose algorithms such as finite difference, finite volume and finite element methods for space discretization, Runge–Kutta and linear multistep methods for time integration, in the last 25 years the focus has shifted to special classes of differential equations and purpose-built algorithms that are tailored to preserve special features of each class. This has given rise to the new fields of geometric numerical integration and of structure preserving discretization. In addition to being quantitatively accurate, these novel methods have the advantage of also being qualitatively accurate as they inherit the key structural properties of their continuum counterparts. This has meant a large-scale introduction of geometric and topological thinking into modern numerical mathematics.

During this scientific programme at the Isaac Newton Institute for Mathematical Sciences, we will address fundamental questions in the field of structure preserving discretizations of differential equations on manifolds in space and time. We will bring together two communities that have been pursuing their science along parallel tracks to endeavour breakthroughs in some major scientific applications, which call for advanced numerical simulation techniques. This will lead to the development of a new generation of space-time discretizations for evolutionary equations.

During the programme we intend to organise three workshops and two focused study periods lasting two weeks on selected application areas.

The core themes of the programme are:

  • Compatible discretizations.
  • Geometric numerical integration.
  • Structure preservation and numerical relativity.
  • Applications to computations in quantum mechanics.
Final Scientific Report: 

GCS Programme: organisers Elizabeth Mansfield, Elena Celledoni and Arieh Iserles

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons