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2005 HMC Mathematics Conference on Scientific Computing: Abstracts

Randy LeVeque (University of Washington, Seattle): Finite-Volume Methods and Software for Hyperbolic PDEs and Conservation Laws

Hyperbolic systems of partial differential equations frequently arise when modeling phenomena involving wave propagation or advective flow. Finite-volume methods are a natural approach for conservation laws since they are based directly on integral formulations and are applicable to problems involving shock waves and other discontinuities. High-resolution shock-capturing methods developed originally for compressible gas dynamics can also be applied to other hyperbolic systems, even if not in conservation form. I will describe a robust class of wave-propagation methods that have been implemented in the CLAWPACK (Conservation LAW PACKage) software for solving hyperbolic problems in one, two, and three space dimensions. Adaptive mesh refinement capabilities are also included. This software has been applied to a variety of problems in diverse fields, including gas dynamics, multiphase flows, linear and nonlinear elasticity, combustion, biological flows, and numerical relativity. Some examples will be shown from recent work on geophysical flows modeling volcanoes and tsunamis.

Adrian Lew (Stanford University): Modeling and Simulation of Highly Deformable Materials

How is the design of a car tire optimized? How can we learn about the relation between the structure of muscle and its function, in healthy and diseased tissue? Will a barrier or container withstand an explosive impact? The assortment and intricacy of behaviors observed in these examples is originated in strong nonlinearities present in each problem. Complex material response capable of undergoing important shape changes is underlying all these questions, and numerical methods to capture them are the subjects of this talk.

I will begin by presenting a class of time-integration algorithms termed Asynchronous Variational Integrators (AVI), designed to accelerate the simulation of multi-physics problems with multiple time scales, as well as their very recent parallel versions (PAVI). The algorithms have a beautiful geometric origin wich endows them with a number of desirable properties, to be discussed. I will later comment very briefly about some general aspects of nonlinear elasticity, and show recent developments in my group in the creation of discontinuous Galerkin methods for its simulation. I will demonstrate the performance of the new methods through selected examples.

Linda Petzold (University of California, Santa Barbara): Bridging the Scales in Biochemical Simulation

In microscopic systems formed by living cells, the small numbers of reactant molecules can result in dynamical behavior that is discrete and stochastic rather than continuous and deterministic. An analysis tool that respects these dynamical characteristics is the stochastic simulation algorithm (SSA), a numerical simulation procedure that is essentially exact for chemical systems that are spatially homogeneous or well stirred. Despite recent improvements, as a procedure that simulates every reaction event, the SSA is necessarily inefficient for most realistic problems. There are two main reasons for this, both arising from the multiscale nature of the underlying problem: (1) stiffness, i.e. the presence of multiple timescales, the fastest of which are stable; and (2) the need to include in the simulation both species that are present in relatively small quantities and should be modeled by a discrete stochastic process, and species that are present in larger quantities and are more efficiently modeled by a deterministic differential equation (or at some scale in between). We will describe several recently developed techniques for multiscale simulation of biochemical systems, and outline some of the technical challenges that still need to be addressed.

Jane Wang (Cornell University): Computational Modeling of Insect Flight

Most living species, include protozoa, bacteria, insects, birds, and fish, locomote via the interactions between fluids and moving surfaces. Whether using cilia, flagella, wings, or fins, the locomotion is governed by the Navier-Stokes equation coupled to moving boundaries. The computational methods I will describe are two examples of solving this class of problems. The first method is designed to solve a single flapping wing using a conformal mesh to resolve the wing tip, and the second method is an immersed interface method to simulate multiple wing interactions. I will discuss some of our recent results on the efficiency of flapping flight, the trajectories of a piece of falling paper, and the fore-hind wing interactions in dragonfly flight.