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

A brand new Lagrangian cell-centered scheme for two-dimensional compressible flows in planar geometry is proposed via 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 phone interfaces are all evaluated in a coherent demeanour opposite to straightforward techniques. during this paper the strategy brought by means of Maire et al. is prolonged for the equations of Lagrangian gasoline dynamics in cylindrical symmetry. assorted schemes are proposed, whose distinction is that one makes use of quantity weighting and the opposite region 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 an analogous 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 show our theoretical issues and the robustness of the hot approach.

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Proof. 2 ) shows that X , KjC |z KjC |z , Y 2 = z , e2j C − z , e1j C C e1j , X z , e2j e1j , Y C + z , e1j e1j , X C e2j , Y C C 2 e2j , X C e2j , Y C + e2j , X C e1j , Y C X , e2j C C C . 2 we have eµj , z 2 C C 2 2 z , ej C X , e2j 2 C z , e1j +2 √ 2 + 1 + 2a(a + 2) eµj , z0 √ 1 ≤ 1 + 2a(a + 2) 1 + 2xj (z0 ) . 6) yield eµj , X 2 C ≤ 1 + 2 eµj , z0 2 X ,X ≤ 1 + 2xj (z0 ) X , X h . 8). ´ ` 1 SEMINAIRES & CONGRES 2 C h C 2 . ANALYTIC MANIFOLDS OF NONPOSITIVE CURVATURE 23 Now, we collect the information.

5) it is clear that πI−1 {p} is a totally geodesic product torus in WIU × (R/2πZ) #I equipped with the metric πI∗ g0 + gI . If η is suﬃciently small, then the function x → x + η 2 h(x), x ≥ 0, takes its absolute minimum precisely at x = 0. Hence, for these values of η all closed geodesics of the torus are absolutely minimizing elements in their homotopy classes in WIU × (R/2πZ) metric πI∗ g0 + gI #I . In order to pass from the partial to πI∗ (g), we add a positive semideﬁnite term which vanishes on the torus.

1 (i). ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1996 44 U. ABRESCH V. SCHROEDER Proof. 7 (ii) that β(η, xi1 ) β(η, xi2 ) ξi1 , LI ξi2 +η 2 h (xi1 ) β(η, xi2 ) vi1 , LI ξi2 + η 2 β(η, xi1 ) h (xi2 ) ξi1 , LI vi2 + η 4 h (xi1 ) h (xi2 ) vi1 , LI vi2 1/2 1/2 ≤ η 2 (1 + 2c4 η 2 )2 cˆ12 (η)xi1 xi2 (1 + xi1 ) −1/2 (1 + xi2 ) −1/2 hence the claim. 13) ∧ LI ∧ pξi vanish identically and that pbi ˆi = B ∧ pbi = −pi 1 + x−1 β(η, xi )2 ξi , LI ξi pi i ∧ ∧ pi pi . 2 (i), we can ﬁnish the proof setting cˆ14 := c0 (1 + c4 η)(1 + c4 η 2 ) and cˆ15 := c0 c4 (1 + η + c4 η)(1 + c4 η 2 ) .