A high-order time formulation of the RBC schemes for unsteady compressible Euler equations

Residual-Based Compact (RBC) schemes can approximate the compressible Euler equations with a high space-accuracy on a very compact stencil. For instance on a 2-D Cartesian mesh, the 5th- and 7th-order accuracy can be reached on a 5×5-point stencil. The time integration of the RBC schemes uses a fully implicit method of 2nd-order accuracy (Gear method) usually solved by a dual-time approach. This method is efficient for computing compressible flows in slow unsteady regimes, but for quick unsteady flows, it may be costly and not accurate enough. A new time-formulation is proposed in the present paper. Unusually, in a RBC scheme the time derivative occurs, through linear discrete operators due to compactness, not only in the main residual but also in the other two residuals (in 2-D) involved in the numerical dissipation. To extract the time derivative, a space-factorization method which preserves the high accuracy in space is developed for reducing the algebra to the direct solution of simple linear systems on the mesh lines. Then a time-integration of high accuracy is selected for the RBC schemes by comparing the efficiency of four classes of explicit methods. The new time-formulation is validated for the diagonal advection of a Gaussian shape, the rotation of a hump, the advection of a vortex for a long time and the interaction of a vortex with a shock.