Implicit High-Order-Accurate-in-Space Algorithms for the Navier-Stokes Equations

Abstract

High-order-accurate, alternating direction implicit schemes have been developed. These schemes can be used to obtain efe ciently high-resolution numerical solutions of the compressible Euler and Navier ‐Stokes equations. Implicit solvers compatible with both centered and upwind schemes for the discretization of the convective terms arediscussed.Forcenteredspacediscretizationoftheconvectivee uxes,fourth-orderspatialaccuracyoftheimplicit operators is obtained at no additional computing cost by performing compact space differentiation. This implicit compact scheme, which requires block-tridiagonal matrix inversions, is used to obtain viscous and inviscid e ow solutions in curvilinear coordinates. In addition, high-order-accurate-in-space implicit operators are obtained for diagonalized schemes, which require scalar matrix inversions. Both high-order algorithms improve convergence characteristics for subsonic and transonic e ow compared to second-order-accurate-in-space implicit schemes. High-order-accurate-in-space implicit algorithms, compatible with upwind, high-resolution shock-capturing total variation diminishing schemes, are also developed. Third-order-accurate, upwind-biased, diagonalized algorithms were found to be as accurate and more efe cient than the standard block-tridiagonal, implicit solvers.