An implicit Galerkin finite element RungeKutta algorithm for shock-structure investigations
This paper introduces an implicit high-order Galerkin finite element RungeKutta algorithm for efficient computational investigations of shock structures. The algorithm induces no spatial-discretization artificial diffusion, relies on cubic and higher-degree elements for an accurate resolution of the steep shock gradients, uses an implicit time integration for swift convergence to steady states, and employs original Neumann-type outlet boundary conditions in the form of generalized RankineHugoniot conditions on normal stress and balance of heat flux and deviatoric-stress work per unit time. The formulation automatically calculates the spatial extent of the shock and employs the single non-dimensional (0,1) computational domain for the determination of any shock structure. Since it is implicit, the algorithm rapidly generates steady shock structures, in at most 150 time steps for any upstream Mach number considered in this study. The finite element discretization is shown to be asymptotically convergent under progressive grid refinements, in respect of both the H0 H 0 and H1 H 1 error norms, with an H0 H 0 accuracy order as high as 6 and reduction of the discretization error to the round-off-error threshold of 110?9 with just 420 computational cells and 5th-degree elements. For upstream Mach numbers in the range 1.05?M?10.0, the computational results satisfy the RankineHugoniot conditions and reflect independently published NavierStokes results.
finite elements normal shocks shock-wave structures computational stability implicit rungekutta