A Remark on Collocation and Upwinding in First Order Hyperbolic Systems
Keenan, Philip T.
Keenan defines and analyzes a new numerical method for coupled systems of nonlinear first order hyperbolic partial differential equations with one degenerate eigenvalue. That work extends in a certain direction the collocation method described by Luskin, which applies to systems with all the eigenvalues uniformly bounded away from zero. Luskin's method and Keenan's method both have direct application to the study of one dimensional fluid flow through pipelines. The pressure and velocity of an isothermal fluid in a pipeline can be described by a coupled pair of nonlinear first order hyperbolic partial differential equations. When thermal effects are important a third equation for temperature is added. While Luskin's method works well for the isothermal situation he discussed, it does not apply in certain common cases when thermal effects are modeled. The analysis of the new method shows how the difficulties that come from the application of standard collocation can be overcome by using upwinded piecewise constant functions for the degenerate component of the solution. Experiments indicate that this method is a substantial improvement over standard collocation. A number of technical details obscure the analysis presented in Keenan , because that work treats the general nonlinear case. The present paper describes and analyzes the method in the context of a linear, constant coefficient system of equations. In this special case the proof simplifies considerably.
Citable link to this pagehttps://hdl.handle.net/1911/101778
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