An implicit Galerkin finite element Runge–Kutta algorithm for shock-structure investigations

Abstract

This paper introduces an implicit high-order Galerkin finite element Runge–Kutta 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 Rankine–Hugoniot 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 H 0 and H 1 error norms, with an H 0 accuracy order as high as 6 and reduction of the discretization error to the round-off-error threshold of 1 × 10 −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 Rankine–Hugoniot conditions and reflect independently published Navier–Stokes results.