Convergence of the Euler–Maruyama method for stochastic differential equations with Markovian switching

Stochastic differential equations with Markovian switching (SDEwMSs), one of the important classes of hybrid systems, have been used to model many physical systems that are subject to frequent unpredictable structural changes. The research in this area has been both theoretical and applied. Most of SDEwMSs do not have explicit solutions so it is important to have numerical solutions. It is surprising that there are not any numerical methods established for SDEwMSs yet, although the numerical methods for stochastic differential equations (SDEs) have been well studied. The main aim of this paper is to develop a numerical scheme for SDEwMSs and estimate the error between the numerical and exact solutions. This is the first paper in this direction and the emphasis lies on the error analysis.     

Parallel iteratively regularized Gauss–Newton method for systems of nonlinear ill-posed equations

We propose a parallel version of the iteratively regularized Gauss–Newton method for solving a system of ill-posed equations. Under certain widely used assumptions, the convergence rate of the parallel method is established. Numerical experiments show that the parallel iteratively regularized Gauss–Newton method is computationally convenient for dealing with underdetermined systems of nonlinear equations on parallel computers, especially when the number of unknowns is much larger than that of equations.     

Jacobi wavelet collocation method for the modified Camassa–Holm and Degasperis–Procesi equations

Matrices representations of integrations of wavelets have a major role to obtain approximate solutions of integral, differential and integro-differential equations. In the present work, operational matrix representation of rth integration of Jacobi wavelets is introduced and to find these operational matrices, all details of the processes are demonstrated for the first time. Error analysis of offered method is also investigated in present study. In the planned method, approximate solutions are constructed with the truncated Jacobi wavelets series. Approximate solutions of the modified Camassa–Holm equation and Degasperis–Procesi equation linearized using quasilinearization technique are obtained by presented method. Applicability and accuracy of presented method is demonstrated by examples. The proposed method is also convergent even when a minor number of grid points. The numerical results obtained by offered technique are compatible with those in the literature.

     

On a collocation B-spline method for the solution of the Navier–Stokes equations

A novel B-spline collocation method for the solution of the incompressible Navier–Stokes equations is presented. The discretization employs B-splines of maximum continuity, yielding schemes with high-resolution power. The Navier–Stokes equations are solved by using a fractional step method, where the projection step is considered as a Div–Grad problem, so that no pressure boundary conditions need to be prescribed. Pressure oscillations are prevented by introducing compatible B-spline bases for the velocity and pressure, yielding efficient schemes of arbitrary order of accuracy. The method is applied to two-dimensional benchmark flows, and mass lumping techniques for cost-effective computation of unsteady problems are discussed.     

A multilevel hybrid Newton–Krylov–Schwarz method for the Bidomain model of electrocardiology

A multilevel hybrid Newton–Krylov–Schwarz (NKS) method is constructed and studied numerically for implicit time discretizations of the Bidomain reaction–diffusion system in three dimensions. This model describes the bioelectrical activity of the heart by coupling two degenerate parabolic equations with a stiff system of ordinary differential equations. The NKS Bidomain solver employs an outer inexact Newton iteration to solve the nonlinear finite element system originating at each time step of the implicit discretization. The Jacobian update during the Newton iteration is solved by a Krylov method employing a multilevel hybrid overlapping Schwarz preconditioner, additive within the levels and multiplicative among the levels. Several parallel tests on Linux clusters are performed, showing that the convergence of the method is independent of the number of subdomains (scalability), the discretization parameters and the number of levels (optimality).     

A rigorous proof for the equivalence of the projective Newton–Euler equations and the Lagrange equations of second kind for spatial rigid multibody systems

Multibody System Dynamics - It is well known that the projective Newton–Euler equation and the Lagrange equation of second kind lead to the same result when deriving the dynamical equations...     

Discretization of the Navier–Stokes equations on non-smooth grids using auxiliary points

The colocated scheme for the incompressible Navier–Stokes equations is improved on structured non-Cartesian grids. The method relies on a finite volume discretization and on the use of auxiliary points to locally approximate gradients following a two-point discretization. Enhanced accuracy is demonstrated for two-dimensional cases on strongly distorted meshes by computing Poiseuille flow and a flow in a differentially heated cavity. Received: 23 February 1999 / Accepted: 17 June 1999     

Convergence analysis of the Neumann–Neumann waveform relaxation method for time-fractional RC circuits

The classical waveform relaxation (WR) methods rely on decoupling the large-scale ODEs system into small-scale subsystems and then solving these subsystems in a Jacobi or Gauss–Seidel pattern. However, in general it is hard to find a clever partition and for strongly coupled systems the classical WR methods usually converge slowly and non-uniformly. On the contrary, the WR methods of longitudinal type, such as the Robin-WR method and the Neumann–Neumann waveform relaxation (NN-WR) method, possess the advantages of simple partitioning procedure and uniform convergence rate. The Robin-WR method has been extensively studied in the past few years, while the NN-WR method is just proposed very recently and does not get much attention. It was shown in our previous work that the NN-WR method converges much faster than the Robin-WR method, provided the involved parameter, namely β, is chosen properly. In this paper, we perform a convergence analysis of the NN-WR method for time-fractional RC circuits, with special attention to the optimization of the parameter β. For time-fractional PDEs, this work corresponds to the study of the NN-WR method at the semi-discrete level. We present a detailed numerical test of this method, with respect to convergence rate, CPU time and asymptotic dependence on the problem/discretization parameters, in the case of two- and multi-subcircuits.     

Note on helicity balance of the Galerkin method for the 3D Navier–Stokes equations

We study the helicity balance of the Galerkin method for the 3D Navier–Stokes equations, and show that although it does not appear to correctly balance helicity in the usual sense, it instead admits a slightly altered helicity balance that matches that of the underlying physics, up to boundary conditions.     

Relaxation method for unsteady convection–diffusion equations

We propose and implement a relaxation method for solving unsteady linear and nonlinear convection–diffusion equations with continuous or discontinuity-like initial conditions. The method transforms a convection–diffusion equation into a relaxation system, which contains a stiff source term. The resulting relaxation system is then solved by a third-order accurate implicit–explicit (IMEX) Runge–Kutta method in time and a fifth-order finite difference WENO scheme in space. Numerical results show that the method can be used to effectively solve convection–diffusion equations with both smooth structures and discontinuities.     

Mean-square stability of the Euler–Maruyama method for stochastic differential delay equations with jumps

This paper deals with the mean-square (MS) stability of the Euler–Maruyama method for stochastic differential delay equations (SDDEs) with jumps. First, the definition of the MS-stability of numerical methods for SDDEs with jumps is established, and then the sufficient condition of the MS-stability of the Euler–Maruyama method for SDDEs with jumps is derived, finally a class scalar test equation is simulated and the numerical experiments verify the results obtained from theory.     

Characteristic stabilized finite element method for the transient Navier–Stokes equations

Based on the lowest equal-order conforming finite element subspace (X_h, M_h) (i.e. P₁–P₁ or Q₁–Q₁ elements), a characteristic stabilized finite element method for transient Navier–Stokes problem is proposed. The proposed method has a number of attractive computational properties: parameter-free, avoiding higher-order derivatives or edge-based data structures, and averting the difficulties caused by trilinear terms. Existence,uniqueness and error estimates of the approximate solution are proved by applying the technique of characteristic finite element method. Finally, a serious of numerical experiments are given to show that this method is highly efficient for transient Navier–Stokes problem.     

Two-level stabilized finite element method for the transient Navier–Stokes equations

In this article, a two-level stabilized finite element method based on two local Gauss integrations for the two-dimensional transient Navier–Stokes equations is analysed. This new stabilized method presents attractive features such as being parameter-free, or being defined for nonedge-based data structures. Some new a priori bounds for the stabilized finite element solution are derived. The two-level stabilized method involves solving one small Navier–Stokes problem on a coarse mesh with mesh size 0<H<1, and a large linear Stokes problem on a fine mesh with mesh size 0<h?H. A H ¹-optimal velocity approximation and a L ²-optimal pressure approximation are obtained. If we choose h=O(H ²), the two-level method gives the same order of approximation as the standard stabilized finite element method.     

Convergence analysis of the two-stage¶multisplitting method

An example is given which shows that the asymptotic convergence rate of the two-stage multisplitting method (see D.B. Szyld and M.T. Jones, SIAM J. Matrix Anal. Appl. 13, 671–679 (1992)) with one inner iteration is, generally, either faster or slower than that with many inner iterations. When the coefficient matrix is an H-matrix and a monotone matrix, respectively, we formulate the convergence as well as the monotone convergence theories for this two-stage multisplitting method under suitable constraints on the two-stage multisplitting. Furthermore, the corresponding comparison theorem in the sense of monotonicity for this method is established and several concrete applications are discussed. Received: April 1996 / Accepted: April 1998     

An improved convergence analysis for the Newton–Kantorovich method using recurrence relations

We generate a sequence using the Newton–Kantorovich method in order to approximate a locally unique solution of an operator equation on a Banach space under Hölder continuity conditions. Using recurrence relations, Hölder as well as centre-Hölder continuity assumptions on the operator involved, we provide a semilocal convergence analysis with the following advantages over the elegant work by Hernánde? in (The Newton method for operators with Hölder continuous first derivative, J. Optim. Theory Appl. 109(3) (2001), pp. 631–648.) (under the same computational cost): finer error bounds on the distances involved, and a more precise information on the location of the solution. Our results also compare favourably with recent and relevant ones in (I.K. Argyros, Concerning the “terra incognita” between convergence regions of two Newton methods, Nonlinear Anal. 62 (2005), pp. 179–194; I.K. Argyros, Computational Theory of Iterative Methods, in Studies in Computational Mathematics, Vol. 15, C.K. Chui and L. Wuytack, eds., Elsevier Publ. Co., New York, USA, 2007; I.K. Argyros, On the gap between the semilocal convergence domain of two Newton methods, Appl. Math. 34(2) (2007), pp. 193–204; I.K. Argyros, On the convergence region of Newton's method under Hölder continuity conditions, submitted for publication; I.K. Argyros, Estimates on majorizing sequences in the Newton–Kantorovich method, submitted for publication; F. Cianciaruso and E. DePascale, Newton–Kantorovich approximations when the derivative is Hölderian: Old and new results, Numer. Funct. Anal. Optim. 24 (2003), pp. 713–723; F. Cianciaruso and E. DePascale, Estimates of majorizing sequences in the Newton–Kantorovich method, Numer. Funct. Anal. Optim. 27(5–6) (2006), pp. 529–538; F. Cianciaruso and E. DePascale, Estimates of majorizing sequences in the Newton–Kanorovich method: A further improvement, J. Math. Anal. Appl. 322 (2006), pp. 329–335; N.T. Demidovich, P.P. Zabreiko, and Ju.V. Lysenko, Some remarks on the Newton–Kantorovich mehtod for nonlinear equations with Hölder continuous linearizations, Izv. Akad. Nauk Belorus 3 (1993), pp. 22–26 (in Russian). (E. DePascale and P.P. Zabreiko, The convergence of the Newton–Kantorovich method under Vertgeim conditions, A new improvement, Z. Anal. Anwendvugen 17 (1998), pp. 271–280.) and (L.V. Kantorovich and G.P. Akilov, Functional Analysis in Normed Spaces, Pergamon Press, Oxford, 1982; J.V. Lysenko, Conditions for the convergence of the Newton–Kantorovich method for nonlinear equations with Hölder linearizations, Dokl. Akad. Nauk BSSR 38 (1994), pp. 20–24 (in Russian); B.A. Vertgeim, On some methods for the approximate solution of nonlinear functional equations in Banach spaces, Uspekhi Mat. Nauk 12 (1957), pp. 166–169 (in Russian); Amer. Math. Soc. Transl. 16 (1960), pp. 378–382. (English Trans.).)     

Convergence of linearized backward Euler–Galerkin finite element methods for the time-dependent Ginzburg–Landau equations with temporal gauge

We present error analysis of fully discrete Galerkin finite element methods for the time-dependent Ginzburg–Landau equations with the temporal gauge, where a linearized backward Euler scheme is used for the time discretization. We prove that the convergence rate is O(τ+h^r) if the finite element space of piecewise polynomials of degree r is used. Due to the degeneracy of the problem, the convergence rate is one order lower than the optimal convergence rate of finite element methods for parabolic equations. Numerical examples are provided to support our theoretical analysis.