2015/05/13 by Claus Goetz, Claus R. Goetz, Armin Iske
Engineering · Mathematics · #Computational Fluid Dynamics and Aerodynamics #Fluid Dynamics and Turbulent Flows #Navier-Stokes equation solutions
paper · pdf · doi:10.1090/mcom/2970
We study analytical properties of the Toro-Titarev solver for generalized Riemann problems (GRPs), which is the heart of the flux computation in ADER generalized Godunov schemes. In particular, we compare the Toro-Titarev solver with a local asymptotic expansion developed by LeFloch and Raviart. We show that for nonlinear scalar problems in 1D the Toro-Titarev solver reproduces the truncated Taylor series expansion of LeFloch-Raviart exactly, whereas for nonlinear systems the Toro-Titarev solver introduces an error whose size depends on the height of the jump in the initial data. Thereby, our analysis answers open questions concerning the justification of simplifying steps in the Toro-Titarev solver. We illustrate our results by giving the full analysis for a nonlinear <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -by- <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> system and numerical results for shallow water equations and a system from traffic flow.