| Contents |
| I will describe a framework for constructing discrete models of
infinite-dimensional systems which preserve underlying geometry. These models
lead to numerical methods that capture the dynamics of the system without
energy or momenta loss and preserve momentum maps in discrete realm. This work
starts with developing a matrix-based exterior calculus. This calculus extends
the classical Discrete Exterior Calculus providing us with the notions of
discrete Lie derivative, interior product etc., while preserving many
properties of their continuous counterparts. This matrix-based exterior
calculus has been used to create structure-preserving discretizations of
various systems, such as fluid dynamics, magnetohydrodynamics, complex fluids
etc. I will show how it can be used to construct a variational integrator for
the Euler and EPDiff equations. I will also describe how these methods can be
extended to create a new model of discrete differential geometry. This
approach uses ideas of noncommutative geometry and can lead to a
structure-preserving discretization of general relativity. |
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