Transmission of progressing waves across a moving interface

We describe global time existence nd uniqueness results for the wave equations with boundary conditions of Dirichlet type on a characteresitc cone and either Dirichlet or Neumann type on a timelike tube. We find that the solution is in general only half as regular as the data and we provide estimates which describe the differing differentiabilities of the solution in directions which are either tangent or transvers to the characteristic cone.     

Numerical solution to a stokes interface problem

A numerical algorithm is described for solving the generalized eigenvalue problem arising in the study of the spectrum of a preconditioned operator in the pressure equation derived from a Stokes interface problem. The algorithm is implemented for two finite element schemes. It is tested for a problem with an analytical solution and is applied to spectrum computations in the case of a piecewise constant viscosity. A large number of numerical experiments are analyzed, and recommendations are given for solving the Stokes interface problem in practice.     

Numerical studies of optimal grid construction

This article is concerned with the construction of optimal grids for convection dominated problems. We consider the redistribution approach to generate the optimal grids. With an optimal grid, the number of unknowns can be dramatically reduced and this generally gives a computationally efficient model. Both Galerkin finite-element and least-squares finite-element approximations are considered with mesh redistribution. Numerical results of models problems illustrating the efficiency of the minimum mesh size approach are presented. Discussions of the capabilities and limitations of the schemes are also provided. © 1996 John Wiley & Sons, Inc.     

Assessing the impact of inventory inaccuracies within a Newsvendor framework

This paper examines a Newsvendor framework in which a wholesaler who sells products to retailers is subject to inaccuracies in inventory data. The wholesaler’s decision regarding the management of his warehouse inventory is based on inventory data recorded in the information system. We assume that the wholesaler uses barcode labels and scanners in order to gather information concerning the available inventory in the warehouse. Because of errors arising during the manual barcode scanning process, the information on the available inventory level can deviate from the physical quantity and can result in an additional cost. This paper quantifies the economic impact of having uncertainty on the inventory level. We first analyze the case of a wholesaler that is not aware of inventory errors or chooses to ignore them in order to evaluate the efficiency loss due to errors compared with an error free situation. We then assess the effect of various actions enabling to tackle the inventory inaccuracy issue with a particular focus on actions such as the deployment of a new data capture technology.     

Numerical solution of the synthesis problem of a linear equidistant grid in a given frequency range

Using the example of a linear grid we study problems of phase and amplitude-phase synthesis in a given frequency range: finding the frequency-dependent currents in radiators that guarantee optimal mean-square approximation of the amplitudes of the given and the synthesized diagrams; finding the averaged frequency-independent currents. The problems reduce to nonlinear integral equations that can be solved numerically by the method of successive approximations.Translated fromMatematicheskie Melody i Fiziko-Mekhanicheskie Polya, Issue 33, 1991, pp. 91–95.     

Two‐grid methods for semilinear interface problems

In this article, we consider two‐grid finite element methods for solving semilinear interface problems in d space dimensions, for d = 2 or d = 3. We consider semilinear problems with discontinuous diffusion coefficients, which includes problems containing subcritical, critical, and supercritical nonlinearities. We establish basic quasioptimal a priori error estimates for Galerkin approximations. We then design a two‐grid algorithm consisting of a coarse grid solver for the original nonlinear problem, and a fine grid solver for a linearized problem. We analyze the quality of approximations generated by the algorithm and show that the coarse grid may be taken to have much larger elements than the fine grid, and yet one can still obtain approximation quality that is asymptotically as good as solving the original nonlinear problem on the fine mesh. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Numerical simulations for the contraction flow using grid generation

We study the incomprssible Navier Stokes equations for the flow inside contraction geometry. The governing equations are expressed in the vorticity-stream function formulations. A rectangular computational domain is arised by elliptic grid generation technique. The numerical solution is based on a technique of automatic numerical generation of acurvilinear coordinate system by transforming the governing equation into computational plane. The transformed equations are approximated using central differences and solved simultaneously by successive over relaxation iteration. The time dependent of the vorticity equation solved by using explicit marching procedure. We will apply the technique on several irregularshapes.     

Computational tests as a means of assessing the inaccuracies of index numbers

In this paper, we propose using computational tests for appraisingthe inaccuracies of index numbers (for example, price and wageindices). This is done by programming a hypothetical situationwhere the true average index value is known and the variationof prices and quantities and the relation between them can bechosen. The indices are then calculated using the formulae ofLaspeyres, Paasche, and Fischer, and the unit and normalizedunit index formulae. By comparing the average values of theabove formulae with the true average index value, we are ableto obtain a quantitative indication of the errors in these formulaeunder various situations. The normalized unit and Fischer indexformulae were found to be consistently accurate for all ourtests, whereas the formulae of Laspeyres and Paasche were foundto have very poor accuracy in many of our tests. The unit indexwas also consistently accurate in all our tests but its useis appropriate only for a single item, thus making this formulasuitable for measuring wages.     

New Cartesian grid methods for interface problems using the finite element formulation

Summary New finite element methods based on Cartesian triangulations are presented for two dimensional elliptic interface problems involving discontinuities in the coefficients. The triangulations in these methods do not need to fit the interfaces. The basis functions in these methods are constructed to satisfy the interface jump conditions either exactly or approximately. Both non-conforming and conforming finite element spaces are considered. Corresponding interpolation functions are proved to be second order accurate in the maximum norm. The conforming finite element method has been shown to be convergent. With Cartesian triangulations, these new methods can be used as finite difference methods. Numerical examples are provided to support the methods and the theoretical analysis. Mathematics Subject Classification (2000):65L10, 65L60, 65L70In this research, Zhilin Li is supported in part by USA ARO grants, 39676-MA and 43751-MA, USA NSF grants DMS-0073403 and DMS-0201094; USA North Carolina State University FR&PD grant; Tao Lin is supported in part a USA NSF grant DMS-97-04621. Special thanks to Thomas Hou for his participation and contribution to this project. We are also grateful to R. LeVeque, K. Bube, and T. Chan for useful discussions.     

Numerical analysis of a Stokes interface problem based on formulation using the characteristic function

Numerical analysis of a model Stokes interface problem with the homogeneous Dirichlet boundary condition is considered. The interface condition is interpreted as an additional singular force field to the Stokes equations using the characteristic function. The finite element method is applied after introducing a regularization of the singular source term. Consequently, the error is divided into the regularization and discretization parts which are studied separately. As a result, error estimates of order h^1/2 in H¹ × L² norm for the velocity and pressure, and of order h in L² norm for the velocity are derived. Those theoretical results are also verified by numerical examples.     

Numerical grid generator based on Moser's deformation method

A deformation method of grid generation has been recently proposed. The method is based on Moser's deformation scheme in his study of the volume elements of a Riemannian manifold. In this paper, this deformation method is improved so that the grid adaption takes place according to the physical coordinates. Some numerical results are included to demonstrate the method. © 1994 John Wiley & Sons, Inc.     

Numerical model for the shallow water equations on a curvilinear grid with the preservation of the Bernoulli integral

Methodological aspects concerning the construction of a two-dimensional numerical model for reservoir flows based on the shallow water equations are considered. A numerical scheme is constructed by applying the control volume method on staggered grids in combination with the Bernoulli integral, which is used to interpolate the desired fields inside a grid cell. The implementation of the method yields a monotone numerical scheme. The results of numerical integration are compared with the exact solution.     

Numerical aspects in the SGBEM solution of softening cohesive interface problems

Some numerical aspects of a new symmetric Galerkin boundary integral formulation presented in [A. Salvadori, A symmetric boundary integral formulation for cohesive interface problems, Comput. Mech. 32 (2003) 381–391] connected with a like arc-length technique for softening cohesive interface problems introduced in [F. Freddi, Cohesive interface analysis via boundary integral equations, Ph.D. thesis, University of Bologna, Italy, 2004] are here investigated. Further, if the problem is invariant with respect to a finite group of congruences of the Euclidean space , we propose to reduce the computational cost of the symmetric Galerkin boundary element method (SGBEM) matrix evaluation and linear system solution using suitable restriction matrices strictly related to a system of irreducible matrix representation of . Some numerical simulations are presented and discussed.     

The nested Sinkhorn divergence to learn the nested distance

Computational Management Science - The nested distance builds on the Wasserstein distance to quantify the difference of stochastic processes, including also the evolution of information modelled by...     

Numerical interface curves for some nonlinear diffusion equations

An initial value problem for the generalized Kolmogorov-Petrowsky-Piscunov (nonlinear degenerate reaction-diffusion) equation is studied numerically by the help of a slightly modified finite difference scheme of Douglas-Yanenko-Mimura type. If the initial function has compact support, the solution also will have compact support and an interface appears between the domains where the solution is positive and where it is zero. We present some examples for different parameter values where the numerical solution as well as numerical interfaces behave according to the analytical theory. This revised version was published online in June 2006 with corrections to the Cover Date.     

The analysis of a nested dissection algorithm

Summary Nested dissection is an algorithm invented by Alan George for preserving sparsity in Gaussian elimination on symmetric positive definite matrices. Nested dissection can be viewed as a recursive divide-and-conquer algorithm on an undirected graph; it usesseparators in the graph, which are small sets of vertices whose removal divides the graph approximately in half. George and Liu gave an implementation of nested dissection that used a heuristic to find separators. Lipton and Tarjan gave an algorithm to findn ^1/2-separators in planar graphs and two-dimensional finite element graphs, and Lipton, Rose, and Tarjan used these separators in a modified version of nested dissection, guaranteeing bounds ofO (n logn) on fill andO(n ^3/2) on operation count. We analyze the combination of the original George-Liu nested dissection algorithm and the Lipton-Tarjan planar separator algorithm. This combination is interesting because it is easier to implement than the Lipton-Rose-Tarjan version, especially in the framework of existïng sparse matrix software. Using some topological graph theory, we proveO(n logn) fill andO(n ^3/2) operation count bounds for planar graphs, twodimensional finite element graphs, graphs of bounded genus, and graphs of bounded degree withn ^1/2-separators. For planar and finite element graphs, the leading constant factor is smaller than that in the Lipton-Rose-Tarjan analysis. We also construct a class of graphs withn ^1/2-separators for which our algorithm does not achieve anO(n logn) bound on fill.The work of this author was supported in part by the Hertz Foundation under a graduate fellowship and by the National Science Foundation under Grant MCS 82-02948The work of this author was supported in part by the National Science Foundation under Grant MCS 78-26858 and by the Office of Naval Research under Contract N00014-76-C-0688     

Numerical experiments in finite element grid optimization by direct energy search

A series of numerical experiments is conducted to assess the feasibility and practical value of finite element grid optimization based on direct minimization of the total potential energy of the discrete model with respect to the node locations. An implementation relying upon non-linear programming techniques is found to be numerically reliable and to lead to improved grids in accord with engineering intuition. This rigorous approach is hampered, however, by the excessive computational effort required by the energy minimization process. A combination of related techniques is therefore proposed to make dynamic node distribution a useful tool within the framework of large-scale finite element analysis. The combined strategy involves use of substructuring methods, application of a local energy-balancing optimality criterion for fast node distribution, and automatic refinement of previously-improved coarse grids.     

How to tangle with a nested radical

Supported by NSF grant DMS-8807202.     

GMRESR: a family of nested GMRES methods

Recently Eirola and Nevanlinna have proposed an iterative solution method for unsymmetric linear systems, in which the preconditioner is updated from step to step. Following their ideas we suggest variants of GMRES, in which a preconditioner is constructed per iteration step by a suitable approximation process, e.g., by GMRES itself. Our numerical experiments indicate that this may lead to considerable savings in CPU-time and memory requirements in typical CFD applications.