By ZhiJunt S., GuangWei Y., JingYan Y.

A brand new Lagrangian cell-centered scheme for two-dimensional compressible flows in planar geometry is proposed by means of Maire et al. the most new characteristic of the set of rules is that the vertex velocities and the numerical puxes in the course of the mobile interfaces are all evaluated in a coherent demeanour opposite to straightforward techniques. during this paper the tactic brought by means of Maire et al. is prolonged for the equations of Lagrangian gasoline dynamics in cylindrical symmetry. various schemes are proposed, whose distinction is that one makes use of quantity weighting and the opposite quarter weighting within the discretization of the momentum equation. within the either schemes the conservation of overall power is ensured, and the nodal solver is followed which has a similar formula as that during Cartesian coordinates. the quantity weighting scheme preserves the momentum conservation and the area-weighting scheme preserves round symmetry. The numerical examples reveal our theoretical concerns and the robustness of the hot procedure.

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10) Setting —. k we hove inequalities (20, No. 12) For k a non-negative integer, polarization gives m = = 01(Ak). 1=1 Then (9] — 0k-1°1 + •.. 13) k > m. +r -1)! m 1 r11 ... r ... ,rm) of non-negative Integers with m E 1 r = k. 1=1 We have liapounoff's inequalities (20, No. 18], equality 1ff A1 ... = Am• 10k+1' with Bilinear Maps If g denotes the inner product of V viewed as a bilinear form on V, its nondegeneracy provides an inverse linear map : V. Now any symmetric bilinear form B : VxV has a canonical interpretation as a linear map V V*; consequently the o B is a seif-adjoint endomorphism of V, with composition o B(u),v> = B(u,v) for all u,v C V.

6. Now, let N1be an integral submanifold of D1 . 5). Thus N is a CR-product. K. 15) foranyXEbandZED1. 2) we have - Ja(Z,X) + Vz(JX) + o(Z,JX). 16) yields: = 0. 13). 5). Thus Now, take X, V D, Z g(JVxZ,Y) = _g(AjzX,Y) = -g(o(X,Y),JZ) = 0. 6. 5 we have g(AjwZ,X) - g(AjzW,X) = =0 + and hence g(JvxZ,W) = 0. 18). lc). 13) we have for any X C and U tangent to N. 'a ED such that ish at x. Therefore P is not parallel. if dim D = 2, then the right—lar. 21) = 0 for any X E 0 and U tangent to N. 14) implies = 0.

17) a(B)(w) using the canonical identifications (the use of which QkW suggested by J. w with polynomial functions of W. In particular, that, that polynomial function identification assigns to v ® ... ® v ... ® e w>. 18) W. by means of is det A uk, V1 A ... A Vk) A The adjugate ekw, : Vx V + of 8 : Vx for all u,v B(u1,v3). given by V V. 22) 0 gives the real number f(ci(B)). 25) In case o is a 2k—homogeneous polynomial, where zation of 3. o2kw is the the induced inner product of ekw*. Qkh, CERTAIN VARIATIONAL INTEGRALS In General Let V -t N a Riemannian vector bundle of fibre dimension m over a RiemanaRm -+ nian manifold N and be a syimietric function.