Semi‐strong dynamic style analysis with time‐varying selectivity measurement: Applications to Brazilian exchange‐rate funds

This paper deals with restricted linear state space models for dynamic style analysis with time‐varying selectivity measurement. Implementation and interpretation of the models are pertinently discussed. Empirical contributions lie on the understanding of how managers of Brazilian US Dollar/Real exchange‐rate funds behaved along 2001 and 2002, a period of some political turbulence especially due to the 2002 Brazilian presidential election. Copyright © 2007 John Wiley & Sons, Ltd.     

Stability and convergence of optimum spectral non‐linear Galerkin methods

Our objective in this article is to present some numerical schemes for the approximation of the 2‐D Navier–Stokes equations with periodic boundary conditions, and to study the stability and convergence of the schemes. Spatial discretization can be performed by either the spectral Galerkin method or the optimum spectral non‐linear Galerkin method; time discretization is done by the Euler scheme and a two‐step scheme. Our results show that under the same convergence rate the optimum spectral non‐linear Galerkin method is superior to the usual Galerkin methods. Finally, numerical example is provided and supports our results. Copyright © 2001 John Wiley & Sons, Ltd.     

Numerical analysis and computational testing of a high accuracy Leray‐deconvolution model of turbulence

We study a computationally attractive algorithm (based on an extrapolated Crank‐Nicolson method) for a recently proposed family of high accuracy turbulence models, the Leray‐deconvolution family. First we prove convergence of the algorithm to the solution of the Navier‐Stokes equations and delineate its (optimal) accuracy. Numerical experiments are presented which confirm the convergence theory. Our 3d experiments also give a careful comparison of various related approaches. They show the combination of the Leray‐deconvolution regularization with the extrapolated Crank‐Nicolson method can be more accurate at higher Reynolds number that the classical extrapolated trapezoidal method of Baker (Report, Harvard University, 1976). We also show the higher order Leray‐deconvolution models (e.g. N = 1,2,3) have greater accuracy than the N = 0 case of the Leray‐α model. Numerical experiments for the 2d step problem are also successfully investigated. Around the critical Reynolds number, the low order models inhibit vortex shedding behind the step. The higher order models, correctly, do not. To estimate the complexity of using Leray‐deconvolution models for turbulent flow simulations we estimate the models' microscale.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

An enhanced‐physics‐based scheme for the NS‐α turbulence model

We study a new enhanced‐physics‐based numerical scheme for the NS‐alpha turbulence model that conserves both energy and helicity. Although most turbulence models (in the continuous case) conserve only energy, NS‐alpha is one of only a very few that also conserve helicity. This is one reason why it is becoming accepted as the most physically accurate turbulence model. However, no numerical scheme for NS‐alpha, until now, conserved both energy and helicity, and thus the advantage gained in physical accuracy by modeling with NS‐alpha could be lost in a computation. This report presents a finite element numerical scheme, and gives a rigorous analysis of its conservation properties, stability, solution existence, and convergence. A key feature of the analysis is the identification of the discrete energy and energy dissipation norms, and proofs that these norms are equivalent (provided a careful choice of filtering radius) in the discrete space to the usual energy and energy dissipation norms. Numerical experiments are given to demonstrate the effectiveness of the scheme over usual (helicity‐ignoring) schemes. A generalization of this scheme to a family of high‐order NS‐alpha‐deconvolution models, which combine the attractive physical properties of NS‐alpha with the high accuracy gained by combining α‐filtering with van Cittert approximate deconvolution. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Parallel characteristic finite element method for time‐dependent convection–diffusion problem

Based on the overlapping‐domain decomposition and parallel subspace correction method, a new parallel algorithm is established for solving time‐dependent convection–diffusion problem with characteristic finite element scheme. The algorithm is fully parallel. We analyze the convergence of this algorithm, and study the dependence of the convergent rate on the spacial mesh size, time increment, iteration times and sub‐domains overlapping degree. Both theoretical analysis and numerical results suggest that only one or two iterations are needed to reach to optimal accuracy at each time step. Copyright © 2010 John Wiley & Sons, Ltd.     

Parallel split least‐squares mixed finite element method for parabolic problem

Based on overlapping domain decomposition, a new class of parallel split least‐squares (PSLS) mixed finite element methods is presented for solving parabolic problem. The algorithm is fully parallel. In the overlapping domains, the partition of unity is applied to distribute the corrections reasonably, which makes that the new method only needs one or two iteration steps to reach given accuracy at each time step while the classical Schwarz alternating methods need many iteration steps. The dependence of the convergence rate on the spacial mesh size, time increment, iteration times, and subdomains overlapping degree is analyzed. Some numerical results are reported to confirm the theoretical analysis.     

The element‐free Galerkin method for a quasistatic contact problem with the Tresca friction in elastic materials

This paper is proposed for the error estimates of the element‐free Galerkin method for a quasistatic contact problem with the Tresca friction. The penalty method is used to impose the clamped boundary conditions. The duality algorithm is also given to deal with the non‐differentiable term in the quasistatic contact problem with the Tresca friction. The error estimates indicate that the convergence order is dependent on the nodal spacing, the time step, the largest degree of basis functions in the moving least‐squares approximation, and the penalty factor. Numerical examples demonstrate the effectiveness of the element‐free Galerkin method and verify the theoretical analysis.     

One‐level and two‐level finite element methods for the Boussinesq problem

Previous works on the convergence of numerical methods for the Boussinesq problem were conducted, while the optimal L²‐norm error estimates for the velocity and temperature are still lacked. In this paper, the backward Euler scheme is used to discrete the time terms, standard Galerkin finite element method is adopted to approximate the variables. The MINI element is used to approximate the velocity and pressure, the temperature field is simulated by the linear polynomial. Under some restriction on the time step, we firstly present the optimal L² error estimates of approximate solutions. Secondly, two‐level method based on Stokes iteration for the Boussinesq problem is developed and the corresponding convergence results are presented. By this method, the original problem is decoupled into two small linear subproblems. Compared with the standard Galerkin method, the two‐level method not only keeps good accuracy but also saves a lot of computational cost. Finally, some numerical examples are provided to support the established theoretical analysis.     

Analytical and numerical solutions of a two‐dimensional multi‐term time‐fractional Oldroyd‐B model

In this paper, we consider a two‐dimensional multi‐term time‐fractional Oldroyd‐B equation on a rectangular domain. Its analytical solution is obtained by the method of separation of variables. We employ the finite difference method with a discretization of the Caputo time‐fractional derivative to obtain an implicit difference approximation for the equation. Stability and convergence of the approximation scheme are established in the L_∞ ‐norm. Two examples are given to illustrate the theoretical analysis and analytical solution. The results indicate that the present numerical method is effective for this general two‐dimensional multi‐term time‐fractional Oldroyd‐B model.     

A BGK‐penalization‐based asymptotic‐preserving scheme for the multispecies Boltzmann equation

An asymptotic‐preserving (AP) scheme is efficient in solving multiscale problems where kinetic and hydrodynamic regimes coexist. In this article, we extend the BGK‐penalization‐based AP scheme, originally introduced by Filbet and Jin for the single species Boltzmann equation (Filbet and Jin, J Comput Phys 229 (2010) 7625–7648), to its multispecies counterpart. For the multispecies Boltzmann equation, the new difficulties arise due to: (1) the breaking down of the conservation laws for each species and (2) different convergence rates to equilibria for different species in disparate masses systems. To resolve these issues, we find a suitable penalty function—the local Maxwellian that is based on the mean velocity and mean temperature and justify various asymptotic properties of this method. This AP scheme does not contain any nonlinear nonlocal implicit solver, yet it can capture the fluid dynamic limit with time step and mesh size independent of the Knudsen number. Numerical examples demonstrate the correct asymptotic‐behavior of the scheme. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

An explicit second order spectral element method for acoustic waves

The acoustic wave equation is here discretized by conforming spectral elements in space and by the second order leap-frog method in time. For simplicity, homogeneous boundary conditions are considered. A stability analysis of the resulting method is presented, providing an upper bound for the allowed time step that is proportional to the size of the elements and inversely proportional to the square of their polynomial degree. A convergence analysis is also presented, showing that the convergence error decreases when the time step or the size of the elements decrease or when the polynomial degree increases. Several numerical results illustrating these results are presented. Mathematics subject classifications (2000) 65M06, 65M70, 65M12 This work was supported by MIUR (PRIN 2001 “Metodi numerici avanzati per equazioni alle derivate parziali di interesse applicativo”).     

Convergence and error bound analysis for the space‐time CESE method

In this work, we study the convergence behavior of a recently developed space‐time conservation element and solution element method for solving conservation laws. In particular, we apply the method to a one‐dimensional time‐dependent convection‐diffusion equation possibly with high Peclet number. We prove that the scheme converges and we obtain an error bound. This method performs well even for strong convection dominance over diffusion with good long‐time accuracy. Numerical simulations are performed to verify the results. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 64–78, 2001     

Strong convergence of the split‐step θ‐method for stochastic age‐dependent population equations

In this paper, we constructed the split‐step θ (SSθ)‐method for stochastic age‐dependent population equations. The main aim of this paper is to investigate the convergence of the SS θ‐method for stochastic age‐dependent population equations. It is proved that the proposed method is convergent with strong order 1/2 under given conditions. Finally, an example is simulated to verify the results obtained from the theory, and comparative analysis with Euler method is given, the results show the higher accuracy of the SS θ‐method. Copyright © 2014 John Wiley & Sons, Ltd.     

A linearized Crank–Nicolson–Galerkin FEM for the time‐dependent Ginzburg–Landau equations under the temporal gauge

We propose a decoupled and linearized fully discrete finite element method (FEM) for the time‐dependent Ginzburg–Landau equations under the temporal gauge, where a Crank–Nicolson scheme is used for the time discretization. By carefully designing the time‐discretization scheme, we manage to prove the convergence rate , where τ is the time‐step size and r is the degree of the finite element space. Due to the degeneracy of the problem, the convergence rate in the spatial direction is one order lower than the optimal convergence rate of FEMs for parabolic equations. Numerical tests are provided to support our error analysis. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1279–1290, 2014     

Mesh‐independent convergence of the modified inexact Newton method for a second order non‐linear problem

In this paper, we consider an inexact Newton method applied to a second order non‐linear problem with higher order non‐linearities. We provide conditions under which the method has a mesh‐independent rate of convergence. To do this, we are required, first, to set up the problem on a scale of Hilbert spaces and second, to devise a special iterative technique which converges in a higher than first order Sobolev norm. We show that the linear (Jacobian) system solved in Newton's method can be replaced with one iterative step provided that the initial non‐linear iterate is accurate enough. The closeness criteria can be taken independent of the mesh size. Finally, the results of numerical experiments are given to support the theory. Published in 2005 by John Wiley & Sons, Ltd.     

A fourth‐order method for numerical integration of age‐ and size‐structured population models

In many applications of age‐ and size‐structured population models, there is an interest in obtaining good approximations of total population numbers rather than of their densities. Therefore, it is reasonable in such cases to solve numerically not the PDE model equations themselves, but rather their integral equivalents. For this purpose quadrature formulae are used in place of the integrals. Because quadratures can be designed with any order of accuracy, one can obtain numerical approximations of the solutions with very fast convergence. In this article, we present a general framework and a specific example of a fourth‐order method based on composite Newton‐Cotes quadratures for a size‐structured population model. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

The Crank‐Nicolson/Adams‐Bashforth scheme for the time‐dependent Navier‐Stokes equations with nonsmooth initial data

In this article, we study the stability and convergence of the Crank‐Nicolson/Adams‐Bashforth scheme for the two‐dimensional nonstationary Navier‐Stokes equations with a nonsmooth initial data. A finite element method is applied for the spatial approximation of the velocity and pressure. The time discretization is based on the implicit Crank‐Nicolson scheme for the linear terms and the explicit Adams‐Bashforth scheme for the nonlinear term. Moreover, we prove that the scheme is almost unconditionally stable for a nonsmooth initial data u₀ with div u₀ = 0, i.e., the time step τ satisfies: τ ≤ C₀ if u₀ ∈ H¹ ∩ L^∞; τ |log h| ≤ C₀ if u₀ ∈ H¹ for the mesh size h and some positive constant C₀. Finally, we obtain some error estimates for the discrete velocity and pressure under the above stability condition. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 28: 155‐187, 2012     

Semi‐implicit schemes for transient Navier–Stokes equations and eddy viscosity models

This study presents two computational schemes for the numerical approximation of solutions to eddy viscosity models as well as transient Navier–Stokes equations. The eddy viscosity model is one example of a class of Large Eddy Simulation models, which are used to simulate turbulent flow. The first approximation scheme is a first order single step method that treats the nonlinear term using a semi‐implicit discretization. The second scheme employs a two step approach that applies a Crank–Nicolson method for the nonlinear term while also retaining the semi‐implicit treatment used in the first scheme. A finite element approximation is used in the spatial discretization of the partial differential equations. The convergence analysis for both schemes is discussed in detail, and numerical results are given for two test problems one of which is the two dimensional flow around a cylinder. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Numerical solution to a one‐dimensional thermoplastic problem with unilateral constraint

In this article, we present a numerical simulation of one‐dimensional problem of quasi‐static contact with an elastic obstacle. A finite difference scheme is derived by the method of reduction of order on uniform meshes. The stability and convergence are proved. The convergence order is of O(τ² + h²), where τ and h are the time step size and the space step size, respectively. Some numerical examples demonstrate the theoretical results. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

An Eulerian‐Lagrangian substructuring domain decomposition method for unsteady‐state advection‐diffusion equations

We develop an Eulerian‐Lagrangian substructuring domain decomposition method for the solution of unsteady‐state advection‐diffusion transport equations. This method reduces to an Eulerian‐Lagrangian scheme within each subdomain and to a type of Dirichlet‐Neumann algorithm at subdomain interfaces. The method generates accurate and stable solutions that are free of artifacts even if large time‐steps are used in the simulation. Numerical experiments are presented to show the strong potential of the method. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:565–583, 2001