Multidimensional systems of hyperbolic conservation laws, numerical schemes, and characteristic theory : connections, differences, and numerical comparison

摘要

Many physical problems, for example the behaviour of a compressible fluid, can be modelled as systems of hyperbolic conservation laws, d/dt U + div.F(U) = 0, if certain effects (for instance viscosity) are neglected. One of the first computational schemes for systems of conservation laws was introduced by Godunov in 1959. This scheme is based on exactly solving a Riemann problem at each cell interface and then projecting the solution back onto the space of piecewise constant functions after some finite time step. A major drawback of Godunov's scheme is the necessity to solve all the Riemann problems exactly, which usually consists of an iterative process. To overcome this handicap, there were in the past many ideas for so-called approximate Riemann solvers, i.e. procedures to obtain a suitable approximation to the solution of a Riemann problem. In principle, the idea of Godunov's scheme is at first only applicable to problems in one space dimension. However, problems naturally often arise in two or three space dimensions, and some standard techniques to extend the scheme to more space dimensions have been developped. A detailed description of Riemann solvers and Riemann solver based schemes can for example be found in the textbooks of LeVeque, Kröner, Godlewski and Raviart, and Toro. On the other hand, while the numerical treatment of systems of conservation laws (as well as of other differential equations) has been developped and refined more and more and computers have become increasingly powerful, the classical analytical tool of characteristic theory (see for example Courant and Hilbert) seems to have been sunk little by little into oblivion. Since about 15 years, there is an ongoing discussion (see for example Roe et al) whether one-dimensional Riemann solvers do justice to the multi-dimensional effects arising in systems of conservation laws in multi-dimensions. There were a number of approaches which therefore purposely dispensed with Riemann solvers, and some of which were based on the classical characteristic theory. This thesis includes both a recapitulation of some aspects of the multi-dimensional characteristic theory (and therefore hopefully makes for preventing this beautiful theory from being forgotten) and some new analytical connections and differences between three Riemann solver free approaches. In particular, the thesis includes a new and elementary derivation of Noelle's version of Fey's "Method of Transport" (called MoT-ICE in contrast to Fey's version, which we call MoT-CCE), which on the one hand naturally fits into the framework of so-called state decompositions and flux decompositions (in which also the standard finite volume approach can integrated) and on the other hand is based on the gas-kinetic theory. This etablishes a close connection between the MoT-ICE and the kinetic schemes. Some implementory details for the MoT-ICE and an extensive numerical comparison of the MoT-ICE to a standard, Riemann solver based scheme complete the thesis. This comparison in a way confirms the close connection between the MoT-ICE and the kinetic schemes, but at the same time shows that the MoT-ICE in its current state does not meet the claim of being a reasonable alternative to Riemann solver based schemes.