tailieunhanh - Numerical_Methods for Nonlinear Variable Problems Episode 7

Tham khảo tài liệu 'numerical_methods for nonlinear variable problems episode 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 4 Transonic Flow Calculations by Least-Squares and Finite Element Methods 227 Figure therein in order to eliminate nonphysical shocks these upwinding techniques have been very effective coupled with alternating-direction methods implicit or semi-implicit see Holst 1 and Deconinck and Hirsch 1 if the computational mesh is regular finite differences or regular finite element grids . In particular their application combined with finite element techniques have been limited see Eberle 1 2 and Deconinck and Hirsh 1 to quadrilateral elements on quasiregular quadrangulations fairly close to finite difference methods in our opinion . In Sec. we would like to discuss a method due to M. o. Bristeau which also makes use of an upwinding of the density this method can be used with simplicial10 finite elements triangles in two dimension tetrahedra in three dimensions and has been very effective for computing flows at high Mach numbers and around complicated two- and three-dimensional geometries. . A modified discrete continuity equation by upwinding of the density in the flow direction. Following Jameson l - 4 and Bristeau 2 3 we may write the continuity equation in a system of local coordinates s n where see Fig. for a two-dimensional flow s is the unit vector of the stream direction . s u 1 u I if u 0 and n is the corresponding normal unit vector conventionally oriented . Using s n and setting11 we obtain from Õ2Ộ 1 - u2 c2 b Põỉ Pỉ ỉũ2õĩ 0 10 We use here the terminology of Ciarlet 1 2 11 u is the Mach number M . 228 VII Least-Squares Solution of Nonlinear Problems the elliptic-hyperbolic aspect of the problem is clear from . Actually can also be written Õ2Ộ u2 - 1 n on 2kaU we have l 2fax y l 2 . Exercise . Prove . We use to modify thé discrete continuity equation as follows Find ộh e vh such that r 1 c í Ô P K Vộh dx -hs JJị - 1 Vộh-Vpj vhdx Jii 2k x Jq cs h ghvhdr VvheVh. Jr The approximate .