High-order global differentiability numerical solutions in gas dynamics using k-version of finite element method

Various finite element strategies are considered for time dependent Navier-Stokes equations describing high speed compressible flows with shocks with the ultimate objective of achieving a general mathematical and computational framework for initial value problems (IVP) in which the computed solutions are in agreement with the physics described by the governing differential equations (GDEs), the solutions are admissible in the non-discretized form of the GDEs in the point-wise sense (i.e. anywhere and everywhere) in the entire space-time domain, and the numerical approximations progressively approach the same global differentiability in space and time as the theoretical solutions. The space-time least squares processes presented yield variationally consistent integral forms and hence unconditionally stable, non-degenerate computational processes. When considered in higher order Hilbert space Ĥ^k,p (Ω^e) the method provides a framework for progressively higher order global differentiability approximations in space and time with the same characteristics as the theoretical solutions of the IVP. The form of the governing differential equations, i.e. strong form or weak form, the choice of variables, details of the space-time least-squares integral forms and the choice of approximation spaces are discussed. Upwinding methods are neither needed nor used in the present work. Numerical studies are presented for the Riemann shock tube with special emphasis on achieving true time accuracy of the evolution and resolving the localized details of the shock structure.