Compact Upwind Biased Dispersion Relation Preserving Finite Difference Schemes

A class of higher-order compact upwind biased finite difference (FD) schemes based on the Taylor Series Expansion is developed in this work. The numerical accuracy of this family of compact upwind biased FD schemes is demonstrated by means of Fourier analysis wherein it is shown that these FD schemes have very low dispersion errors over a relatively large bandwidth of wave numbers as compared to explicit or non-compact schemes. The advantage of its use over compact central FD schemes is seen by the presence of inbuilt dissipation which damps only the unresolved high-frequency waves, thereby avoiding the need to use explicit artificial selective damping (ASD) terms. Compact downwind FD schemes are also formulated to obtain boundary closure schemes for expressing the spatial derivatives adjacent to and at the boundary nodes, thereby obtaining an overall FD scheme for the entire computational domain in terms of simple matrix operation. The stability of overall FD scheme in context with its use in the pseudo-characteristic formulation (PCF) of 1-D linearized Euler equations (with super-imposed mean flow) and anechoic boundary conditions is established by means of an eigenvalue analysis.