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 de 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 ow eld contains a strong bow shock wave, where the formally second-order numerical scheme is reduced to rst order via a ux-limiting procedure. The proposed mixed-order error analysis method allows for nonmonotonic behavior in the solution variables as the mesh is re ned. Nonmonotonicity in the local solution variables is shown to arise from a cancellation of rst- and 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 re nement. Nomenclature DE = discretizationerror Fs = factor of safety for error estimators, 3 f = general solution variable gi = i th order error term coef cient h = normalizedmeasure of grid spacing, N1=Nk1=2 M = Mach number N = number of mesh cells p = spatial order of accuracy; pressure, N/m2 R = gas constant, 296.8 J/kg ¢K RN = nose radius, 0.00508m r = grid re nement factor x = axial coordinate,m y = radial coordinate,m ° = ratio of speci c heats, 1:4 k C 1;k = difference between solution variable on mesh kC 1 and mesh k Subscripts and Superscript exact = exact continuumvalue k = mesh level, 1, 2, 3, etc.; ne to coarse n = ow eld node index » = estimated value to order h p C 1