High-Resolution Large Time-Step Schemes for Hyperbolic Conservation Laws
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- Institutt for fysikk 
This thesis is concerned with numerical methods for solving hyperbolic conservation laws. A generalization of large time-step schemes (LTSS) to high resolution is presented. The generalization is based on the previous work in [Lindquist, 2014; Harten, 1986]. Starting from a general LTSS, a set of sufficient conditions for conservative, consistent, and total variation diminishing (TVD) LTSS are derived. Second-order accuracy away from discontinuities is achieved by a modified flux approach. Such an approach is shown to be TVD whenever a supplementary condition is satisfied. The full set of criteria constitutes a new framework of sufficient conditions for high-resolution LTSS. By application of this framework on the large time-step Roe scheme (LTS-Roe1), a new second-order version (LTS-Roe2) is proposed. Further, to overcome the problems of LTS-Roe1 and LTS-Roe2 with transonic rarefaction, a hybrid scheme of LTS-Roe and Lax-Friedrichs is proposed (Hybrid). The methods are investigated and compared against the second-order LTS-Harten. This is done by numerical studies on Burgers equation and on the Euler equations. Numerical tests for continuous initial conditions show second-order convergence for all methods. For discontinuous initial conditions LTS-Roe2 has better accuracy than LTS-Roe1- however, this difference becomes small for high CFL-numbers. LTS-Roe2 is shown to have a very good resolution of discontinuities, but for high CF L-numbers it produces spurious oscillations for the Euler equations. Hybrid is more diffusive, but has no problems with transonic rarefaction. Tests show that LTS-Harten consistently gives good results with less oscillations than LTS-Roe2, but it has, however, a tendency to smear out discontinuities when the CFL-number is increased.