A completely conservative difference scheme of gas dynamics in orthogonal curvilinear coordinates

A completely conservative difference scheme in Eulerian curvilinear orthogonal coordinates is proposed for calculating discontinuous gas-dynamic flows on sufficiently coarse grids. The stability of the linear approximation of the proposed scheme is analyzed. Courant's condition provides a stability condition for this scheme.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 58, pp. 9–15, 1986.     

A conservative semi-Lagrangian method for multidimensional fluid dynamics applications

We describe a finite volume semi-Lagrangian method for the numerical approximation of conservation laws arising in fluid-dynamic applications. A discrete conservation relation is satisfied by using conservative interpolation for the material (or property) being conserved. The method was developed with a view to application in climate prediction. © 1995 John Wiley & Sons, Inc.     

A shock-capturing model based on flux-vector splitting method in boundary-fitted curvilinear coordinates

In this paper, a robust and accurate high-resolution finite-volume scheme is presented which employs flux-vector splitting (FVS) as the building block for solution of shallow water equations in boundary-fitted curvilinear coordinates. Eddy viscosity approach is used to accommodate shear stresses due to turbulence. Splitting of the convective terms is achieved via flux Jacobians whereas Liou–Steffen Splitting (LSS) technique, but in transformed coordinates, is used to split pressure terms. Limited flux gradients are also used to increase the computational accuracy of evaluation of interface fluxes and decrease the excessive numerical dissipation associated with FVS. This will completely remove spurious oscillations in high-gradient regions without introducing too much numerical dissipations. The method is tested for some classic simulations including hydraulic jump, 1D dam break and 2D dam break problems. The results show very satisfactory agreement with experimental data, analytical solutions and other numerical results.     

UPML formulation for truncating conductive media in curvilinear coordinates

This work presents a formulation based on UPML for truncating conductive media by using a local and non-orthogonal coordinate system to solve Maxwell’s equations by the FDTD method. The detailed procedure for obtaining the UPML equations for this case is shown and the complete equation set is provided.     

The obstacle problem for shallow shells in curvilinear coordinates

Starting from the 3D Signorini problem in presence of a plane obstacle, we justify the limit inequation of unilateral contact posed in a 2D domain. In particular, we show that we can uncouple the three covariant components of the limit Kirchhoff–Love displacement field so that the non-penetrability condition involves only the “transverse” component as this is the case in Cartesian framework.     

A particle method to solve the Navier-Stokes system

Summary We extend to the case of the two-dimensional Navier-Stokes equations, a particle method introduced in a previous paper to solve linear convection-diffusion equations. The method is based on a viscous splitting of the operator. The particles move under the effect of the velocity field but are not affected by the diffusion which is taken into account by the weights. We prove the stability and the convergence of the method.     

From gas dynamics to pressureless gas dynamics

This paper is devoted to the convergence of solutions of the compressible Euler equations towards solutions of the pressureless gas dynamics system, when the pressure tends to 0. The goal is to prove accurate uniform bounds for particular solutions of the Euler equations.

     

A multi-iterate method to solve systems of nonlinear equations

We propose an extension of secant methods for nonlinear equations using a population of previous iterates. Contrarily to classical secant methods, where exact interpolation is used, we prefer a least squares approach to calibrate the linear model. We propose an explicit control of the numerical stability of the method.     

A linearization approach to solve the natural gas cash-out bilevel problem

In this article, we discuss a particular imbalance cash-out problem arising in the natural gas supply chain. This problem was created by the liberalization laws that regulate deals between a natural gas shipping company and a pipeline operator. The problem was first modeled as a bilevel nonlinear mixed-integer problem that considers the cash-out penalization for the final imbalance occurring in the system. We extend the original problem’s upper level objective function by including additional terms accounting for the gas shipping company’s daily actions aimed at taking advantage of the price variations. Then we linearize all the constraints at both levels in an equivalent way so as to make easier their numerical solution. The results of numerical experiments are compared with those obtained by the inexact penalization method proposed by the authors in previous papers.     

A Newton-type curvilinear search method for optimization

A new search method is presented for unconstrained optimization. The method requires the evaluation of first and second derivatives and defines a curve along which a undimensional step takes place. For large step-size, the method performs as Newton's method, but it does not fail where the latter fails. For small step-size, the method behaves as the gradient method.     

A graphical method to solve a family of allocation problems

This paper discusses the classical resource allocation problem. By very elementary arguments on Lagrangian duality it is shown that this problem can be reduced to a single one-dimensional maximization of a differentiable concave function. Moreover, a simple graphical method is developed and applied to a family of well-known problems from the literature.     

A feasible conjugate-direction method to solve linearly constrained minimization problems

An iterative procedure is presented which uses conjugate directions to minimize a nonlinear function subject to linear inequality constraints. The method (i) converges to a stationary point assuming only first-order differentiability, (ii) has ann-q step superlinear or quadratic rate of convergence with stronger assumptions (n is the number of variables,q is the number of constraints which are binding at the optimum), (iii) requires the computation of only the objective function and its first derivatives, and (iv) is experimentally competitive with well-known methods.For helpful suggestions, the author is much indebted to C. R. Glassey and K. Ritter.This research has been partially supported by the National Research Council of Canada under Grants Nos. A8189 and C1234.     

A new method to solve the damped nonlinear Klein-Gordon equation

This paper discusses a damped nonlinear Klein-Gordon equation in the reproducing kernel space and provides a new method for solving the damped nonlinear Klein-Gordon equation based on the reproducing kernel space.Two numerical examples are given for illustrating the feasibility and accuracy of the method.     

A curvilinear method based on minimal-memory BFGS updates

We present a new matrix-free method for the computation of negative curvature directions based on the eigenstructure of minimal-memory BFGS matrices. We determine via simple formulas the eigenvalues of these matrices and we compute the desirable eigenvectors by explicit forms. Consequently, a negative curvature direction is computed in such a way that avoids the storage and the factorization of any matrix. We propose a modification of the L-BFGS method in which no information is kept from old iterations, so that memory requirements are minimal. The proposed algorithm incorporates a curvilinear path and a linesearch procedure, which combines two search directions; a memoryless quasi-Newton direction and a direction of negative curvature. Results of numerical experiments for large scale problems are also presented.     

A Newton-type curvilinear search method for constrained optimization

An algorithm is presented that minimizes a nonlinear function in many variables under equality constraints by generating a monotonically improving sequence of feasible points along curvilinear search paths obeying an initialvalue system of differential equations. The derivation of the differential equations is based on the idea of a steepest descent curve for the objective function on the feasible region. Our method for small stepsize behaves as the generalized reduced gradient algorithm, whereas for large enough stepsize the constrained equivalent of Newton's method for unconstrained minimization is obtained.     

求矩阵A的Jordan标准形的另一方法

本提出了求矩阵A的Jordan标准形的另一方法：利用rank(λ(E-A)^P的结果，得出了对应于特征(λi的Jordan块的阶数和个数，然后求出矩阵A的Jordan标准形．     

Using Lin's method to solve Bykov's problems

We consider nonwandering dynamics near heteroclinic cycles between two hyperbolic equilibria. The constituting heteroclinic connections are assumed to be such that one of them is transverse and isolated. Such heteroclinic cycles are associated with the termination of a branch of homoclinic solutions, and called T-points   in this context. We study codimension-two T-points and their unfoldings in Rⁿ${\mathbb{R}}^n$. In our consideration we distinguish between cases with real and complex leading eigenvalues of the equilibria. In doing so we establish Lin's method as a unified approach to (re)gain and extend results of Bykov's seminal studies and related works. To a large extent our approach reduces the study to the discussion of intersections of lines and spirals in the plane.     

Piecewise parabolic method on a local stencil in cylindrical coordinates for fluid dynamics simulations

A PPML algorithm for the simulation of hydrodynamic flows on a cylindrical grid is described. The algorithm is based on a local stencil variant of the popular piecewise parabolic method (PPM). Numerical results obtained a point explosion in a homogeneous medium (Sedov self-similar solution) are presented.