SPATIALLY FOURTH-ORDER-ACCURATE SCHEME FOR UNSTEADY-CONVECTION PROBLEMS

Abstract

This work introduces a method for solving advection and advection-diffusion equations which is fourth-order-accurate in space and second-order-accurate in time. MacCarmack ( MC) time-splitting schemes are used with various differencing schemes in discretizing the advection terms. By using various differencing schemes, it is found that not all differencing schemes produce accurate results even though the truncation orders are same. The suggested method is compared with an analytical solution and conventional MCfor pure advection and advection-diffusion equations. For advection problems the method is more accurate than the MC method. Also, two-dimensioned Navier-Stokes equations are solved for forced( Re = 1,000, 3,200, and 5000) and natural convection ( Ra = 1 × 10⁴and 1 × 10⁵) in a closed cavity. The results are compared with benchmark solutions. The method is attractive as far as accuracy and simplicity of application is concerned, but it is conditionally stable, as stability analysis revealed.