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NA Seminar: Franziska Weber, U Maryland, Structure preserving numerical methods for nonlinear partial differential equations in material science
Nonlinear partial differential equations (PDEs) emerge as mathematical descriptions of many phenomena in physics, biology, engineering, and other fields. Despite lots of research efforts, there are still many open questions in the understanding of nonlinear PDEs. This can be attributed to the complex behavior that solutions of nonlinear PDEs exhibit: They develop singularities of various type, such as shock waves, blow-ups and rapid oscillations.
This also poses a challenge for the design of efficient numerical methods for nonlinear PDEs: The methods should be stable and 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 constraint or energy balances – at the discrete level.
In this talk, I 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 PDEs that emerge in material sciences. In particular, I 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). I will also show that these numerical methods converge.