Toward robust linear implicit schemes for steady state hypersonic flows

Implicit time discretization in computational fluid dynamics dedicated to compute steady state solution of hypersonic flows was an intense field of research in the 70's-80's. It is suitable for computational efficiency to use implicit schemes that do not suffer from time step restriction to guarantee stability, unlike explicit ones. Unfortunately time step restriction is still required in practice, especially for stiff numerical test cases such as high Mach number flows around objects. A method introduced by Yee et al. (1985) [34] is commonly used to simulate computational fluid dynamics problems in an implicit fashion. However this method has no formal theoretical basis for systems of conservation laws. Consequently the practical time step is driven by a ad-hoc user-given profile. The purpose of this work is first to study the mathematical properties of such linearized implicit finite volume schemes to enlighten their weaknesses and exhibit more adequate linearization processes. We rely on the hyperbolicity of the Euler equations to establish a general framework to design implicit schemes. Secondly, we propose a correction of the system matrix to adapt to any given finite volume scheme. The obtained linearized implicit finite volume methods are more robust and less constrained with regards to ad-hoc user-given profile. Numerical results in 1D and 2D will provide evidences to confirm the analysis on relevant and challenging hypersonic test cases.