Computational Aeroacoustics (CAA) and Direct numerical simulations (DNS) require an accurate finite difference scheme which have a high-order of truncation and high-resolution characteristics in the evaluation of spatial derivatives. The compact finite difference schemes are optimized to obtain maximum resolution characteristics in space for various spatial truncation orders. An analytic method with a systematic procedure to achieve maximum resolution characteristics is devised for multidiagonal schemes, based on the idea of the minimization of dispersive (phase) errors in the wavenumber domain, and these are applied to the analytic optimization of multidiagonal compact schemes. Actual performances of the optimized compact schemes with a variety of truncation orders are compared by means of numerical simulations of simple wave convections, and in this way the most effective compact schemes are found for tridiagonal and pentadiagonal cases respectively. From these comparisons, the usefulness of an optimized high-order tridiagonal compact scheme which is more efficient than a pentadiagonal scheme is discussed.
The feasibility of using the optimized high-order compact (OHOC) schemes for the numerical computations of nonlinear wave propagation is investigated. The high-order compact finite difference schemes which are optimized for high-resolution characteristics are less dissipative, less dispersive and require less grid points to resolve a wave profile than the other low-order standard schemes. They are well adapted to linear problems with smooth wave solutions. However, they inevitably generate spurious spatial oscillations when applied to nonlinear problems with highly discontinuous wave solutions. Thus, they require an effective artificial dissipation algorithm to damp out only the spurious oscillations, while keeping the wave components in low wavenumber range unaffected. They also require an efficient high-order low dissipation and dispersion time advancing method to produce long-time accurate nonlinear wave speeds and profiles. For the nonlinear computations, the OHOC schemes are coupled with the artificial selective damping (ASD) terms and the fourth-order low dissipation and dispersion Runge-Kutta (LDDRK) scheme which is more efficient within a certain accuracy limit than the classical fourth-order Runge-Kutta scheme. It is shown that the application of these schemes to the numerical computations of the nonlinear wave propagation presents long-time accurate solutions with well-resolved shocks or contact surfaces and without spurious oscillations.
