AbstractsEngineering

This work is devoted to the development of a hybrid, explicit-implicit, scheme for simulation of unsteady compressible flows with shock waves. The proposed scheme is of the second-order accuracy in space and time for both explicit and implicit modes, while satisfying the TVD (Total Variation Diminishing) property. The scheme is designed for simulation of the compressible flows with temporal stiffness. In this situation,the numerical time step of explicit schemes is severely limited by particular conditions in a relatively small part of the computational domain, while the rest of the domain admits much higher time steps. The hybrid scheme is designed to operate in its implicit mode in the small areas causing temporal stiffness, thus allowing to proceed with higher time steps and reduce the computational time. In this study, a new hybridization approach is suggested. On its basis, the hybrid scheme is first introduced for hyperbolic conservation laws in one dimension. In order to satisfy the TVD property and obtain monotone solutions in the presence ofdiscontinuities, TVD limiters are applied to both spatial and temporal reconstructions. The second-order accuracy in time for the implicit mode, which is the main distinction of the proposed hybrid scheme in comparison with the existing methods, is achieved through a reconstruction of the solution in time. To make the reconstruction TVD preserving, a novel time limiter is derived. The stability condition and the relation to the hybridization factor of the new scheme are obtained. Moreover, the relationship of the proposed scheme with another existing hybrid method is revealed and analyzed. A set of one-dimensional test problems is used to demonstrate the accuracy and efficiency of the new hybrid scheme as well as to show its advantagesin comparison with the existing hybrid schemes. For two-dimensional problems, the hybrid scheme is generalized for the Euler and Navier-Stokes equations on unstructured grids. Similar to one-dimensional problems, reconstructions in space and time are applied with the corresponding TVD limiters. The newly proposed time limiter of the hybrid scheme is further generalized for unstructured grids. The non-linear system of discretized equations is solved using the Lower-Upper-Symmetric-Gauss-Seidel (LU-SGS) approximate factorization method for unstructured grids. In order to eliminate the factorization and linearization errors, internal iterations are introduced at each time step. The lower and upper matrices in the LU-SGS scheme are formed via reordering of grid nodes at each time step. In addition, local transient grid adaptation is applied near solution peculiarities, such as shock waves and contact surfaces. Viscous terms of the Navier-Stokes equations are also approximated with the second order of accuracy both in space and time in both explicit and implicit modes. The new hybrid scheme is applied to a number of demonstrative problems chosen to represent typical classes of gasdynamic problems with specific source of stiffness. The…