Title: Positivity-preserving methods for ideal magnetohydrodynamics Abstract: The density and pressure are positive physical quantities in magnetohydrodynamics (MHD). Design of provably positivity-preserving (PP) numerical schemes for ideal compressible MHD is highly desired, but remains a big challenge. The difficulties mainly arise from the intrinsic complexity of the MHD equations as well as the indeterminate relation between the PP property and the divergence-free condition on the magnetic field. In this talk, I am going to introduce our recent efforts on the design and rigorous analysis of PP methods for ideal MHD. The analysis theoretically reveals a tight connection between the PP property of conservative MHD schemes and a discrete divergence-free condition. Provably PP discontinuous Galerkin methods for symmetrizable MHD equations will also be introduced, and are joint work with Professor Chi-Wang Shu.
Title: Entropy stable high order discontinuous Galerkin methods for nonlinear conservation laws Abstract: High order discontinuous Galerkin (DG) methods offer several advantages in the approximation of solutions of nonlinear conservation laws, such as geometric flexibility, improved accuracy, and low numerical dispersion/dissipation. However, these methods also tend to suffer from instability in practice, requiring filtering, limiting, or artificial dissipation to prevent solution blow up. Entropy stable schemes address one primary cause of this instability by utilizing summation-by-parts (SBP) finite difference operators and an approach called flux differencing to ensure that the solution satisfies a semi-discrete entropy inequality. In this talk, we show that high order DG methods can be re-interpreted within an SBP framework using discrete projection and “decoupled” SBP operators, and utilize this equivalence to construct semi-discretely entropy stable schemes on meshes of simplicial and tensor product elements.
Friday, Sep 28 11:30 am - 12:30 pm
Title: Modeling and Simulation for Solid-State Dewetting Problems Abstract: In this talk, I present sharp interface models with anisotropic surface energy and a phase field model for simulating solid-state dewetting and the morphological evolution of patterned islands on a substrate in two and three dimensions. We show how to derive the sharp interface model via thermovariation dynamics, i.e. variation of the interfacial energy via an open curve with two triple points moving along a fixed substrate. The sharp interface model tracks the moving interface explicitly and it is very easy to be handled in two dimensions via arc-length parametrization. The phase field model is governed by the Cahn-Hilliard equation with isotropic surface tension and variable scalar mobility and it easily deals with the complex boundary conditions and/or complicated geometry arising in the solid-state dewetting problem. Since the phase field model does not explicitly track the moving surface, it naturally captures the topological changes that occur during film/island morphology evolution. Efficient and accurate numerical methods for both sharp interface models and phase field models are proposed. They are applied to study numerically different setups of solid-state dewetting including short and long island films, pinch-off, hole dynamics, semi-infinite film, etc. Our results agree with experimental results very well. This talk is based on joint works with Wei Jiang, David J. Srolovitz, Carl V. Thompson, Yan Wang and Quan Zhao.