Past Event: Oden Institute Seminar

Development and Applications of Numerical Solvers for Nonlinear Partial Differential Equations on Octree Adaptive Grids

Frederic Gibou, Professor, Departments of Mechanical Engineering and Mathematics, University of California Santa Barbara

Abstract

Several phenomena in the physical and the life sciences can be modeled as time dependent free boundary problems and nonlinear partial differential equations. Examples include the study of electro-osmotic flows, solidification of binary alloys and polymeric materials. One of the main difficulties in solving numerically these equations is associated with the fact that such problems involve dissimilar length scales, with smaller scales influencing larger ones so that nontrivial pattern formation dynamics can be expected to occur at all intermediate scales. Uniform grids are limited in their ability to resolve small scales and are in such situations extremely inefficient in terms of memory storage and CPU requirements. Another difficulty stems from the fact that the geometry of the problems is often arbitrary and special care is needed to correctly apply boundary conditions. In this talk, I will present recent advances in the numerical treatment of interface problems and describe new numerical solvers for nonlinear partial differential equations in the context of adaptive mesh refinement based on Octree grids and some applications. If time permits, I will also present a method for accurately simulating fluid-solid two-way coupling. Bio Professor Gibou is also a core faculty member in the Computational Science and Engineering program. He received his PhD from the Applied Mathematics Department at UCLA, and did his post-doctoral research in the Departments of Mathematics and Computer Science at Stanford University. His research is at the interface between Applied Mathematics, Computer Science and Engineering Sciences. It is focused on the design of a novel paradigm for high resolution computational methods for large scale computations and their use for a variety of applications including Computational Materials Science, Computational Fluid Dynamics and Computational Image Analysis. The main commonality in these applications is that they are described by complex/free boundaries and similar classes of nonlinear partial differential equations. In addition, the multiscale nature of most interesting phenomena in the physical and life sciences motivate the development and use of versatile computational strategies on spatially adaptive grids and on massively parallel environments. Professor Gibou leads a multidisciplinary research group, named Computational Applied Science Laboratory (CASL). Strong collaborations exist between CASL and other research groups at UCSB and worldwide, especially with experimentalists. It is therefore standard that researchers in CASL benefit from a truly interdisciplinary training in modern science and engineering.