Aspects of viscous shocks

• Main Author: Siklosi, Malin
• Format: Doctoral Thesis
• Language: English
• Published: KTH, Numerisk analys och datalogi, NADA 2004
• Subjects: Numerical analysis; hyperbolic conservation laws; viscous shocks; modified equation; shock capturing; computer-assisted proofs; Numerisk analys
• Online Access: http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-89; http://nbn-resolving.de/urn:isbn:91-7283-922-8

This thesis consists of an introduction and five papers concerning different numerical and mathematical aspects of viscous shocks. Hyperbolic conservation laws are used to model wave motion and advect- ive transport in a variety of physical applications. Solutions of hyperbolic conservation laws may become discontinuous, even in cases where initial and boundary data are smooth. Shock waves is one important type of discontinu- ity. It is also interesting to study the corresponding slightly viscous system, i.e., the system obtained when a small viscous term is added to the hyper- bolic system of equations. By a viscous shock we denote a thin transition layer which appears in the solution of the slightly viscous system instead of a shock in the corresponding purely hyperbolic problem. A slightly viscous system, a so called modified equation, is often used to model numerical solutions of hyperbolic conservation laws and their beha- vior in the vicinity of shocks. Computations presented elsewhere show that numerical solutions of hyperbolic conservation laws obtained by higher order accurate shock capturing methods in many cases are only first order accurate downstream of shocks. We use a modified equation to model numerical solu- tions obtained by a generic second order shock capturing scheme for a time dependent system in one space dimension. We present analysis that show how the first order error term is related to the viscous terms and show that it is possible to eliminate the first order downstream error by choosing a special viscosity term. This is verified in computations. We also extend the analysis to a stationary problem in two space dimensions. Though the technique of modified equation is widely used, rather little is known about when (for what methods etc.) it is applicable. The use of a modified equation as a model for a numerical solution is only relevant if the numerical solution behaves as a continuous function. We have experimentally investigated a range of high resolution shock capturing methods. Our experiments indicate that for many of the methods there is a continuous shock profile. For some of the methods, however, this not the case. In general the behavior in the shock region is very complicated. Systems of hyperbolic conservation laws with solutions containing shock waves, and corresponding slightly viscous equations, are examples where the available theoretical results on existence and uniqueness of solutions are very limited, though it is often straightforward to find approximate numerical solu- tions. We present a computer-assisted technique to prove existence of solu- tions of non-linear boundary value ODEs, which is based on using an approx- imate, numerical solution. The technique is applied to stationary solutions of the viscous Burgers' equation.We also study a corresponding method suggested by Yamamoto in SIAM J. Numer. Anal. 35(5)1998, and apply also this method to the viscous Burgers' equation.