PROJECTS SUMMARY

1. PROJECTS SUMMARY MATHEMATICS DEPT. PENN STATE PENGTAO SUN 6/25/2003

2. BLACK HOLE SIMULATIONParticipators: Pablo Laguna, Jinchao Xu, Pengtao Sun • Numerical evolutions of black holes have been improved slowly but steadily over the last few years and now first attempts are being made to extract physical information from these evolutions. Most notably one wants to predict the gravitational radiation emitted during black hole coalescence • Initial data are the starting point for any numerical simulation. In the case of numerical relativity, Einstein's equations constrain our choices of these initial data. • The quality of the initial data will be crucial to the success of the predictions of the gravitational wave forms. Unphysical gravitational radiation present in the initial data will contribute to the gravitational waves computed in an evolution and might overwhelm the true gravitational wave signature of the physical process under consideration. Therefore an important question is how to control the gravitational wave content of initial-data sets, and how to specify astrophysically relevant initial data with the appropriate gravitational wave content, for e.g. two black holes orbiting each other.

3. BLACK HOLE IN 3D

4. BLACK HOLE IN 2D • Extreme mass ratios binary systems, binaries involving compact objects such as stellar mass black holes or neutron stars orbiting super-massive black holes, are considered to be a primary source of gravitational radiation to be detected by the space-based interferometer LISA. • The numerical modeling of these binary systems is extremely challenging because the scales involved expand orders of magnitude. One needs to handle large wavelength scales comparable to the super-massive black hole and, at the same time, to resolve the scales in the vicinity of the small compact object where radiation reaction effects play a crucial role. • Finite element methods are a natural choice to achieve this high level of adaptivity. To demonstrate this, we present results of a toy problem consisting of a point-like source orbiting a black hole in scalar gravitation.

5. TWO PHASE FLOWSExample 1— Thermo-induced Marangoni effects • One of two-phase fluids flow example, is two-fluid Marangoni convection in which the heated boundary is embedded in the free surface between two liquid with different densities (with the lighter one on the top), the induced temperature gradient on the surface drives convective motion, and induces vorticity in the bulk fluid. This motion deforms the free surface and lead to a complicated flow pattern. Preliminary numerical experiments for a two dimensional model have already been performed using grid adaptation techniques • The experimental counter part has been constructed in the Pritchard Lab by Belmonte.

6. TWO PHASE FLOWSExample 2 — Gel fingering phenomenon • It is well known that certain surfactants can form long cylindrical micelles in aqueous solution at low concentration, if they are mixed with certain large and semi-hydrophobic organic counterions such as sodium salicylate. Such solutions, known as wormlike micellar fluids, are similar in some ways to polymer solutions. • We found a thick gel-like micellar phase at the interface between an aqueous surfactant solution and an aqueous organic salt solution. When mixed homogeneously, these two solutions are known to form a highly elastic fluid. This observation are made during the slow injection of one fluid into the other by a tube or pipette.

7. Fingering of Gel • Numerical experiment is ongoing, the primary results are as follows.

8. TWO PHASE FLOWSExample 3 — Surface water waves • Another even more challenging example is in the area of surface water waves. The Pritchard Lab contains a wave basin with a segmented, programmable wave-maker system that is capable of generating both 2D and 3D water waves. These wave motions are typically modeled by the (inviscid) Euler equations assuming the flow to be irrotational. Yet, both viscous and rotational effects have been observed in many experiments. In particular, a remarkably stable 2D vortex has been observed in 2D and weakly 3D experiments carried out by J. Hammack and D. Henderson. The vortex forms near the center of the basin, spanning its width,and then propagates slowly to the wave-maker where it is extinguished. • We will use our grid adaptation and multigrid techniques to develop a \numerical wave basin" based on the Navier-Stokes equations that will be used to predict wave motions as well as resulting vortical motions.

9. FUEL CELL MODEL • We propose a model of liquid and heat flux, ignoring the gas dynamics. More specifically we assume that the pressure and vapour pressure are constant and solve for the water volume fraction and the temperature as functions of space and time. The water motion is driven by capillary pressure, and a heat flux is generated by boundary conditions. The two equations are coupled by condensation, which exchanges heat for liquid, generating a liquid flux opposite that of the heat flux. • Numerical simulation is done with the adaptive finite element method

10. CONVECTION-DIFFUSION PROBLEM • multilevel discretization and grid adaptation

11. ANISOTROPIC ADAPTIVITY • All edges are equal under some metric which depends on p norm, which give us a criteria to construct a nonlinear functional which satisfies the equidistribution and isotropy simultaneously.