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 function of the set of rules is that the vertex velocities and the numerical puxes throughout the telephone interfaces are all evaluated in a coherent demeanour opposite to straightforward ways. during this paper the tactic brought through Maire et al. is prolonged for the equations of Lagrangian fuel dynamics in cylindrical symmetry. diverse 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 strength is ensured, and the nodal solver is followed which has an analogous 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 display our theoretical issues and the robustness of the recent process.

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With effort, the exponent h(a) can be sharpened— and made more explicit. • Indeed, for a = b or even a ≈ b we now have many other, rapidly convergent options. 45 Proof: With a view to induction, assume that for some constants (n-independent) d(a), g(a) and for n ∈ [1, N − 1] we have qn < dng . 1) mean that An > f (a)/n for an n-independent f . Then we have a bound for the next qN : f qN > d(N − 1)g + d (N − 2)g . N Since for g < 1, 0 < x ≤ 1/2 we have (1 − x)g > 1 − gx − gx2, and the constants d, g can be arranged so that qN > dN g and the induction goes through.

11: The precise domain of convergence for R1(a, b) is D0 = {(a, b) ∈ C×C : (a/b ∈ C ) or (a2 = b2, b ∈ I)}. In particular, for a/b ∈ C we have divergence. Moreover, provably, R1(a, b) converges to an analytic function of both a or b on the domain D2 := {(a, b) ∈ C × C : |a/b| = 1} ⊂ D0. 6 because neither H nor K intersects C . 11 is very subtle. 2) with all three fractions converging. 11 we have sufficient analyticity c to apply Berndt’s technique. 12 follows: Equivalently, a/b belongs to the closed exterior of ∂H, which in polar-coordinates is given by the cardioid-knot r2 + (2 cos φ − 4)r + 1 = 0 drawn in the complex plane (r = |a/b|).

Define • H := {z ∈ C : √ 2 z 1+z < 1}, 2z • K := {z ∈ C : 1+z 2 < 1}. 4: If a/b ∈ K then both R1(a, b) and R1(b, a) converge. 5: H ⊂ K (properly). 6: If a/b ∈ H then R1(a, b) & R1(b, a) converge, and the arithmetic mean √ (a + b)/2 dominates the geometric mean ab in modulus. 2) · · · . • We performed “scatter diagram” analysis to find computationally where the AGM relation holds in the parameter space. The results were quite spectacular! And lead to the Theorems above. 11: The precise domain of convergence for R1(a, b) is D0 = {(a, b) ∈ C×C : (a/b ∈ C ) or (a2 = b2, b ∈ I)}.