Geometry, compatibility and structure preservation in computational differential equations

 Geometry, compatibility and structure preservation in computational differential equations's image
Created: 2019-07-17 14:42
Institution: Isaac Newton Institute for Mathematical Sciences
Description: 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.

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