Spectral properties of high-order residual-based compact schemes for unsteady compressible flows

The wave propagation (spectral) properties of high-order Residual-Based Compact (RBC) discretizations are analyzed to obtain information on the evolution of the Fourier modes supported on a grid of finite size. For these genuinely multidimensional and intrinsically dissipative schemes, a suitable procedure is used to identify the modified wave number associated to their spatial discretization operator, and their dispersive and dissipative behaviors are investigated as functions of a multidimensional wave number. For RBC schemes of higher orders (5 and 7), both dissipation and dispersion errors take very low values up to reduced wave numbers close to the grid resolvability limit, while higher frequencies are efficiently damped out. Thanks to their genuinely multidimensional formulation, RBC schemes conserve good dissipation and dispersion properties even for flow modes that are not aligned with the computational grid. Numerical tests support the theoretical results. Specifically, the study of a complex nonlinear problem dominated by energy transfer from large to small flow scales, the inviscid Taylor-Green vortex flow, confirms numerically the interest of a well-designed RBC dissipation to resolve accurately fine scale flow structures.