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Superconvergence of discontinuous Galerkin method for linear hyperbolic equations
Rahmati, Sirvan
Rahmati, Sirvan
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dissertation
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2020-12
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Abstract
This thesis is concerned with the investigation of the superconvergence of the Discontinuous
Method for linear conservation laws. We use Fourier analysis to study the superconvergence
of the semi-discrete discontinuous Galerkin method for scalar linear advection
equations in one spatial dimension.
We provide the error bounds and asymptotic errors for initial di erent initial discretizations.
For the pedagogical purpose, the errors are computed in two di erent ways. In
the rst approach, we compute the di erence between the numerical solution and a special
interpolation of the exact solution, and show that it consists of an asymptotic error of order
2k + 1 (where k is the order of polynomial approximation) and a transient error of lower
order.
In the second approach, we compute the error directly by decomposing it into physical
and nonphysical modes, and obtain agreement with the rst approach. We then extend the
analysis to vector conservation laws, solved using the Lax-Friedrichs
ux. We prove that the
superconvergence holds with the same order. The error bounds and asymptotic errors are
demonstrated by various numerical experiments for scalar and vector advection equations.
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Thesis (Ph.D.)-- Wichita State University, College of Liberal Arts and Sciences, Dept. of Mathematics, Statistics and Physics
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Wichita State University
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© Copyright 2020 by Sirvan Rahmati
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