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A class of upwind flux splitting methods in the Euler equations of compressible flow is considered in this paper. Using the property that Euler flux F(U) is a homogeneous function of degree one in U, we reformulate the splitting fluxes with F^{+}=A^{+} U, F^{-}=A^{-} U, and the corresponding matrices are either symmetric or symmetrizable and keep only non-negative and non-positive eigenvalues. That leads to the conclusion that the first order schemes are positive in the sense of Lax-Liu [18], which implies that it is L^2-stable in some suitable sense. Moreover, the second order scheme is a stable perturbation of the first order scheme, so that the positivity of the second order schemes is also established, under a CFL-like condition. In addition, these splitting methods preserve the positivity of density and energy.

In this paper, we study the geometry of compact complex manifolds with Levi-Civita Ricci-flat metrics and prove that compact complex surfaces admitting Levi-Civita Ricci-flat metrics are Khler Calabi-Yau surfaces or Hopf surfaces.

We propose efficient Eulerian methods for approximating the finite-time Lyapunov exponent (FTLE). The idea is to compute the related flow map using the Level Set Method and the Liouville equation. There are several advantages of the proposed approach. Unlike the usual Lagrangian-type computations, the resulting method requires the velocity field defined only at discrete locations. No interpolation of the velocity field is needed. Also, the method automatically stops a particle trajectory in the case when the ray hits the boundary of the computational domain. The computational complexity of the algorithm is <i>O</i>(<i>x</i>^(<i>d</i>+1)) with <i>d</i> the dimension of the physical space. Since there are the same number of mesh points in the <i>x</i><i>t</i> space, the computational complexity of the proposed Eulerian approach is optimal in the sense that each grid point is visited for only <i>O</i>(1) time. We also extend the algorithm to compute the FTLE

In this paper, we present an adaptive, analysis of variance (ANOVA)-based data-driven stochastic method (ANOVA-DSM) to study the stochastic partial differential equations (SPDEs) in the multi-query setting. Our new method integrates the advantages of both the adaptive ANOVA decomposition technique and the data-driven stochastic method. To handle high-dimensional stochastic problems, we investigate the use of adaptive ANOVA decomposition in the stochastic space as an effective dimension-reduction technique. To improve the slow convergence of the generalized polynomial chaos (gPC) method or stochastic collocation (SC) method, we adopt the data-driven stochastic method (DSM) for speed up. An essential ingredient of the DSM is to construct a set of stochastic basis under which the stochastic solutions enjoy a compact representation for a broad range of forcing functions and/or boundary conditions

We show that direct limit completions of vertex tensor categories inherit vertex and braided tensor category structures, under conditions that hold for example for all known Virasoro and affine Lie algebra tensor categories. A consequence is that the theory of vertex operator (super)algebra extensions also applies to infinite-order extensions. As an application, we relate rigid and non-degenerate vertex tensor categories of certain modules for both the affine vertex superalgebra of osp(1|2) and the N=1 super Virasoro algebra to categories of Virasoro algebra modules via certain cosets.

No abstract uploaded!

No abstract uploaded!

We study the collapsing behavior of the Kähler–Ricci flow on a compact Kähler manifold X admitting a holomorphic submersion inherited from its canonical bundle. We show that the flow metric degenerates at exactly the rate of e^{-t} as predicted by the cohomology information, and so the fibres collapse at the optimal rate. Consequently, it leads to some analytic and geometric extensions to the regular case of works by J. Song and G. Tian. Its applicability to general Calabi–Yau fibrations will also be discussed in local settings.