Applied Mathematics Seminar
Monday, April 20, 1998, 4:00pm, 120 Physics
Christina Marliani (Courant Institute of Math. Sci., NYU)
Adaptive Mesh Computations for Fluid and Plasma Flows
Abstract:
The formation of singular structures in incompressible hydrodynamic and related systems is still a controversial issue. Although there exist high resolution numerical experiments the question whether the three dimensional incompressible Euler equations develop a finite time singularity in the vorticity is still under discussion. A main problem with non-adaptive three-dimensional simulations is the insufficient spatial resolution due to limited computer resources. The situation is even worse for the ideal incompressible magnetohydrodynamic equations. Already the question whether global solutions do exist in two dimensions is not solved analytically, although numerical simulations indicate that the growth of vorticity and current density is only exponential in time. Here we investigate such problems using adaptive mesh refinement techniques as introduced by Berger and Colella [J. Comput. Phys. 82, 64 (1989)] and Bell et al. [SIAM J. Sci. Comput. 15, 127 (1994)]. Our algorithm for this purpose was recently described in Friedel et al. [J. Comput. Phys. 134, 190 (1997)]. First, I will show results for the magnetohydrodynamic equations in two dimensions. Simulations show exponential growth of vorticity and current density in accordance with a simple scaling ansatz. The necessary depletion of the nonlinear production term is demonstrated numerically. However, the question why the flow develops this type of alignment is still not understood. In addition, I show recent results for the coalescence instability (compare Longcope and Strauss [Phys. Fluids B, 5, 2858, 1993]) obtained with the same adaptive mesh refinement technique. Simulations for the ideal case and the case of small, non-vanishing resistivity are discussed. Finally, I present simulation for the three-dimensional incompressible Euler equations. The initial conditions are those of Bell and Marcus [Comm. Math. Phys. 147, 371 (1992)] and constitute a cylindrical shear layer. Results of the adaptive mesh treatment indicate a finite time singularity of vorticity.

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