(200p) Entropy-Stable Finite Difference Schemes for the Non-Linear Euler Equations with Boundary Conditions

In fluid dynamics, the Euler equations are set of equations representing conservation of mass, momentum and energy. Since finding analytical solutions for such equations is extremely challenging, the approximate solutions are often derived numerically.

In this study, we consider numerical schemes for the compressible, non-linear Euler equations in one space dimension. Finite difference numerical schemes, which satisfy an entropy stability condition, have been derived to obtain approximate solution including far-field and wall boundary conditions. Entropy functions provide symmetry in the system of Euler equations and limit the growth of entropy in order to infer stability on approximate solution. In addition, a stable and a conservative numerical treatment of interfaces between two connected different grid domains are proposed. Finally, we present numerical computations with the second and fourth order schemes to demonstrate robustness even when shocks hit the boundaries.   

References: M.Svard, H.Ozcan, Entropy-Stable Schemes for the Euler Equations with Far-Field and Wall Boundary Conditions, J. Sci. Comput.1-29,2013