WENO schemes based on upwind and centred TVD fluxes

In this paper we propose to use second-order TVD fluxes, instead of first-order monotone fluxes, in the framework of finite-volume weighted essentially non-oscillatory (WENO) schemes. We call the new improved schemes the WENO-TVD schemes. They include both upwind and centred schemes on non-staggered meshes. Numerical results suggest that our schemes are superior to the original schemes used with first-order monotone fluxes. This is especially so for long time evolution problems containing both smooth and non-smooth features.     

Application of Second Order TVD and WENO Schemes in Internal Aerodynamics

We deal with the comparison of several finite volume TVD schemes and finite difference ENO schemes and we describe a second order finite volume WENO scheme which was developed for the case of general unstructured meshes. The proposed second order WENO reconstruction is much simpler than the original ENO scheme introduced in [Harten and Chakravarthy 1991]. Moreover, the proposed WENO method is very easily extendible for unstructured meshes in 3D. All above mentioned schemes are applied for the solution of 2D and 3D transonic flows in the turbines and channels and the numerical solution is compared to experimental results or to the results obtained by other authors.     

Efficient solutions for discrete Asian options

While in the literature most studies on pricing focus on continuous Asian options, in this paper we provide efficient solutions for both European and American discrete average price Asian options. The method used for deriving the approximation formula for European Asian options is based on the idea of Bouaziz et al. (J Bank Finance 18:823–839, 1994) and Taso et al. (J Futures Mark 23:487–516, 2003) in which the Taylor expansion is used to obtain the approximation formula for continuous average strike Asian options. By using the Taylor expansion to the second order, a simple and accurate solution can be obtained. The approximation formula for the European Asian option can further be used to enhance the efficiency of the pricing of the American Asian options when using the numerical method.     

An exact subexponential-time lattice algorithm for Asian options

Asian options are popular financial derivative securities. Unfortunately, no exact pricing formulas exist for their price under continuous-time models. Asian options can also be priced on the lattice, which is a discretized version of the continuous- time model. But only exponential-time algorithms exist if the options are priced on the lattice without approximations. Although efficient approximation methods are available, they lack accuracy guarantees in general. This paper proposes a novel lattice structure for pricing Asian options. The resulting pricing algorithm is exact (i.e., without approximations), converges to the value under the continuous-time model, and runs in subexponential time. This is the first exact, convergent lattice algorithm to break the long-standing exponential-time barrier. An early version of this paper appeared in the Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms, 2004. T.-S. Dai was supported in part by NSC grant 94-2213-E-033-024. Y.-D. Lyuu was supported in part by NSC grant 94-2213-E-002-088.     

Dispersion,group velocity,and multisymplectic discretizations

This paper examines the dispersive properties of multisymplectic discretizations of linear and nonlinear PDEs. We focus on a leapfrog in space and time scheme and the Preissman box scheme. We find that the numerical dispersion relations are monotonic and determine the relationship between the group velocities of the different numerical schemes. The group velocity dispersion is used to explain the qualitative differences in the numerical solutions obtained with the different schemes. Furthermore, the numerical dispersion relation is found to be relevant when determining the ability of the discretizations to resolve nonlinear dynamics.     

Fuzzy pricing of geometric Asian options and its algorithm

Owing to the fluctuations of the financial market, input data in the options pricing formula cannot be expected to be precise. This paper discusses the problem of pricing geometric Asian options under the fuzzy environment. We present the fuzzy price of the geometric Asian option under the assumption that the underlying stock price, the risk-free interest rate and the volatility are all fuzzy numbers. This assumption makes the financial investors to pick any geometric Asian option price with an acceptable belief degree. In order to obtain the belief degree, the interpolation search algorithm has been proposed. Some numerical examples are presented to illustrate the rationality and practicability of the model and the algorithm. Finally, an empirical study is performed based on the real data. The empirical study results indicate that the proposed fuzzy pricing model of geometric Asian option is a useful tool for modeling the imprecise problem in the real world.     

BDDC and FETI-DP preconditioners for spectral element discretizations

Two of the most recent and important nonoverlapping domain decomposition methods, the BDDC method (Balancing Domain Decomposition by Constraints) and the FETI-DP method (Dual-Primal Finite Element Tearing and Interconnecting) are here extended to spectral element discretizations of second-order elliptic problems. In spite of the more severe ill-conditioning of the spectral element discrete systems, compared with low-order finite elements and finite differences, these methods retain their good properties of scalability, quasi-optimality and independence on the discontinuities of the elliptic operator coefficients across subdomain interfaces.     

A meshless method for Asian style options pricing under the Merton jump-diffusion model

In this paper, we consider the partial integro-differential equation arising when a stock follows a Poisson distributed jump process, for the pricing of Asian options. We make use of the meshless radial basis functions with differential quadrature for approximating the spatial derivatives and demonstrate that the algorithm performs effectively well as compared to the commonly employed finite difference approximations. We also employ Strang splitting with the exponential time integration technique to improve temporal efficiency. Throughout the numerical experiments covered in the paper, we show how the proposed scheme can be efficiently employed for the pricing of American style Asian options under both the Black–Scholes and the Merton jump-diffusion models.     

Explicit and Implicit TVD Schemes for Conservation Laws with Caputo Derivatives

In this paper, we investigate numerical approximations of the scalar conservation law with the Caputo derivative, which introduces the memory effect. We construct the first order and the second order explicit upwind schemes for such equations, which are shown to be conditionally \(\ell ^1\) contracting and TVD. However, the Caputo derivative leads to the modified CFL-type stability condition, \( (\Delta t)^{\alpha } = O(\Delta x)\), where \(\alpha \in (0,1]\) is the fractional exponent in the derivative. When \(\alpha \) is small, such strong constraint makes the numerical implementation extremely impractical. We have then proposed the implicit upwind scheme to overcome this issue, which is proved to be unconditionally \(\ell ^1\) contracting and TVD. Various numerical tests are presented to validate the properties of the methods and provide more numerical evidence in interpreting the memory effect in conservation laws.     

A scaled thickness conditioning for solid- and solid-shell discretizations of thin-walled structures

The numerical discretization of thin shell structures yields ill-conditioned stiffness matrices due to an inherent large eigenvalue spectrum. Finite element parametrization that depends on shell thickness, like relative displacement shells, solid shells and other solid finite elements even add to the ill-conditioning by introducing high eigenmodes.To overcome this numerical issue we present a scaled thickness conditioning (STC) approach, a mechanically motivated preconditioner for thin-walled structures discretized with continuum based element formulations. The proposed approach is motivated by the scaled director conditioning (SDC) method for relative displacement shell elements. In contrast to SDC, the novel STC approach yields a preconditioner for the effective linear system. It is applicable independently of element technology employed, coupling to other physical fields, boundary conditions applied and additional algebraic constraints and can be easily extended to multilayer shell formulations.The effect of the proposed preconditioner on the conditioning of the effective stiffness matrix and its eigenvalue spectrum is studied. It is shown that the condition number of the modified system becomes almost independent from the aspect ratio of the employed elements. The improved conditioning has a positive influence on the convergence behavior of iterative linear solvers. In particular, in combination with algebraic multigrid preconditioners the number of iterations could be decreased by more than 85% for some examples and the computation time could be reduced by about 60%.     

Truncation error,dissipation and dispersion terms of fifth order WENO and of WCS for 1D conservation law

In this paper, a derivation and a comparison of the truncation errors and the dissipation and dispersion terms of the fifth-order weighted essentially non-oscillatory scheme and of the weighted compact scheme are presented. The schemes are compared for smooth functions (by Fourier analysis), and near a shock.     

On different flux splittings and flux functions in WENO schemes for balance laws

In this paper we focus our attention on obtaining well-balanced schemes for balance laws by using Marquina’s flux in combination with the finite difference and finite volume WENO schemes. We consider also the Rusanov flux splitting and the HLL approximate Riemann solver. In particular, for the presented numerical schemes we develop corresponding discretizations of the source term, based on the idea of balancing with the flux gradient. When applied to the open-channel flow and to the shallow water equations, we obtain the finite difference WENO scheme with Marquina’s flux splitting, which satisfies the approximate conservation property, and also the balanced finite volume WENO scheme with Marquina’s solver satisfying the exact conservation property. Finally, we also present an improvement of the balanced finite difference WENO scheme with the Rusanov (locally Lax–Friedrichs) flux splitting, we previously developed in [Vuković S, Sopta L. ENO and WENO schemes with the exact conservation property for one-dimensional shallow water equations. J Comput Phys 2002;179:593–621].     

A mesh-independence principle for nonlinear operator equations and their discretizations under mild differentiability conditions

In this note we extend the validity of the mesh-independence principle for nonlinear operator equations and their discretizations to include operators whose derivatives are only Hölder continuous.     

Hardware options,evaluation metrics,and a design sequence for interactive information systems

Interactive information systems must satisfy a wide variety of users, serve a broad range of tasks, and be suited to diverse hardware environments. This paper concentrates on three aspects of interactive information systems design: hardware options, evaluation metrics, and a possible design sequence. Rigorous pilot studies are emphasized, and supporting experimental evidence is offered.     

Cell-centered discontinuous Galerkin discretizations for two-dimensional scalar conservation laws on unstructured grids and for one-dimensional Lagrangian hydrodynamics

We present cell-centered discontinuous Galerkin discretizations for two-dimensional scalar conservation laws on unstructured grids and also for the one-dimensional Lagrangian hydrodynamics up to third-order. We also demonstrate that a proper choice of the numerical fluxes allows to enforce stability properties of our discretizations.     

Solving Initial Value Problems for Ordinary Differential Equations by two approaches: BDF and piecewise-linearized methods

Many scientific and engineering problems are described using Ordinary Differential Equations (ODEs), where the analytic solution is unknown. Much research has been done by the scientific community on developing numerical methods which can provide an approximate solution of the original ODE. In this work, two approaches have been considered based on BDF and Piecewise-linearized Methods. The approach based on BDF methods uses a Chord-Shamanskii iteration for computing the nonlinear system which is obtained when the BDF schema is used. Two approaches based on piecewise-linearized methods have also been considered. These approaches are based on a theorem proved in this paper which allows to compute the approximate solution at each time step by means of a block-oriented method based on diagonal Padé approximations. The difference between these implementations is in using or not using the scale and squaring technique.Five algorithms based on these approaches have been developed. MATLAB and Fortran versions of the above algorithms have been developed, comparing both precision and computational costs. BLAS and LAPACK libraries have been used in Fortran implementations. In order to compare in equality of conditions all implementations, algorithms with fixed step have been considered. Four of the five case studies analyzed come from biology and chemical kinetics stiff problems. Experimental results show the advantages of the proposed algorithms, especially when they are integrating stiff problems.     

WENO Schemes and Their Application as Limiters for RKDG Methods Based on Trigonometric Approximation Spaces

In this paper, we present a class of finite volume trigonometric weighted essentially non-oscillatory (TWENO) schemes and use them as limiters for Runge-Kutta discontinuous Galerkin (RKDG) methods based on trigonometric polynomial spaces to solve hyperbolic conservation laws and highly oscillatory problems. As usual, the goal is to obtain a robust and high order limiting procedure for such a RKDG method to simultaneously achieve uniformly high order accuracy in smooth regions and sharp, non-oscillatory shock transitions. The major advantage of schemes which are based on trigonometric polynomial spaces is that they can simulate the wave-like and highly oscillatory cases better than the ones based on algebraic polynomial spaces. We provide numerical results in one and two dimensions to illustrate the behavior of these procedures in such cases. Even though we do not utilize optimal parameters for the trigonometric polynomial spaces, we do observe that the numerical results obtained by the schemes based on such spaces are better than or similar to those based on algebraic polynomial spaces.     

On options for interdisciplinary analysis and design optimization

The interdisciplinary optimization of engineering systems is discussed from the standpoint of the computational alternatives available to the designer. The analysis of such systems typically requires the solution of coupled systems of nonlinear algebraic equations. The solution procedure is necessarily iterative in nature. It is shown that the system can be solved by fixed point iteration, by Newton's method, or by a combination of the two. However, the need for sensitivity analysis may affect the choice of analysis solution method. Similarly, the optimization of the system can be formulated in several ways that are discussed in the paper. It is shown that the effect of the topology of the interaction between disciplines is a key factor in the choice of analysis, sensitivity and optimization methods. Several examples are presented to illustrate the discussion.     

Wireless Authentication options for up and down the Stack

This is a series dedicated to the issues of protecting wireless networks. The column deals with new threats, standards, products, auditing of wireless networks and secure network architectures. Allowing authorized users onto your network and keeping attackers out is a cornerstone of network security. With wireless, the issue of authentication is even more critical. Your users and infrastructure are at heightened risk because attackers have complete physical access to the network medium.