Department of Mathematics
University of Maryland
"Structure Peserving Numerical Methods for Nonlinear Partial Differential Equations Modeling Complex Fluids"
Nonlinear partial differential equations (PDEs) emerge as mathematical descriptions of many phenomena in physics, biology, engineering, and other fields. No unified mathematical theory is available as of now, and showing existence and uniqueness of solutions to nonlinear PDEs is often challenging. Their solutions can develop singularities of various type, such as shock waves, rapid oscillations, and blow-ups.
This complex behavior complicates the task of developing numerical methods to approximate nonlinear PDEs. Good numerical schemes should be stable and efficient but at the same time capture the true physical behavior and singularities that the solution may display. To achieve this, it is crucial to mimic properties that the continuous solution of the PDE has – for example, physical constraints or energy balances – at the discrete level.
In this talk, we will examine the procedure of constructing numerical methods that preserve the underlying structure of the solution to the PDE at the discrete level at the example of some nonlinear PDEs modeling complex fluids. In particular, we will focus on numerical methods for nonlinear PDEs that arise as simplified models for liquid crystal dynamics and the Rosensweig model for ferrofluids (magnetically conducting particles in a carrier fluid). We will also prove that these numerical methods converge.