EAS-PHD Proposal Defense by Zachary J. Grant
When: Monday,
March 5, 2018
11:00 AM
-
12:00 PM
Where: Textiles Building 105
Description: TITLE: ON IMPLICIT AND EXPLICIT HIGH ORDER SSP METHODS WITH A VARIETY STAGES, STEPS, AND DERIVATIVES FOR LINER 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 were worked out in terms of the method coefficients, and we formulated 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 found 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 were used to find optimal SSP multistage two-derivative methods. These methods were tested on a variety of scalar hyperbolic partial differential equations, which demonstrate the need for the SSP condition and the sharpness of the SSP time-step.
Advisor: Dr. Sigal Gottlieb, Mathematics
Committee members: Prof. Gaurav Khanna, Physics; Prof. Yanlai Chen, Mathematics; and
Prof. Alfa Heryudono, Mathematics
For further information, please contact Dr. Sigal Gottlieb at 508-999-8205 or by email sgottlieb@umassd.edu.
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 were worked out in terms of the method coefficients, and we formulated 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 found 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 were used to find optimal SSP multistage two-derivative methods. These methods were tested on a variety of scalar hyperbolic partial differential equations, which demonstrate the need for the SSP condition and the sharpness of the SSP time-step.
Advisor: Dr. Sigal Gottlieb, Mathematics
Committee members: Prof. Gaurav Khanna, Physics; Prof. Yanlai Chen, Mathematics; and
Prof. Alfa Heryudono, Mathematics
For further information, please contact Dr. Sigal Gottlieb at 508-999-8205 or by email sgottlieb@umassd.edu.
Contact: > See Description for contact information
Topical Areas: University Community, Mathematics, College of Engineering, Physics