k-version of finite element method in gas dynamics: higher-order global differentiability numerical solutions

In this paper, we consider and examine alternate finite element computational strategies for time-dependent Navier–Stokes equations describing high-speed compressible flows with shocks in a viscous and conducting medium, with the ultimate objective of establishing the desired features of a general mathematical and computational framework for such initial value problems (IVP) in which: (a) the numerically computed solutions are in agreement with the physics of evolution described by the governing differential equations (GDEs) i.e. the IVP, (b) the solutions are admissible in the non-discretized form of the GDEs in the pointwise sense (i.e. anywhere and everywhere) in the entire space–time domain, and hence in the integrated sense as well, (c) the numerical approximations progressively approach the same global differentiability in space and time as the theoretical solutions, (d) it is possible to time march the solutions (this is essential for efficiency as well as ensuring desired accuracy of the computed solution for the current increment of time, i.e. to minimize the error build up in the time marching process), (e) the computational process is unconditionally stable and non-degenerate regardless of the choice of discretization, nature of approximations and their global differentiability and the dimensionless parameters influencing the physics of the process, (f) there are no issues of stability, CFL number limitations and (g) the mathematical and computational methodology is independent of the nature of the space–time differential operators. We consider one-dimensional compressible flow in a viscous and conducting medium with shocks as model problems to illustrate various features of the general mathematical and computational framework used here and to demonstrate that the proposed framework is general and is applicable to all IVP. The Riemann shock tube with a single diaphragm serves as a model problem. The specific details presented in the paper discuss: (1) Choice of the form of the GDEs, i.e. strong form or weak form. (2) Various choices of variables. The paper establishes and considers density, velocity and temperature as variables of choice. (3) Details of the space–time least squares (LS) integral forms (meritorious over all others in all aspects) are presented and choice of approximation spaces are discussed. (4) In all numerical studies we consider a viscous and conducting medium with ideal gas law, however results are also presented for non-conducting medium. Extension of this work to real gas models will be presented in a separate paper. It is worth noting that when the medium is viscous and conducting, the solutions of gas dynamics equations are analytic. (5) It is also significant to note that upwinding methods based on addition of artificial diffusion such as SUPG, SUPG/DC, SUPG/DC/LS and their many variations are neither needed nor used in this present work. (6) Numerical studies are aimed at resolving the localized details of the shock structure, i.e. shock relations, shock width, shock speed, etc. as well as the over all global behaviour of the solution in the entire space–time domain. (7) Numerical studies are presented for Riemann shock tube for high Mach number flows with special emphasis also on time accuracy of the evolution which is ensured by requiring that the approximations for each increment of time satisfy non-discretized form of the GDEs in the pointwise sense, and hence in the integrated sense as well. (8) Comparisons are made with published results as well as theoretical solutions (when possible). It is established that space–time least squares processes are the only processes that yield variationally consistent space–time integral forms, and hence unconditionally non-degenerate space–time computational processes, which when considered in higher-order scalar product spaces provide the desired mathematical framework in which progressively higher-order global differentiability solutions in space and time yield the same characteristics as the theoretical solutions of the IVP in all aspects. Copyright © 2006 John Wiley & Sons, Ltd.