Grid Convergence Error Analysis for Mixed-Order Numerical Schemes

New developments are presented in the area of grid convergence error analysis for mixed-order numerical schemes. A mixed-order scheme is deŽ ned as a numerical method where the formal order of the truncation error varies either spatially, for example, at a shock wave, or for different terms in the governing equations, for example, third-order convection with second-order diffusion. The case examined is the Mach 8 inviscid  ow of a calorically perfect gas over a spherically blunted cone. This  owŽ eld contains a strong bow shock wave, where the formally second-order numerical scheme is reduced to Ž rst order via a  ux-limiting procedure. The proposed mixedorder error analysis method allows for nonmonotonic behavior in the solution variables as the mesh is reŽ ned. Nonmonotonicity in the local solution variables is shown to arise from a cancellation of Ž rstand second-order error terms for the present case. An error estimator is proposed based on the mixed-order analysis and is shown to provide good estimates of the actual error when the solution converges nonmonotonicallywith grid reŽ nement.