tailieunhanh - Numerical_Methods for Nonlinear Variable Problems Episode 9

Tham khảo tài liệu 'numerical_methods for nonlinear variable problems episode 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 5 Numerical Solution of the Navier-Stokes Equations 307 Let us prove using the notation in the proof of Theorem we have p i p _ pV-ũ aũ - vAũ - rV V ũ - Vp in Q ũ e Hi Q N. In fact can also be written aũ - vAũ - V p - rV Ũ in Í2 ũ e Hẳ í2 JV since V ũ 6 H from and from the definition of Jt we have V Ũ jaZ p rV Ũ . V Ũ Wp . Combining with we obtain pn 1 I p I XrMp . We have I p J r Z _ w rl 5 -1 -1 r p I -i 1 and yields I - p I r rMl r - Pl ik - . r From it follows that for the classical choice p r we have llp 1 - pIIl2 O - lip - pIIl2 Q . Therefore if r is large enough and if p r the convergence ratio of algorithm - is of order 1 r. Remark . The system is closely related to the linear elasticity system. Once it is discretized by finite differences or finite elements as in Sec. it can be solved using a Cholesky factorization LL or LDL done once and for all. Remark . Algorithm - has the drawback of requiring the solution of a system of N partial differential equations coupled if r 0 by rV V- while this is not so for algorithms of Secs. and . Hence much more computer storage is required. Remark . By inspecting it seems that one should take p r and r as large as possible. However and its discrete forms would be ill conditioned if r is too large. In practice if is solved by a direct 308 VII Least-Squares Solution of Nonlinear Problems method Gauss Cholesky one should take r in the range of 102v to 105v. In such cases and if p r the convergence of is extremely fast about three iterations . Under such conditions it is not necessary to use a conjugate gradient accelerating scheme. . Numerical experiments In this section we shall present the results of some numerical experiments obtained using the methods of the above sections. Further numerical results obtained using .