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EAS Doctoral Proposal Defense by Zachary Grant

When: Thursday, September 29, 2016
9:20 AM - 11:20 AM
Where: Textiles Building 105
Description: TOPIC: ON IMPLICIT & EXPLICIT HIGH ORDER SSP METHODS WITH A VARIETY OF STAGES, STEPS, AND DERIVATIVES FOR LINEAR AND NONLINEAR PROBLEMS.

Abstract: High order spatial discretizations with monotonicity properties are often desirable for the solution of hyperbolic PDEs. These methods can advantageously be coupled with high order strong stability preserving time discretizations. The search for high order strong stability time-stepping methods with large allowable strong stability coefficient has been an active area of research over the last two decades. This research has shown that explicit SSP Runge--Kutta methods exist only up to fourth order, and implicit ones only up to sixth order, while multistep SSP methods have small allowable time step. To break these order barriers and SSP time-step bounds, we turn to general (multistep, multi-stage) linear methods of order two and above. Order conditions and monotonicity conditions for such methods are worked out in terms of the method coefficients, and we propose a numerical optimization to find optimal methods of up to five steps, eight stages, and tenth order.

Another approach to breaking the order barrier is to restrict ourselves to solving only linear autonomous problems. In this case the order conditions simplify and this order barrier is lifted: explicit SSP Runge--Kutta methods of any linear order exist, but these methods reduce to second order when applied to nonlinear problems. We propose to seek explicit SSP Runge--Kutta methods with large allowable time-step, that feature high linear order and simultaneously have the optimal fourth order nonlinear order. Finally, we turn to multiderivative time-stepping methods which have recently been implemented with hyperbolic PDEs. We describe sufficient conditions for a two-derivative multistage method to be SSP, and formulate an optimization problem that will be used to find optimal SSP multistage two-derivative methods. These methods will be tested on a variety of scalar hyperbolic partial differential equations, to demonstrate the need for the SSP condition and investigate the sharpness of the SSP time-step.
Topical Areas: University Community, Mathematics, College of Engineering