Analysis of Fluid Motion

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DMS-9870091 J. Thomas Beale Abstract The purpose of the proposed work is to analyze the errors in numerical methods for the time-dependent motion of water waves, to design improved methods, and to verify the reliability of the proposed methods with error estimates, convergence proofs, and computational tests. The methods of interest are called boundary integral methods. They have been widely used in ocean engineering as well as in applied mathematics. The motion of the water surface is tracked by markers representing material particles which are moved according to a velocity field computed from quantities on the surface. The velocity is expressed in terms of singular integrals, since the governing equations are formulated through potential theory. The evolution has the character of nonlinear, nonlocal wave motion on the boundary. Numerical instabilities have long been observed in such methods, but analysis of the two-dimensional case has shown that they can be ruled out by proper design. The first goal of the proposed research is to develop stable, convergent numerical methods for the simulation of exact, time-dependent, three-dimensional water wave motion which is periodic in the horizontal directions. Thus the full nonlinear motion will be dealt with, but interactions with solid objects will be neglected. The method of calculation of the singular integrals is a primary issue, one which may have relevance to other problems. A second goal is to deal with the errors from edges where a solid object meets the water surface. Practical predictions of the motion of fluids and solids are often made by numerical solution of the equations which express the basic physical laws of the motion. Methods of the type studied here have been used in ocean engineering in order to predict the motion of large water waves, the interaction of water waves with ships or stationary objects, and the force exerted by a wave on a wall or object. Calculations of these probl ems and others have encountered numerical instabilities; that is, because of the numerical approximation, small scales may grow in a way unrelated to the physical problem. Mathematical analysis can be used to understand such errors and to design improved methods. Previous work for two-dimensional waves (those not varying in the third direction) has identified the sources of errors for this special case. The purpose of the present work is to carry out such analysis in more difficult and realistic cases where the sources of errors are likely to be more serious. It is hoped that improvements in the numerical methods could contribute to reliable predictions of water wave motion.