A self-similar premixed turbulent flame model

This paper is devoted to premixed combustion modelling in turbulent flow. First, we derive a model for the turbulent flame velocity based on the observed self-similarity of the turbulent flame. The model uses the local flame brush width as a fundamental parameter and, therefore, we show how it can be retrieved for numerical implementation. The diffusive property of the brush width is treated in such a way as to theoretically let the brush have a clearly defined boundary propagating at finite velocity. The model, implemented in Star-CD CFD software through user programming, is then numerically tested on three configurations for which another model, the Turbulent Flame Closure model, is known to give very good agreement. Some effects of numerics are commented and results for both models are compared. While based on very different approaches the two models lead to substantially similar results. In this way, we have shown that the local brush width can effectively be used, giving an additional degree of freedom for premixed turbulent combustion modelling.     

A free boundary problem arising in flame propagation

We study a free boundary problem describing the propagation of laminar flames. The problem arises as the limit of a singular perturbation problem. We introduce the notion of viscosity solutions for the problem to show the maximum principle-type property of the solutions. Using this property we show the uniform convergence of the approximating solutions and the uniqueness of the viscosity solution under several geometric conditions on the initial data.     

A sequentially optimal algorithm for numerical integration

For the class of functions of one variable, satisfying the Lipschitz condition with a fixed constant, an optimal passive algorithm for numerical integration (an optimal quadrature formula) has been found by Nikol'skii. In this paper, a sequentially optimal algorithm is constructed; i.e., the algorithm on each step makes use in an optimal way of all relevant information which was accumulated on previous steps. Using the algorithm, it is necessary to solve an integer program at each step. An effective algorithm for solving these problems is given.The author is indebted to Professor S. E. Dreyfus, Department of Industrial Engineering and Operations Research, University of California, Berkeley, California, for his helpful attention to this paper.     

A bisection algorithm for the numerical Mountain Pass

We propose a constructive proof for the Ambrosetti-Rabinowitz Mountain Pass Theorem providing an algorithm, based on a bisection method, for its implementation. The efficiency of our algorithm, particularly suitable for problems in high dimensions, consists in the low number of flow lines to be computed for its convergence; for this reason it improves the one currently used and proposed by Y.S. Choi and P.J. McKenna in [3]. Susanna Terracini: This work is partially supported by M.I.U.R. project “Metodi Variazionali ed Equazioni Differenziali Nonlineari”.     

A numerical algorithm for computing tsunami wave amplitudes

A numerical multistep algorithm for computing tsunami wave front amplitudes is proposed. The first step consists in solving an appropriate eikonal equation. The eikonal equation is solved with Godunov’s approach and a bicharacteristic method. A qualitative comparison of the two methods is done. A change of variables is made with the eikonal equation solution at the second step. At the last step, using an expansion of the fundamental solution to the shallow water equations in the new variables, we obtain a Cauchy problem of lesser dimension for the leading edge wave amplitude. The results of numerical experiments are presented.     

A numerical scheme for the one‐dimensional pressureless gases system

In this work, we investigate the numerical approximation of the one‐dimensional pressureless gases system. After briefly recalling the mathematical framework of the duality solutions introduced by Bouchut and James (Comm. Partial Differential Equations 24 (1999), 2173–2189), we point out that the upwind scheme for density and momentum does not satisfy the one‐sided Lipschitz (OSL) condition on the expansion rate required for the duality solutions. Then we build a diffusive scheme which allows the OSL condition to be recovered by following the strategy described by Boudin (SIAM J Math Anal 32 (2000), 172–193) for the continuous model. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

A numerical algorithm for the nonlinear Timoshenko beam system

An initial boundary value problem is considered for the dynamic beam system Its solution is found by means of an algorithm, the constituent parts of which are the finite element method, the implicit symmetric difference scheme used to approximate the solution with respect to the spatial and time variables, and also a Picard type iteration process for solving the system of nonlinear equations obtained by discretization. Errors of three parts of the algorithm are estimated and, as a result, its total error estimate is obtained. A numerical example is solved.     

A numerical algorithm for the nonlinear Kirchhoff string equation

The initial boundary value problem is considered for the dynamic string equation . Its solution is found by means of an algorithm, the constituent parts of which are the Galerkin method, the modified Crank-Nicolson difference scheme used to perform approximation with respect to spatial and time variables, and also a Picard type iteration process for solving the system of nonlinear equations obtained by discretization. Errors of the three parts of the algorithm are estimated and, as a result, its total error estimate is obtained.     

A numerical algorithm for nonlinear multi-point boundary value problems

In this paper, an algorithm is presented for solving second-order nonlinear multi-point boundary value problems (BVPs). The method is based on an iterative technique and the reproducing kernel method (RKM). Two numerical examples are provided to show the reliability and efficiency of the present method.     

A numerical algorithm for singular optimal LQ control systems

A numerical algorithm to obtain the consistent conditions satisfied by singular arcs for singular linear–quadratic optimal control problems is presented. The algorithm is based on the Presymplectic Constraint Algorithm (PCA) by Gotay-Nester (Gotay et al., J Math Phys 19:2388–2399, 1978; Volckaert and Aeyels 1999) that allows to solve presymplectic Hamiltonian systems and that provides a geometrical framework to the Dirac-Bergmann theory of constraints for singular Lagrangian systems (Dirac, Can J Math 2:129–148, 1950). The numerical implementation of the algorithm is based on the singular value decomposition that, on each step, allows to construct a semi-explicit system. Several examples and experiments are discussed, among them a family of arbitrary large singular LQ systems with index 2 and a family of examples of arbitrary large index, all of them exhibiting stable behaviour. Research partially supported by MEC grant MTM2004-07090-C03-03. SIMUMAT-CM, UC3M-MTM-05-028 and CCG06-UC3M/ESP-0850.     

A numerical algorithm for the solution of Signorini problems

In this paper we propose a new iterative algorithm for the solution of a certain class of Signorini problems. Such problems arise in the modelling of a variety of physical phenomena and usually involve the determination of an unknown free boundary. Here we describe a way of locating the free boundary directly and provide a proof that the algorithm converges when used with analytic methods. The advantage of this algorithm is that it can be used in conjunction with any numerical method with minimal development of extra code. We demonstrate its application with the boundary element method to some physical problems in both two and three dimensions.     

A fast algorithm for numerical solutions to Fortet's equation

A fast algorithm for computation of default times of multiple firms in a structural model is presented. The algorithm uses a multivariate extension of the Fortet's equation and the structure of Toeplitz matrices to significantly improve the computation time. In a financial market consisting of M1 firms and N discretization points in every dimension the algorithm uses O(nlogn·M·M!·N^M(M-1)/2) operations, where n is the number of discretization points in the time domain. The algorithm is applied to firm survival probability computation and zero coupon bond pricing.     

A staggered-grid numerical algorithm for the extended Boussinesq equations

A second-order accurate numerical scheme is developed to solve Nwogu’s extended Boussinesq equations. A staggered-grid system is introduced with the first-order spatial derivatives being discretized by the fourth-order accurate finite-difference scheme. For the time derivatives, the fourth-order accurate Adams predictor–corrector method is used. The numerical method is validated against available analytical solutions, other numerical results of Navier–Stokes equations, and experimental data for both 1D and 2D nonlinear wave transformation problems. It is shown that the new algorithm has very good conservative characteristics for mass calculation. As a result, the model can provide accurate and stable results for long-term simulation. The model has proven to be a useful modeling tool for a wide range of water wave problems.     

A scattering‐based algorithm for wave propagation in one dimension

We present an explicit numerical scheme to solve the variable coefficient wave equation in one space dimension with minimal restrictions on the coefficient and initial data.     

Diffusion-controlled smoulder propagation in a semi-infinite solid: A numerical solution

The propagation of a thin two-dimensional smouldering reactionfront (SRF) parallel to a plane surface in a semi-infinite solidfuel is considered. Combustion is controlled by oxidizer diffusionthrough the porous burnt material behind the SRF. The modelgives rise to a free boundary problem as the position of theSRF is to be determined as part of the solution. The Burke-Schumanncondition of the oxidizer concentration vanishing on the SRFis adopted. Different forms of the plane surface boundary conditionare considered. The governing linear p.d.e., which holds inan unknown domain, is converted into a nonlinear p.d.e. overa fixed domain by Alt's method, in which the free boundary conditionsare satisfied by a jump condition. The position of the SRF andthe oxidizer concentration distribution behind it are determined.It is shown that the shape of the SRF is asymptotically parabolic,consistent with analytic and experimental results. Contoursof equal oxygen concentration are presented.     

A fast numerical algorithm for the integration of rational functions

A new iterative method for high-precision numerical integration of rational functions on the real line is presented. The algorithm transforms the rational integrand into a new rational function preserving the integral on the line. The coefficients of the new function are explicit polynomials in the original ones. These transformations depend on the degree of the input and the desired order of the method. Both parameters are arbitrary. The formulas can be precomputed. Iteration yields an approximation of the desired integral with mth order convergence. Examples illustrating the automatic generation of these formulas and the numerical behaviour of this method are given.     

A Bayesian model and numerical algorithm for CBM availability maximization

In this paper, we consider an availability maximization problem for a partially observable system subject to random failure. System deterioration is described by a hidden, continuous-time homogeneous Markov process. While the system is operational, multivariate observations that are stochastically related to the system state are sampled through condition monitoring at discrete time points. The objective is to design an optimal multivariate Bayesian control chart that maximizes the long-run expected average availability per unit time. We have developed an efficient computational algorithm in the semi-Markov decision process (SMDP) framework and showed that the availability maximization problem is equivalent to solving a parameterized system of linear equations. A numerical example is presented to illustrate the effectiveness of our approach, and a comparison with the traditional age-based replacement policy is also provided.