High Order Modified Weighted Compact Scheme for High Speed Flow (FOR AFOSR GRANT FA9550-05-1-0136)

The critical problem of CFD is perhaps an accurate approximation of derivatives for a given discrete data set. Based on our previous work on the weighted compact scheme (WCS), a modified weighted compact scheme (MWCS) has been developed. Similar to WENO, three high order candidates, left, right, and central, are constructed by Hermite polynomials. According to the smoothness, three weights are derived and assigned to each candidate. The weights will lead the scheme to be bias when approaching the shock or other discontinuities but quickly becomes central, compact, and of high order just off the shock. Therefore, the new scheme can get a sharp shock without oscillation, but keep central, compact and of high resolution in the smooth area. This feature is particularly important to numerical simulation of the shock-boundary layer interaction, where both shock and small eddies are important. The new scheme has three compact candidate stencils, 2 1 0 , , E E E , with three weights 2 1 0 , , ω ω ω . Candidate stencils 2 1 0 , , E E E have third order, fourth order, and third order of accuracy respectively. Picking 18 1 , 9 8 , 18 1 2 1 0 = = = ω ω ω , MWCS has sixth order of accuracy in the smooth area. For near shock grid points, 0 , 0 , 1 2 1 0 = = = ω ω ω , MWCS still has fourth order of accuracy. Comparing with 5 th order WENO which has 5 th order in the smooth area and third order near the shock, MWCS is super with smaller stencils and higher order of accuracy. The necessary dissipation is provided by weights and some high order bias up-winding scheme. The new scheme has been successfully applied for 1-D shock tube and shock-entropy interaction and 2-D incident shock. The new scheme has obtained sharper shock, no deformation for expansion wave, and much higher resolution than 5 th order WENO for small length scales. A first version of a black-box type subroutine with MWCS scheme has been developed and given to AFRL collaborators. The subroutine can give a high order derivative for any discrete data sets, no matter they are smooth, highly oscillatory, or no-differentiable. The scheme is currently applied for shock-boundary layer interaction with double cones for validation. A variety of cases including shock-boundary interaction with incident shock, double angles and double-cones is being tested. The preliminary numerical solution is encouraging while the numerical simulation with double angles and double cones are still running. The new effort will also focus on new versions of the black-box type subroutines. The first version requires explicit boundary values. The second version will be able to treat variety of boundary conditions implicitly and keep high order for the REPORT DOCUMENTATION PAGE Form Approved OMB No. 0704-0188 1. REPORT DATE (DD-MM-YYYY) 2. REPORT TYPE 4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER