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• W213426403 abstract "In this dissertation we discuss the numerical solution of systems of hyperbolic partial differential equations with lower order terms and step function initial data. These equations arise in modeling the propagation of a signal with loss, such as a signal in a resistive co-axial cable, or the flow of neutrons in a reactor. Majda and Osher have shown that dissipative finite difference approximations to such problems display a numerical artifact which is not encountered for scalar equations. Namely, noise from an initial discontinuity propagates into a large region behind the discontinuity. Their results do not apply in the vicinity of a discontinuity, and our goal is to discover the detailed behavior in this region. This information will be of use in constructing algorithms that attempt to accurately approximate solutions with discontinuities or shocks.We analyze the behavior of finite difference approximations to a particular model problem with step function initial data. We use Fourier transforms to express the solution of the difference approximation as a Fourier integral. The behavior of this integral is then examined by using the theory of uniform asymptotic estimation of integrals. We show that the solution of the difference approximation is modeled by a particular class of integrals, which we call generalized Bessel functions. With these results, we are able to show that there is a smearing of the discontinuity which is the same as that which one obtains when approximating the scalar wave equation. Thus, the behavior near the discontinuity is qualitatively similar to the near-front behavior of the scalar wave equation. These results show that the artifact discovered by Majda and Osher is overwhelmed near the discontinuity by the numerical dispersion and/or dissipation introduced by the difference approximation. Graphs of the generalized Bessel functions are provided. Finally, we discuss the relevance of our results to automatic mesh refinement and give an example." @default.
• W213426403 created "2016-06-24" @default.
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• W213426403 date "1981-01-01" @default.
• W213426403 modified "2023-09-25" @default.
• W213426403 title "Numerical solution of transport equations" @default.
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