The th moment stability for the stochastic pantograph equation

In this paper, we investigate the αth moment asymptotical stability of the analytic solution and the numerical methods for the stochastic pantograph equation by using the Razumikhin technique. Especially the linear stochastic pantograph equations and the semi-implicit Euler method applying them are considered. The convergence result of the semi-implicit Euler method is obtained. The stability conditions of the analytic solution of those equations and the numerical method are given. Finally, some experiments are given.     

The composite Milstein methods for the numerical solution of Stratonovich stochastic differential equations

In this paper, we present two composite Milstein methods for the strong solution of Stratonovich stochastic differential equations driven by d-dimensional Wiener processes. The composite Milstein methods are a combination of semi-implicit and implicit Milstein methods. The criterion for choosing either the implicit or the semi-implicit method at each step of the numerical solution is given. The stability and convergence properties of the proposed methods are analyzed for the linear test equation. It is shown that the proposed methods converge to the exact solution in Stratonovich sense. In addition, the stability properties of our methods are found to be superior to those of the Milstein and the composite Euler methods. The convergence properties for the nonlinear case are shown numerically to be the same as the linear case. Hence, the proposed methods are a good candidate for the solution of stiff SDEs.     

A High-Order Kernel-Free Boundary Integral Method for Incompressible Flow Equations in Two Space Dimensions

This paper presents a fourth-order kernel-free boundary integral method for the time-dependent, incompressible Stokes and Navier-Stokes equations defined on irregular bounded domains. By the stream function-vorticity formulation, the incompressible flow equations are interpreted as vorticity evolution equations. Time discretization methods for the evolution equations lead to a modified Helmholtz equation for the vorticity, or alternatively, a modified biharmonic equation for the stream function with two clamped boundary conditions. The resulting fourth-order elliptic boundary value problem is solved by a fourth-order kernel-free boundary integral method, with which integrals in the reformulated boundary integral equation are evaluated by solving corresponding equivalent interface problems, regardless of the exact expression of the involved Green's function. To solve the unsteady Stokes equations, a four-stage composite backward differential formula of the same order accuracy is employed for time integration. For the Navier-Stokes equations, a three-stage third-order semi-implicit Runge-Kutta method is utilized to guarantee the global numerical solution has at least third-order convergence rate. Numerical results for the unsteady Stokes equations and the Navier-Stokes equations are presented to validate efficiency and accuracy of the proposed method.     

MULTILEVEL AUGMENTATION METHODS FOR SOLVING OPERATOR EQUATIONS

We introduce multilevel augmentation methods for solving operator equations based on direct sum decompositions of the range space of the operator and the solution space of the operator equation and a matrix splitting scheme. We establish a general setting for the analysis of these methods, showing that the methods yield approximate solutions of the same convergence order as the best approximation from the subspace. These augmentation methods allow us to develop fast, accurate and stable nonconventional numerical algorithms for solving operator equations. In particular, for second kind equations, special splitting techniques are proposed to develop such algorithms. These algorithms are then applied to solve the linear systems resulting from matrix compression schemes using wavelet-like functions for solving Fredholm integral equations of the second kind. For this special case, a complete analysis for computational complexity and convergence order is presented. Numerical examples are included to demonstra     

Semi-Lagrangian semi-implicit time-splitting scheme for a regional model of the atmosphere

Semi-Lagrangian semi-implicit (SLSI) method is currently one of the most efficient approaches for numerical solution of the atmosphere dynamics equations. In this research we apply splitting techniques in the context of a two-time-level SLSI scheme in order to simplify the treatment of the slow physical modes and optimize the solution of the elliptic equations related to implicit part of the scheme. The performed numerical experiments show the accuracy and computational efficiency of the scheme.     

Two-Stage Stochastic Runge-Kutta Methods for Stochastic Differential Equations

In this paper we discuss two-stage diagonally implicit stochastic Runge-Kutta methods with strong order 1.0 for strong solutions of Stratonovich stochastic differential equations. Five stochastic Runge-Kutta methods are presented in this paper. They are an explicit method with a large MS-stability region, a semi-implicit method with minimum principal error coefficients, a semi-implicit method with a large MS-stability region, an implicit method with minimum principal error coefficients and another implicit method. We also consider composite stochastic Runge-Kutta methods which are the combination of semi-implicit Runge-Kutta methods and implicit Runge-Kutta methods. Two composite methods are presented in this paper. Numerical results are reported to compare the convergence properties and stability properties of these stochastic Runge-Kutta methods.     

Error analysis of a fully discrete projection method for magnetohydrodynamic system

In this paper, we develop and analyze a finite element projection method for magnetohydrodynamics equations in Lipschitz domain. A fully discrete scheme based on Euler semi-implicit method is proposed, in which continuous elements are used to approximate the Navier–Stokes equations and H ( curl ) conforming Nédélec edge elements are used to approximate the magnetic equation. One key point of the projection method is to be compatible with two different spaces for calculating velocity, which leads one to obtain the pressure by solving a Poisson equation. The results show that the proposed projection scheme meets a discrete energy stability. In addition, with the help of a proper regularity hypothesis for the exact solution, this paper provides a rigorous optimal error analysis of velocity, pressure and magnetic induction. Finally, several numerical examples are performed to demonstrate both accuracy and efficiency of our proposed scheme.     

Semi-implicit finite volume level set method for advective motion of interfaces in normal direction

In this paper a semi-implicit finite volume method is proposed to solve the applications with moving interfaces using the approach of level set methods. The level set advection equation with a given speed in normal direction is solved by this method. Moreover, the scheme is used for the numerical solution of eikonal equation to compute the signed distance function and for the linear advection equation to compute the so-called extension speed [1]. In both equations an extrapolation near the interface is used in our method to treat Dirichlet boundary conditions on implicitly given interfaces. No restrictive CFL stability condition is required by the semi-implicit method that is very convenient especially when using the extrapolation approach. In summary, we can apply the method for the numerical solution of level set advection equation with the initial condition given by the signed distance function and with the advection velocity in normal direction given by the extension speed. Several advantages of the proposed approach can be shown for chosen examples and application. The advected numerical level set function approximates well the property of remaining the signed distance function during whole simulation time. Sufficiently accurate numerical results can be obtained even with the time steps violating the CFL stability condition.     

The application of Rosenbrock-Wanner type methods with stepsize control in differential-algebraic equations

Summary Two Rosenbrock-Wanner type methods for the numerical treatment of differential-algebraic equations are presented. Both methods possess a stepsize control and an index-1 monitor. The first method DAE34 is of order (3)4 and uses a full semi-implicit Rosenbrock-Wanner scheme. The second method RKF4DA is derived from the Runge-Kutta-Fehlberg 4(5)-pair, where a semi-implicit Rosenbrock-Wanner method is embedded, in order to solve the nonlinear equations. The performance of both methods is discussed in artificial test problems and in technical applications.     

Block boundary value methods for solving Volterra integral and integro-differential equations

Reducible quadrature rules generated by boundary value methods are considered in block version and applied to solve the second kind Volterra integral equations and Volterra integro-differential equations. These extended block boundary value methods are shown to possess both excellent stability properties and high accuracy for Volterra-type equations. Numerical experiments are presented and the efficiency, accuracy and stability of the schemes are confirmed.     

A hybrid technique for computing the power distribution generated in a lossy medium during microwave heating

Over the years researchers in the field of computational electromagnetics (CEM) have investigated and explored a number of different techniques to resolve electromagnetic fields inside waveguide and cavity structures. The equations that govern the fundamental behaviour of electromagnetic wave propagation in such structures are Maxwell's equations. In the literature, a number of different techniques have been employed to solve these equations and out of these methods, the classical finite-difference time-domain (FD-TD) scheme, which uses a staggered time and space discretisation, is the most well-known and widely used. However, this scheme is complicated to implement on an irregular computational domain using unstructured meshes.This research work builds upon previous work undertaken for a waveguide, where a coupled method was introduced for the solution of the governing electromagnetic equations. In that work, the free-space component of the solution was computed in the time-domain, whilst the power distribution in the load was resolved using the frequency dependent electric field Helmholtz equation. This methodology resulted in a time-frequency domain hybrid scheme. In this paper, the hybrid method has been tested further for both waveguide and cavity configurations that are loaded with a lossy dielectric material. Numerical tests highlight both the accuracy and computational efficiency of the proposed hybrid strategy for predicting the power distribution generated during microwave heating processes. The accuracy of the hybrid scheme is gauged by direct comparison with the FD-TD numerical solutions and previously published thermal images.     

A fourth-order approximate projection method for the incompressible Navier–Stokes equations on locally-refined periodic domains

In this follow-up of our previous work [30], the author proposes a high-order semi-implicit method for numerically solving the incompressible Navier–Stokes equations on locally-refined periodic domains. Fourth-order finite-volume stencils are employed for spatially discretizing various operators in the context of structured adaptive mesh refinement (AMR). Time integration adopts a fourth-order, semi-implicit, additive Runge–Kutta method to treat the non-stiff convection term explicitly and the stiff diffusion term implicitly. The divergence-free condition is fulfilled by an approximate projection operator. Altogether, these components yield a simple algorithm for simulating incompressible viscous flows on periodic domains with fourth-order accuracies both in time and in space. Results of numerical tests show that the proposed method is superior to previous second-order methods in terms of accuracy and efficiency. A major contribution of this work is the analysis of a fourth-order approximate projection operator.     

An improved solution for diffusion waves to overland flow

Diffusion wave equation describes the flood wave propagation which is used in solving overland and open channel flow problems. Therefore, it is important to understand and solve the diffusion wave equations accurately. For this purpose, researchers have previously developed several analytical and numerical methods for the solution of the partial differential equation of diffusion waves. The solution derived by Kazezy?lmaz-Alhan and Medina (2007) [12] can be used to solve overland flow problems during rainfall events with both constant and variable rainfall intensity, and with constant hydraulic diffusivity and wave celerity. In this paper, this method is improved by employing the De Hoog algorithm instead of Stehfest algorithm for Laplace inversion and adapting the solution to variable hydraulic diffusivity and wave celerity. In addition, the performances of the Stehfest and the De Hoog algorithms are compared. Synthetic examples are solved by using both Stehfest and De Hoog algorithms incorporated into the existing analytical solution to present the accuracy of the De Hoog algorithm over the Stehfest algorithm. The examples are also solved by using the new method in order to demonstrate the improvement over the existing method.     

Implications of the Choice of Quadrature Nodes for Picard Integral Deferred Corrections Methods for Ordinary Differential Equations

This paper concerns a class of deferred correction methods recently developed for initial value ordinary differential equations; such methods are based on a Picard integral form of the correction equation. These methods divide a given timestep [t_n,t_n+1] into substeps, and use function values computed at these substeps to approximate the Picard integral by means of a numerical quadrature. The main purpose of this paper is to present a detailed analysis of the implications of the location of quadrature nodes on the accuracy and stability of the overall method. Comparisons between Gauss-Legendre, Gauss-Lobatto, Gauss-Radau, and uniformly spaced points are presented. Also, for a given set of quadrature nodes, quadrature rules may be formulated that include or exclude function values computed at the left-hand endpoint t_n. Quadrature rules that do not depend on the left-hand endpoint (which are referred to as right-hand quadrature rules) are shown to lead to L(α)-stable implicit methods with α≈π/2. The semi-implicit analog of this property is also discussed. Numerical results suggest that the use of uniform quadrature nodes, as opposed to nodes based on Gaussian quadratures, does not significantly affect the stability or accuracy of these methods for orders less than ten. In contrast, a study of the reduction of order for stiff equations shows that when uniform quadrature nodes are used in conjunction with a right-hand quadrature rule, the form and extent of order-reduction changes considerably. Specifically, a reduction of order to is observed for uniform nodes as opposed to for non-uniform nodes, where Δt denotes the time step and ε a stiffness parameter such that ε→0 corresponds to the problem becoming increasingly stiff. AMS subject classification (2000) 65B05     

Solution to the Boltzmann kinetic equation for high-speed flows

The Boltzmann kinetic equation is solved by a finite-difference method on a fixed coordinate-velocity grid. The projection method is applied that was developed previously by the author for evaluating the Boltzmann collision integral. The method ensures that the mass, momentum, and energy conservation laws are strictly satisfied and that the collision integral vanishes in thermodynamic equilibrium. The last property prevents the emergence of the numerical error when the collision integral of the principal part of the solution is evaluated outside Knudsen layers or shock waves, which considerably improves the accuracy and efficiency of the method. The differential part is approximated by a second-order accurate explicit conservative scheme. The resulting system of difference equations is solved by applying symmetric splitting into collision relaxation and free molecular flow. The steady-state solution is found by the relaxation method.     

Fast solvers with block‐diagonal preconditioners for linear FEM–BEM coupling

The purpose of this paper is to present optimal preconditioned iterative methods to solve indefinite linear systems of equations arising from symmetric coupling of finite elements and boundary elements. This is a block‐diagonal preconditioner together with a conjugate residual method and a preconditioned inner–outer iteration. We prove the efficiency of these methods by showing that the number of iterations to preserve a given accuracy is bounded independent of the number of unknowns. Numerical examples underline the efficiency of these methods. Copyright © 2008 John Wiley & Sons, Ltd.     

Convergence of the semi-implicit Euler method for neutral stochastic delay differential equations with phase semi-Markovian switching

Recently, in the numerical analysis for stochastic differential equations (SDEs), it is a new topic to study the numerical schemes of neutral stochastic functional differential equations (NSFDEs) (see Wu and Mao [1]). Especially when Markovian switchings are taken into consideration, these problems will be more complicated. Although Zhou and Wu [2] develop a numerical scheme to neutral stochastic delay differential equations with Markovian switching (short for NSDDEwMSs), their method belongs to explicit Euler–Maruyama methods which are in general much less accurate in approximation than their implicit or semi-implicit counterparts. Therefore, to propose an implicit method becomes imperative to fill the gap. In this paper we will extend Zhou and Wu [2] to the case of the semi-implicit Euler–Maruyama methods and equations with phase semi-Markovian switching rather than Markovian switching. The employment of phase semi-Markovian chains can avoid the restriction of the negative exponential distribution of the sojourn time at a state. We prove the semi-implicit Euler solution will converge to the exact solution to NSDDEwMS under local Lipschitz condition. More precise inequalities and new techniques are put forward to overcome the difficulties for the existence of the neutral part.     

Numerical studies on nonlinear Schrödinger equations by spectral collocation method with preconditioning

In this study, we use the spectral collocation method with preconditioning to solve various nonlinear Schrödinger equations. To reduce round-off error in spectral collocation method we use preconditioning. We study the numerical accuracy of the method. The numerical results obtained by this way have been compared with the exact solution to show the efficiency of the method.     

Efficient numerical schemes for the solution of generalized time fractional Burgers type equations

The main aim of this paper is to propose two semi-implicit Fourier pseudospectral schemes for the solution of generalized time fractional Burgers type equations, with an analysis of consistency, stability, and convergence. Under some assumptions, the unconditional stability of the schemes is shown. In implementation of these schemes, the fast Fourier transform (FFT) can be used efficiently to improve the computational cost. Various test problems are included to illustrate the results that we have obtained regarding the proposed schemes. The results of numerical experiments are compared with analytical solutions and other existing methods in the literature to show the efficiency of proposed schemes in both accuracy and CPU time. As numerical solution of fractional stochastic nonlinear partial differential equations driven by Brownian motions are among current related research interests, we report the performance of these schemes on stochastic time fractional Burgers equation as well.     

A Multilevel Correction Method for Steklov Eigenvalue Problem by Nonconforming Finite Element Methods

In this paper, a multilevel correction scheme is proposed to solve the Steklov eigenvalue problem by nonconforming finite element methods. With this new scheme, the accuracy of eigenpair approximations can be improved after each correction step which only needs to solve a source problem on finer finite element space and an Steklov eigenvalue problem on the coarsest finite element space. This correction scheme can increase the overall efficiency of solving eigenvalue problems by the nonconforming finite element method. Furthermore, as same as the direct eigenvalue solving by nonconforming finite element methods, this multilevel correction method can also produce the lower-bound approximations of the eigenvalues.