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The von Neumann Triple Point Paradox 119 (a) (b) (c) Fig. 4. Closeups of an apparent triple point for the UTSDE using the approach of Tesdall and Hunter. In (a) and (b) the incident shock leaves to the upper right, the reflected shock towards the top, and the Mach stem exits at the bottom. The plot in (a) depicts contour lines of u and shows a sequence of expansions/shocks running down the Mach stem. The plot in (b) shows a detail of v; 1 denotes state v = 0, 2 state v = −ǎ and 3 points to the expansion wave emanating from what appears macroscopically to be a triple point. The dotted line in (b) delineates the supersonic patches within the subsonic zone behind the Mach stem. The Guderley Mach reflection structure can be seen better in the surface plot (c) where the viewer is upstream looking back at the triple point. experiments were carried out on a 15 ◦ ramp with incident shock Mach numbers ranging from 1.05 to 1.1. They present images that “clearly show the existence of an expansion wave immediately behind the reflected wave as proposed by Guderley”, and they found “a distinct sharp contrasting line immediately
120 R. Sanders and A.M. Tesdall (a) (b) Fig. 5. On the left, a schlieren image of an experimental weak shock reflection. The incident shock (vertical) exits at the top and is moving from left to right. The reflected wave exits to the upper left, and an expansion wave is visible immediately behind it. A highly contrasted image is on the right, showing evidence of a second shocklet behind the first. after the expansion wave, indicating the existence of a terminating shock”. In addition, they obtained evidence in some of their images of a second terminating shocklet behind the first, as predicted by the simulations in [TH02]. Professor Beric Skews graciously supplied us with the images which we give here in Figure 5. Further experimental refinements and data acquisition are currently underway. 3 The Nonlinear Wave System Here we consider a problem for the nonlinear wave system which is analogous to the reflection of weak shocks discussed in the previous section. The shock reflection problem consists of the nonlinear wave system ∂ρ + ∇·ρu =0, ∂t ∂ρu +gradp =0, ∂t in the half space x>0 with piecewise constant Riemann data consisting of two states separated by a discontinuity located at x = κy. Again, ρ should be thought of as density, u =(u, v) as velocity having x- andy-components, and p = p(ρ) as pressure. For convenience, we assume p(ρ) =Cρ γ where C is a constant and γ = 2. See [TSK06]. The nonlinear wave system is a simplification of the isentropic Euler equations obtained by dropping the momentum transport terms from the
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Partial Differential Equations
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Partial Differential Equations Mode
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Dedicated to Olivier Pironneau
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VIII Preface computers has been at
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Contents List of Contributors .....
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List of Contributors Yves Achdou UF
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List of Contributors XV Claude Le B
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Discontinuous Galerkin Methods Vive
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Discontinuous Galerkin Methods 5 2
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Discontinuous Galerkin Methods 7 g
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Discontinuous Galerkin Methods 9 (
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Discontinuous Galerkin Methods 15 W
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Let a h and b h denote the bilinear
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Discontinuous Galerkin Methods 19 t
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Discontinuous Galerkin Methods 21 l
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Table 1. Primal DG for transport Di
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Discontinuous Galerkin Methods 25 [
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Mixed Finite Element Methods on Pol
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Mixed FE Methods on Polyhedral Mesh
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4 Hybridization and Condensation Mi
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is symmetric and positive definite,
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with some coefficient α ∈ R wher
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44 E.J. Dean and R. Glowinski so fa
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46 E.J. Dean and R. Glowinski 2 A L
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48 E.J. Dean and R. Glowinski S:T=
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50 E.J. Dean and R. Glowinski minim
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52 E.J. Dean and R. Glowinski 6 On
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54 E.J. Dean and R. Glowinski Fig.
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56 E.J. Dean and R. Glowinski and
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58 E.J. Dean and R. Glowinski 7 Num
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60 E.J. Dean and R. Glowinski Fig.
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62 E.J. Dean and R. Glowinski Assum
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Numerical Analysis of a Finite Elem
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so that |u| 2 1,ω h ≤ 2 Numerica
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188 A. Bonito et al. of the model a
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190 A. Bonito et al. Fig. 1. The sp
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192 A. Bonito et al. 3 1 16 4 1 1 1
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194 A. Bonito et al. ∫ v n+1 h
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196 A. Bonito et al. −pn +2µD(v)
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198 A. Bonito et al. each of its pa
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200 A. Bonito et al. The normal vec
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202 A. Bonito et al. with initial c
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204 A. Bonito et al. Fig. 8. Jet bu
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206 A. Bonito et al. References [AM
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208 A. Bonito et al. [Set96] J. A.
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210 J. Hao et al. due to shear flow
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212 J. Hao et al. The backward reac
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214 J. Hao et al. where u and p den
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216 J. Hao et al. * * * * * * * * *
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218 J. Hao et al. and solve for V n
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220 J. Hao et al. Table 2. The calc
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222 J. Hao et al. References [ASS80
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Computing the Eigenvalues of the La
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Eigenvalues of the Laplace-Beltrami
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A Fixed Domain Approach in Shape Op
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Shape Optimization Problems with Ne
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Reduced-Order Modelling of Dispersi
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Calibration of Lévy Processes with
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We have proved Calibration of Lévy
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Note that p ∗ satisfies Calibrati
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280 S. Ikonen and J. Toivanen the p
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282 S. Ikonen and J. Toivanen Merto
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284 S. Ikonen and J. Toivanen For H
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286 S. Ikonen and J. Toivanen and {
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288 S. Ikonen and J. Toivanen 1.6 1
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290 S. Ikonen and J. Toivanen 8 Con
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292 S. Ikonen and J. Toivanen [Mer7
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