Large Time Step Methods for Hyperbolic Partial Differential Equations
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In this thesis we consider explicit finite volume methods that are not limited by the Courant-Friedrichs-Lewy (CFL) condition, referred to as large time step (LTS) methods. LeVequeproposed the first LTS method in the 1980 s as an extension of the Godunov method. Since then, classic concepts in numerical analysis, such as total variation diminishing (TVD) schemes, modified equation and higher order schemes have been extended to LTS, as well as approximate Riemann solvers, such as the Roe scheme, the Lax-Friedrichs scheme and the HLL scheme. At large time steps, LTS methods often yield entropy violating solutions, and oscillations appear due to interacting waves, especially for systems of equations. Because of this reduction in robustness, the maximum allowable time step is in practice limited for many LTS schemes. We will look at LTS methods from a new angle, by introducing an artificial flux function framework. We show how the flux-difference splitting coefficients and numerical diffusion coefficient can be evaluated numerically from the artificial flux function, which gives us a convenient way of experimenting with new LTS schemes. In his master s thesis, Solberg developed a class of LTS schemes, with an inherent mechanism for adding numerical diffusion. As an extension of this work, we develop anew three parameter LTS scheme, denoted LTS-HLLφ, which is the main original contribution in this thesis. We propose special choices of parameters, which appears to give a good trade off between robustness and accuracy. Numerical simulations are performed on the Burgers equation and the Euler equations, assessing the robustness and accuracy of the new LTS-HLLφ scheme, compared tomore established LTS schemes.