Seminar: Numerical Methods for Partial Differential Equations

Wednesday, December 06, 2023 at 4:30pm to 5:30pm

Building 2, 449
182 MEMORIAL DR, Cambridge, MA 02139

Speaker:  Wim Van Rees  (MIT)

Title:  A high-order sharp immersed method for simulating PDEs with moving interfaces or boundaries on adaptive grids


The performance and safety of ocean systems such as underwater robots, aquaculture installations, and hydrofoils, are ultimately governed by non-linear flows and fluid-structure interactions. Performing high-fidelity simulations of such problems requires the ability to handle complex, moving domains and multiphysics phenomena. Immersed methods are widely used for this category of problems, because of their ability to handle complex domains with moving boundaries on structured grids without the need to generate body-fitted meshes and associated remeshing. A disadvantage of these methods is the complexity involved in extending their accuracy beyond first or second order, especially for quantities near or on the boundary or interface. Another challenge is their generalization to arbitrary boundary and interface conditions. Consequently, the vast majority of existing embedded boundary methods achieve first or second order accuracy in space.

In this talk I will detail our progress towards increasing the efficiency of adaptive-grid flow problems with immersed moving boundaries and interfaces. First, I will discuss our approach for high-order finite-difference based immersed boundary and interface discretization, and share our error and stability analyses. I will demonstrate the effectiveness of the approach in a variety of problems including a second-order 2D vorticity-velocity Navier-Stokes solver, high-order 2D linear elasticity problems, and high-order 3D advection-diffusion problems. Second, I will present our methodology for 3D multiresolution grid adaptation techniques using a wavelet-based analysis implemented within a scalable parallel software framework. This leads to a predictable convergence of the solution error with respect to the adaptation criteria, while the wavelet order effectively controls the compression ratio.


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