An algorithm for computing Bingham flows

A three-parameter iterative method for computing Bingham flows is examined. The method is a generalization of the well-known Arrow-Hurwicz algorithm. The local convergence of the method is proved.     

A minimum-stencil difference scheme for computing two-dimensional axisymmetric gas flows: Examples of pulsating flows with instabilities

A minimum-stencil difference scheme for computing two-dimensional axisymmetric gas flows is described. The scheme is explicit, conservative, and second-order accurate in space and time. The numerical results obtained for pulsating flows and contact discontinuity instabilities are discussed. The mechanisms of flow pulsation and instability generation are described.     

Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type

We establish the existence and stability of multidimensional transonic shocks for the Euler equations for steady potential compressible fluids. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for the velocity, can be written as a second-order, nonlinear equation of mixed elliptic-hyperbolic type for the velocity potential. The transonic shock problem can be formulated into the following free boundary problem: The free boundary is the location of the transonic shock which divides the two regions of smooth flow, and the equation is hyperbolic in the upstream region where the smooth perturbed flow is supersonic. We develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem. Our results indicate that there exists a unique solution of the free boundary problem such that the equation is always elliptic in the downstream region and the free boundary is smooth, provided that the hyperbolic phase is close to a uniform flow. We prove that the free boundary is stable under the steady perturbation of the hyperbolic phase. We also establish the existence and stability of multidimensional transonic shocks near spherical or circular transonic shocks.

     

A second-order dynamical system with Hessian-driven damping and penalty term associated to variational inequalities

Abstract

We consider the minimization of a convex objective function subject to the set of minima of another convex function, under the assumption that both functions are twice continuously differentiable. We approach this optimization problem from a continuous perspective by means of a second-order dynamical system with Hessian-driven damping and a penalty term corresponding to the constrained function. By constructing appropriate energy functionals, we prove weak convergence of the trajectories generated by this differential equation to a minimizer of the optimization problem as well as convergence for the objective function values along the trajectories. The performed investigations rely on Lyapunov analysis in combination with the continuous version of the Opial Lemma. In case the objective function is strongly convex, we can even show strong convergence of the trajectories.     

A variant of the finite superelement method for computing viscous incompressible flows

A variant of the finite superelement method (FSEM) is proposed for computing viscous incompressible convection-dominated flows on a triangular unstructured mesh. To construct the high-order FSEM scheme, vector polynomial test functions (SE basis) are calculated in each grid cell by solving the linearized Navier-Stokes equations with special boundary conditions in the form of basis functions in the trace space.     

A fast minimal residual algorithm for shifted unitary matrices

A new iterative scheme is described for the solution of large linear systems of equations with a matrix of the form A = ρU + ζI, where ρ and ζ are constants, U is a unitary matrix and I is the identity matrix. We show that for such matrices a Krylov subspace basis can be generated by recursion formulas with few terms. This leads to a minimal residual algorithm that requires little storage and makes it possible to determine each iterate with fairly little arithmetic work. This algorithm provides a model for iterative methods for non-Hermitian linear systems of equations, in a similar way to the conjugate gradient and conjugate residual algorithms. Our iterative scheme illustrates that results by Faber and Manteuffel [3,4] on the existence of conjugate gradient algorithms with short recurrence relations, and related results by Joubert and Young [13], can be extended.     

Entropies and Flux-Splittings for the Isentropic Euler Equations

51. IntroductionThe Euler equations for an iselitropic compressible fluid readwhere p 2 0 denotes the density, v the velocity) and p(p) 2 0 the pressure. The equstiope(1.1) form a nonlinear hyperbolic system of conserVation laws. By definition, a msthematicalentropy n = n(p, v) and its corresponding elltropy flux-function q = q(p, v) satisfyfor any smooth solution (p,m) of (1.1). A weak entropy3 by definition, vanishes on thevacuum p = 0. Following Laxlll'lz], we are interested in measuxable…     

A fast and accurate algorithm for solving linear systems associated with a class of negative matrix

A class of negative matrices including Vandermonde-like matrices tends to be extremely ill-conditioned, and linear systems associated with this class of matrices appear in the polynomial interpolation problems. In this article, we present a fast and accurate algorithm with $O(n^2)$ complexity to solve the linear systems whose coefficient matrices belong to the class of negative matrix. We show that the inverse of any such matrix is generated in a subtraction-free manner. Consequently, the solutions of linear systems associated with the class of negative matrix are accurately determined by parameterization matrices of coefficient matrices, and a pleasantly componentwise forward error is provided to illustrate that each component of the solution is computed to high accuracy. Numerical experiments are performed to confirm the claimed high accuracy.     

A globally convergent inexact Newton method with a new choice for the forcing term

In inexact Newton methods for solving nonlinear systems of equations, an approximation to the step s _k of the Newton’s system J(x _k )s=−F(x _k ) is found. This means that s _k must satisfy a condition like ‖F(x _k )+J(x _k )s _k ‖≤η _k ‖F(x _k )‖ for a forcing term η _k ∈[0,1). Possible choices for η _k have already been presented. In this work, a new choice for η _k is proposed. The method is globalized using a robust backtracking strategy proposed by Birgin et al. (Numerical Algorithms 32:249–260, 2003), and its convergence properties are proved. Several numerical experiments with boundary value problems are presented. The numerical performance of the proposed algorithm is analyzed by the performance profile tool proposed by Dolan and Moré (Mathematical Programming Series A 91:201–213, 2002). The results obtained show a competitive inexact Newton method for solving academic and applied problems in several areas. Supported by FAPESP, CNPq, PRONEX-Optimization.     

A polynomial cycle canceling algorithm for submodular flows

Submodular flow problems, introduced by Edmonds and Giles [2], generalize network flow problems. Many algorithms for solving network flow problems have been generalized to submodular flow problems (cf. references in Fujishige [4]), e.g. the cycle canceling method of Klein [9]. For network flow problems, the choice of minimum-mean cycles in Goldberg and Tarjan [6], and the choice of minimum-ratio cycles in Wallacher [12] lead to polynomial cycle canceling methods. For submodular flow problems, Cui and Fujishige [1] show finiteness for the minimum-mean cycle method while Zimmermann [16] develops a pseudo-polynomial minimum ratio cycle method. Here, we prove pseudo-polynomiality of a larger class of the minimum-ratio variants and, by combining both methods, we develop a polynomial cycle canceling algorithm for submodular flow problems. Received July 22, 1994 / Revised version received July 18, 1997? Published online May 28, 1999     

A locally and cubically convergent algorithm for computing 𝒵‐eigenpairs of symmetric tensors

This paper is concerned with computing $\mathrm{??}$‐eigenpairs of symmetric tensors. We first show that computing $\mathrm{??}$‐eigenpairs of a symmetric tensor is equivalent to finding the nonzero solutions of a nonlinear system of equations, and then propose a modified normalized Newton method (MNNM) for it. Our proposed MNNM method is proved to be locally and cubically convergent under some suitable conditions, which greatly improves the Newton correction method and the orthogonal Newton correction method recently provided by Jaffe, Weiss and Nadler since these two methods only enjoy a quadratic rate of convergence. As an application, the unitary symmetric eigenpairs of a complex‐valued symmetric tensor arising from the computation of quantum entanglement in quantum physics are calculated by the MNNM method. Some numerical results are presented to illustrate the efficiency and effectiveness of our method.     

Semilocal and global convergence of the Newton‐HSS method for systems of nonlinear equations

Newton‐HSS methods, which are variants of inexact Newton methods different from the Newton–Krylov methods, have been shown to be competitive methods for solving large sparse systems of nonlinear equations with positive‐definite Jacobian matrices (J. Comp. Math. 2010; 28 :235–260). In that paper, only local convergence was proved. In this paper, we prove a Kantorovich‐type semilocal convergence. Then we introduce Newton‐HSS methods with a backtracking strategy and analyse their global convergence. Finally, these globally convergent Newton‐HSS methods are shown to work well on several typical examples using different forcing terms to stop the inner iterations. Copyright © 2010 John Wiley & Sons, Ltd.     

Steady Compressible Euler Equations of Concentration Layers for Hypersonic-limit Flows Passing Three-dimensional Bodies and Generalized Newton-Busemann Pressure Law*

For stationary hypersonic-limit Euler flows passing a solid body in three-dimensional space, the shock-front coincides with the upwind surface of the body, hence there is an infinite-thin layer of concentrated mass, in which all particles hitting the body move along its upwind surface. By proposing a concept of Radon measure solutions of boundary value problems of the multi-dimensional compressible Euler equations, which incorporates the large-scale of three-dimensional distributions of upcoming hypersonic flows and the small-scale of particles moving on two-dimensional surfaces, the authors derive the compressible Euler equations for flows in concentration layers, which is a stationary pressureless compressible Euler system with source terms and independent variables on curved surface. As a by-product, they obtain a formula for pressure distribution on surfaces of general obstacles in hypersonic flows, which is a generalization of the classical Newton-Busemann law for drag/lift in hypersonic aerodynamics.     

A fast algorithm for computing minimum 3-way and 4-way cuts

For an edge-weighted graph G with n vertices and m edges, we present a new deterministic algorithm for computing a minimum k-way cut for k=3,4. The algorithm runs in O(n ^k-1 F(n,m))=O(mn ^k log(n ² /m)) time and O(n ²) space for k=3,4, where F(n,m) denotes the time bound required to solve the maximum flow problem in G. The bound for k=3 matches the current best deterministic bound ?(mn ³) for weighted graphs, but improves the bound ?(mn ³) to O(n ² F(n,m))=O(min{mn ^8/3,m ^3/2 n ²}) for unweighted graphs. The bound ?(mn ⁴) for k=4 improves the previous best randomized bound ?(n ⁶) (for m=o(n ²)). The algorithm is then generalized to the problem of finding a minimum 3-way cut in a symmetric submodular system. Received: April 1999 / Accepted: February 2000?Published online August 18, 2000     

A globally convergent interval method for computing and bounding real roots

In this paper, we extend the interval Newton method to the case where the interval derivative may contain zero. This extended method will isolate and bound all the real roots of a continuously differentiable function in a given interval. In particular, it will bound multiple roots. We prove that the method never fails to converge.     

A three-step Oseen-linearized finite element method for incompressible flows with damping

This paper introduces a three-step Oseen-linearized finite element method for the 2D/3D steady incompressible Navier–Stokes equations with nonlinear damping term. Within this method, we solve a nonlinear problem over a coarse grid followed by solving two Oseen-linearized problems over a fine grid, which possess the same stiffness matrices with only various right-hand sides. We theoretically analyze the stability of the present method, and derive optimal error estimates of the finite element solutions. We conduct a series of numerical experiments which support the theoretical analysis and test the effectiveness of the proposed method. We demonstrate numerically that there is a significant improvement in the accuracy of the approximate solutions over those for the standard two-level method.     

A relaxation Riemann solver for compressible two-phase flow with phase transition and surface tension

The dynamics of two-phase flows depend crucially on interfacial effects like surface tension and phase transition. A numerical method for compressible inviscid flows is proposed that accounts in particular for these two effects. The approach relies on the solution of Riemann-like problems across the interface that separates the liquid and the vapour phase. Since the analytical solutions of the Riemann problems are only known in particular cases an approximative Riemann solver for arbitrary settings is constructed. The approximative solutions rely on the relaxation technique.The local well-posedness of the approximative solver is proven. Finally we present numerical experiments for radially symmetric configurations that underline the reliability and efficiency of the numerical scheme.     

A parallel finite element variational multiscale method based on fully overlapping domain decomposition for incompressible flows

Based on fully overlapping domain decomposition and a recent variational multiscale method, a parallel finite element variational multiscale method for convection dominated incompressible flows is proposed and analyzed. In this method, each processor computes a local finite element solution in its own subdomain using a global mesh that is locally refined around its own subdomain, where a stabilization term based on two local Gauss integrations is adopted to stabilize the numerical form of the Navier–Stokes equations. Using the technical tool of local a priori estimate for the finite element solution, error bounds of the discrete solution are estimated. Algorithmic parameter scalings are derived. Numerical tests are also given to verify the theoretical predictions and demonstrate the effectiveness of the method. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 856–875, 2015     

An exact Riemann solver and a Godunov scheme for simulating highly transient mixed flows

The current research aims at deriving a one-dimensional numerical model for describing highly transient mixed flows. In particular, this paper focuses on the development and assessment of a unified numerical scheme adapted to describe free-surface flow, pressurized flow and mixed flow (characterized by the simultaneous occurrence of free-surface and pressurized flows). The methodology includes three steps. First, the authors derived a unified mathematical model based on the Preissmann slot model. Second, a first-order explicit finite volume Godunov-type scheme is used to solve the set of equations. Third, the numerical model is assessed by comparison with analytical, experimental and numerical results. The key results of the paper are the development of an original negative Preissmann slot for simulating sub-atmospheric pressurized flow and the derivation of an exact Riemann solver for the Saint-Venant equations coupled with the Preissmann slot.