Qinghai Zhang
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SPECIAL MATHEMATICS COLLOQUIUM
Speaker: Qinghai Zhang Abstract. Suppose you wanted to travel from New York to Tallahassee. You could drive a car, or you could fly in an airplane. But if you want to travel from Beijing to Tallahasse, the airplane will get you there and the car won't! This advantage of an airplane over a car is analogous to that of fourth- and higher-order methods over first- and second-order methods. Low-order methods converge, but they might not converge to the right physics when derivatives of primary variables (e.g. curvature or vorticity) are crucial in determining the physics. Also, high-order methods can be much more accurate and efficient than low-ordermethods: to achieve three decimal digits of accuracy for vorticity in solving the incompressible Navier-Stokes equations (INSE) for a turbulent boundary-layer problem (Re=20,000), it is faster to run my fourth-order projection method on a personal computer than to run a classical second-order projection method on the fastest super-computer in the world! In this talk, I will discuss a new formulation of the INSE, a fourth-order projection method, a new fourth-order method for tracking the moving boundaries, a systematic framework for analyzing interface tracking methods, and some applications of the above high-order methods to geophysics and mathematical biology. |