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 way of Maire et al. the most new characteristic of the set of rules is that the vertex velocities and the numerical puxes during the telephone interfaces are all evaluated in a coherent demeanour opposite to plain methods. during this paper the strategy brought via 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 sector 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 amount weighting scheme preserves the momentum conservation and the area-weighting scheme preserves round symmetry. The numerical examples reveal our theoretical issues and the robustness of the hot approach.

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**Extra resources for A cell-centered lagrangian scheme in two-dimensional cylindrical geometry**

**Example text**

X + η 2 h(x) By deﬁnition B := supY =0 |Y |−2 |B(Y, Y )| = supY =0,Z=0 |Y |−1 |Z|−1 |B(Y, Z)|. t. the metric g0 . 10 ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1996 32 U. ABRESCH V. SCHROEDER Proof. 20) Bi1 ∧ (1l+G )−1 I Bi2 = η 4 h (xi1 )h (xi2 ) pξi1 In the case where i1 = i2 =: i we observe that pξi −pi ∧ ∧ ∧ pξi = pξi pξi2 . 21) = ∧ (1l+G )−1 I Bi 1+η 2 h (xi ) ϕ0 (η, xi ) − ϕ1 (η, xi ) + 2η 2 (1+xi )h (xi ) pi ∧ pi . 3. 18). Of course, −g0 ∧ g0 is negative deﬁnite. 6. Remarks. 4 we may absorb the third, fourth, and nineth term in our expression for R# into −g0 ∧ g0 , provided η > 0 is suﬃciently small.

Iii) Note that p ∈ SI is contained in some domain UI with I ⊂ I ⊂ J. 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.

Hn maps each subspace Hn−2 , i ∈ I, into i , j ∈ J \ I. Hence, itself, and it permutes the other subspaces Hn−2 j ∗ i (gJ\I ) = gJ\I , or equivalently d i GJ\I = GJ\I d i . Since BJ\I is a linear combination of covariant derivatives of gJ\I , we get d i BJ\I = BJ\I d i . , d i . Because of these two identities it is suﬃcient to observe that at any point p ∈ Hn−2 i the diﬀerential d i |p acts as a rotation of order mi = ord i on the 2–dimensional subspace im Pi |p ⊂ Tp Hni , whereas it acts as the identity on its orthogonal complement ker Pi |p ⊂ Tp Hni .