Implicit, High-Resolution, Compact Schemes for Gas Dynamics and Aeroacoustics

Abstract

Implicit, high-order schemes are developed for time-accurate numerical solutions of hyperbolic equation systems. High-order spatial accuracy for the implicit operators is obtained at no additional computing cost by performing compact differentiation. The resulting alternating direction implicit and unfactored algorithms yield improved dispersion characteristics compared to second-order accurate in space implicit schemes which makes them suitable for high-resolution numerical simulations in gas dynamics and computational aeroacoustics. First, a fourth-order accurate in space implicit, factorized scheme, which requires block-tridiagonal matrix inversion, is presented. Next, a class of implicit factorized schemes, which require scalar matrix inversions, is presented. Higher order of accuracy in space of the implicit operators is achieved at the expense of inverting scalar matrices with larger bandwidth. Finally, extensions to unfactored algorithms, which use upwind compact schemes, are obtained. The proposed high-order schemes can be implemented with little modification of existing second-order accurate in space, implicit CFD methods. The efficiency, accuracy, and convergence characteristics of the new, high-resolution implicit schemes are demonstrated by their implementation for test problems.