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We introduce a new class of Discontinuous Galerkin (DG) methods for solving nonlinear conservation laws on unstructured Voronoi meshes that use a nonconforming Virtual Element basis defined within each polygonal control volume. The basis functions are evaluated as an L2 projection of the virtual basis which remains unknown, along the lines of the Virtual Element Method (VEM). Contrarily to the VEM approach, the new basis functions lead to a nonconforming representation of the solution with discontinuous data across the element boundaries, as typically employed in DG discretizations. To improve the condition number of the resulting mass matrix, an orthogonalization of the full basis is proposed. The discretization in time is carried out following the ADER (Arbitrary order DERivative Riemann problem) methodology, which yields one-step fully discrete schemes that make use of a coupled space-time representation of the numerical solution. The space-time basis functions are constructed as a tensor product of the virtual basis in space and a one-dimensional Lagrange nodal basis in time. The resulting space-time stiffness matrix is stabilized by an extension of the dof-dof stabilization technique adopted in the VEM framework, hence allowing an element-local space-time Galerkin finite element predictor to be evaluated. The novel methods are referred to as VEM-DG schemes, and they are arbitrarily high order accurate in space and time. The new VEM-DG algorithms are rigorously validated against a series of benchmarks in the context of compressible Euler and Navier-Stokes equations. Numerical results are verified with respect to literature reference solutions and compared in terms of accuracy and computational efficiency to those obtained using a standard modal DG scheme with Taylor basis functions. An analysis of the condition number of the mass and space-time stiffness matrix is also forwarded.

In this work, a Cole-Hopf transformation based fourth-order multiple-relaxation-time lattice Boltzmann (MRT-LB) model for d-dimensional coupled Burgers' equations is developed. We first adopt the Cole-Hopf transformation where an intermediate variable \theta is introduced to eliminate the nonlinear convection terms in the Burgers' equations on the velocity u=(u_1,u_2,...,u_d). In this case, a diffusion equation on the variable \theta can be obtained, and particularly, the velocity u in the coupled Burgers' equations is determined by the variable \theta and its gradient term \nabla\theta. Then we develop a general MRT-LB model with the natural moments for the d-dimensional transformed diffusion equation and present the corresponding macroscopic finite-difference scheme. At the diffusive scaling, the fourth-order modified equation of the developed MRT-LB model is derived through the Maxwell iteration method. With the aid of the free parameters in the MRT-LB model, we find that not only the consistent fourth-order modified equation can be obtained, but also the gradient term $\nabla\theta$ can be calculated locally by the non-equilibrium distribution function with a fourth-order accuracy, this indicates that theoretically, the MRT-LB model for $d$-dimensional coupled Burgers' equations can achieve a fourth-order accuracy in space. Finally, some simulations are conducted to test the MRT-LB model, and the numerical results show that the proposed MRT-LB model has a fourth-order convergence rate, which is consistent with our theoretical analysis.

In this paper, we are concerned with symmetric integrators for the nonlinear relativistic Klein--Gordon (NRKG) equation with a dimensionless parameter $0<\varepsilon\ll 1$, which is inversely proportional to the speed of light. The highly oscillatory property in time of this model corresponds to the parameter $\varepsilon$ and the equation has strong nonlinearity when $\eps$ is small. There two aspects bring significantly numerical burdens in designing numerical methods. We propose and analyze a novel class of symmetric integrators which is based on some formulation approaches to the problem, Fourier pseudo-spectral method and exponential integrators. Two practical integrators up to order four are constructed by using the proposed symmetric property and stiff order conditions of implicit exponential integrators. The convergence of the obtained integrators is rigorously studied, and it is shown that the accuracy in time is improved to be $\mathcal{O}(\varepsilon^{3} \hh^2)$ and $\mathcal{O}(\varepsilon^{4} \hh^4)$ for the time stepsize $\hh$. The near energy conservation over long times is established for the multi-stage integrators by using modulated Fourier expansions. These theoretical results are achievable even if large stepsizes are utilized in the schemes. Numerical results on a NRKG equation show that the proposed integrators have improved uniform error bounds, excellent long time energy conservation and competitive efficiency.

This paper gives a self-contained introduction to the Hilbert projective metric $\mathcal{H}$ and its fundamental properties, with a particular focus on the space of probability measures. We start by defining the Hilbert pseudo-metric on convex cones, focusing mainly on dual formulations of $\mathcal{H}$ . We show that linear operators on convex cones contract in the distance given by the hyperbolic tangent of $\mathcal{H}$, which in particular implies Birkhoff's classical contraction result for $\mathcal{H}$. Turning to spaces of probability measures, where $\mathcal{H}$ is a metric, we analyse the dual formulation of $\mathcal{H}$ in the general setting, and explore the geometry of the probability simplex under $\mathcal{H}$ in the special case of discrete probability measures. Throughout, we compare $\mathcal{H}$ with other distances between probability measures. In particular, we show how convergence in $\mathcal{H}$ implies convergence in total variation, $p$-Wasserstein distance, and any $f$-divergence. Furthermore, we derive a novel sharp bound for the total variation between two probability measures in terms of their Hilbert distance.

In this work, we develop first-order (Hessian-free) and zero-order (derivative-free) implementations of the Cubically regularized Newton method for solving general non-convex optimization problems. For that, we employ finite difference approximations of the derivatives. We use a special adaptive search procedure in our algorithms, which simultaneously fits both the regularization constant and the parameters of the finite difference approximations. It makes our schemes free from the need to know the actual Lipschitz constants. Additionally, we equip our algorithms with the lazy Hessian update that reuse a previously computed Hessian approximation matrix for several iterations. Specifically, we prove the global complexity bound of $\mathcal{O}( n^{1/2} \epsilon^{-3/2})$ function and gradient evaluations for our new Hessian-free method, and a bound of $\mathcal{O}( n^{3/2} \epsilon^{-3/2} )$ function evaluations for the derivative-free method, where $n$ is the dimension of the problem and $\epsilon$ is the desired accuracy for the gradient norm. These complexity bounds significantly improve the previously known ones in terms of the joint dependence on $n$ and $\epsilon$, for the first-order and zeroth-order non-convex optimization.

The task of natural language inference (NLI) asks whether a given premise (expressed in NL) entails a given NL hypothesis. NLI benchmarks contain human ratings of entailment, but the meaning relationships driving these ratings are not formalized. Can the underlying sentence pair relationships be made more explicit in an interpretable yet robust fashion? We compare semantic structures to represent premise and hypothesis, including sets of contextualized embeddings and semantic graphs (Abstract Meaning Representations), and measure whether the hypothesis is a semantic substructure of the premise, utilizing interpretable metrics. Our evaluation on three English benchmarks finds value in both contextualized embeddings and semantic graphs; moreover, they provide complementary signals, and can be leveraged together in a hybrid model.

Recently, a stability theory has been developed to study the linear stability of modified Patankar--Runge--Kutta (MPRK) schemes. This stability theory provides sufficient conditions for a fixed point of an MPRK scheme to be stable as well as for the convergence of an MPRK scheme towards the steady state of the corresponding initial value problem, whereas the main assumption is that the initial value is sufficiently close to the steady state. Initially, numerical experiments in several publications indicated that these linear stability properties are not only local, but even global, as is the case for general linear methods. Recently, however, it was discovered that the linear stability of the MPDeC(8) scheme is indeed only local in nature. Our conjecture is that this is a result of negative Runge--Kutta (RK) parameters of MPDeC(8) and that linear stability is indeed global, if the RK parameters are nonnegative. To support this conjecture, we examine the family of MPRK22($\alpha$) methods with negative RK parameters and show that even among these methods there are methods for which the stability properties are only local. However, this local linear stability is not observed for MPRK22($\alpha$) schemes with nonnegative Runge-Kutta parameters.

In this paper, we investigate the structure of the Schur complement matrix for the fully-staggered finite-difference discretization of the stationary Stokes equation. Specifically, we demonstrate that the structure of the Schur complement matrix depends qualitatively on a particular characteristic, namely the number of non-unit eigenvalues, and the two limiting cases are of special interest.

Galois self-orthogonal (SO) codes are generalizations of Euclidean and Hermitian SO codes. Algebraic geometry (AG) codes are the first known class of linear codes exceeding the Gilbert-Varshamov bound. Both of them have attracted much attention for their rich algebraic structures and wide applications in these years. In this paper, we consider them together and study Galois SO AG codes. A criterion for an AG code being Galois SO is presented. Based on this criterion, we construct several new classes of maximum distance separable (MDS) Galois SO AG codes from projective lines and several new classes of Galois SO AG codes from projective elliptic curves, hyper-elliptic curves and hermitian curves. In addition, we give an embedding method that allows us to obtain more MDS Galois SO codes from known MDS Galois SO AG codes.

We introduce new control-volume finite-element discretization schemes suitable for solving the Stokes problem. Within a common framework, we present different approaches for constructing such schemes. The first and most established strategy employs a non-overlapping partitioning into control volumes. The second represents a new idea by splitting into two sets of control volumes, the first set yielding a partition of the domain and the second containing the remaining overlapping control volumes required for stability. The third represents a hybrid approach where finite volumes are combined with finite elements based on a hierarchical splitting of the ansatz space. All approaches are based on typical finite element function spaces but yield locally mass and momentum conservative discretization schemes that can be interpreted as finite volume schemes. We apply all strategies to the inf-sub stable MINI finite-element pair. Various test cases, including convergence tests and the numerical observation of the boundedness of the number of preconditioned Krylov solver iterations, as well as more complex scenarios of flow around obstacles or through a three-dimensional vessel bifurcation, demonstrate the stability and robustness of the schemes.