Interdisciplinary Applied Mathematics

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Pass I


Ji-1


v = 53 aqvn-q + At


q=0


Ji-1


nn-q



q=0 Ji-1


xn+1 = > aq



Je~1



Y @q N(vn-q , wn-q )+fn+1


q=0 -1


Y    aqnn-q + At( £ f3qN(vn-q, wn-q, nn-q),


q=0    q=0


Ji-1    J e — 1


Y    ачxn-q + At(Y Pqwn-q),



q=0



q=0



dpn+1


dn


PassII



n



‘ Je-1


~Y eqN(vn-q, wn-q)


q=0



— n



1 Je-1


—    (3q[V X (V x «»-«)]


. e q=0



At


vy+1 = v • (-A.








= — t


At    At


V2wn+1 = 0, where x(X, t) are the coordinates of the moving frame, relative to a fixed set of coordinates X, and


N(v, w) = (v — w) • Vv,


N(v, w,n) = (v — w) • Vn.


In the    first    pass    all    steps are    explicit and    computed    using    the    values    of


в, v, w, which are computed at the quadrature points. In the second pass all steps are computed implicitly. This scheme is of first order in time, but second-order schemes can be constructed based on staggered algorithms or predictor-corrector methods. The constants aq,@q,70 are integration weights (see Table 6.1 in (Karniadakis and Sherwin, 1999)). The mesh velocity is arbitrary, and can be specified explicitly or can be obtained from a Laplace equation, following (Ho, 1989). A better approach is to employ a variable coefficient in the Laplacian to provide enhanced smoothing, thereby preventing sudden distortions in the mesh (Lohner and Yang, 1996).

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