2026-09-18T22:49:51Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/417362022-01-13T07:54:53Zcom_1721.1_7582com_1721.1_7581col_1721.1_131023
W. Gilbert Strang.
Dharmaraja, Sohan
Massachusetts Institute of Technology. Computation for Design and Optimization Program.
Massachusetts Institute of Technology. Computation for Design and Optimization Program
2008-05-19T16:13:17Z
2008-05-19T16:13:17Z
2007
2007
http://hdl.handle.net/1721.1/41736
225094125
Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2007.
Includes bibliographical references (p. 75-76).
We intend to try to better our understanding of how the combined L-stable 'Trapezoidal Rule with the second order Backward Difference Formula' (TR-BDF2) integrator and the standard A-stable Trapezoidal integrator perform on systems of coupled non-linear partial differential equations (PDEs). It was originally Professor KlausJiirgen Bathe who suggested that further analysis was needed in this area. We draw attention to numerical instabilities that arise due to insufficient numerical damping from the Crank-Nicolson method (which is based on the Trapezoidal rule) and demonstrate how these problems can be rectified with the TR-BDF2 scheme. Several examples are presented, including an advection-diffusion-reaction (ADR) problem and the (chaotic) damped driven pendulum. We also briefly introduce how the ideas of splitting methods can be coupled with the TR-BDF2 scheme and applied to the ADR equation to take advantage of the excellent modern day explicit techniques to solve hyperbolic equations.
by Sohan Dharmaraja.
S.M.
76 leaves
eng
Massachusetts Institute of Technology
M.I.T. theses are protected by
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http://dspace.mit.edu/handle/1721.1/7582
Computation for Design and Optimization Program.
An analysis of the TR-BDF2 integration scheme
Analysis of the Trapezoidal Rule with the second order Backward Difference Formula integration scheme
Thesis
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<Abstract>We intend to try to better our understanding of how the combined L-stable 'Trapezoidal Rule with the second order Backward Difference Formula' (TR-BDF2) integrator and the standard A-stable Trapezoidal integrator perform on systems of coupled non-linear partial differential equations (PDEs). It was originally Professor KlausJiirgen Bathe who suggested that further analysis was needed in this area. We draw attention to numerical instabilities that arise due to insufficient numerical damping from the Crank-Nicolson method (which is based on the Trapezoidal rule) and demonstrate how these problems can be rectified with the TR-BDF2 scheme. Several examples are presented, including an advection-diffusion-reaction (ADR) problem and the (chaotic) damped driven pendulum. We also briefly introduce how the ideas of splitting methods can be coupled with the TR-BDF2 scheme and applied to the ADR equation to take advantage of the excellent modern day explicit techniques to solve hyperbolic equations.</Abstract>
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