Three-stage Stiffly Accurate Runge-Kutta Methods for Stiff Stochastic Differential Equations

In this paper we discuss diagonally implicit and semi-implicit methods based on the three-stage stiffly accurate Runge-Kutta methods for solving Stratonovich stochastic differential equations（SDEs）.Two methods,a three-stage stiffly accurate semi-implicit（SASI3） method and a three-stage stiffly accurate diagonally implicit （SADI3） method,are constructed in this paper.In particular,the truncated random variable is used in the implicit method.The stability properties and numerical results show the effectiveness of these methods in the pathwise approximation of stiff SDEs.     

NON SPURIOUS SPECTRAL-LIKE ELEMENT METHODS FOR MAXWELL＇S EQUATIONS

In this paper, we give the state of the art for the so called “mixed spectral elements” for Maxwell＇s equations. Several families of elements, such as edge elements and discon-tinuous Galerkin methods （DGM） are presented and discussed. In particular, we show the need of introducing some numerical dissipation terms to avoid spurious modes in these methods. Such terms are classical for DGM but their use for edge element methods is novel approach described in this paper. Finally, numerical experiments show the fast and low-cost character of these elements.     

FINITE ELEMENTS WITH LOCAL PROJECTION STABILIZATION FOR INCOMPRESSIBLE FLOW PROBLEMS

In this paper we review recent developments in the analysis of finite element methods for incompressible flow problems with local projection stabilization （LPS）. These methods preserve the favourable stability and approximation properties of classical residual-based stabilization （RBS） techniques but avoid the strong coupling of velocity and pressure in the stabilization terms. LPS-methods belong to the class of symmetric stabilization techniques and may be characterized as variational multiscale methods. In this work we summarize the most important a priori estimates of this class of stabilization schemes developed in the past 6 years. We consider the Stokes equations, the Oseen linearization and the NavierStokes equations. Furthermore, we apply it to optimal control problems with linear（ized） flow problems, since the symmetry of the stabilization leads to the nice feature that the operations ＂discretize＂ and ＂optimize＂ commute.     

DISSIPATIVE NUMERICAL METHODS FOR THE HUNTER-SAXTON EQUATION

In this paper, we present further development of the local discontinuous Galerkin （LDG） method designed in [21] and a new dissipative discontinuous Galerkin （DG） method for the HuntermSaxton equation. The numerical fluxes for the LDG and DG methods in this paper are based on the upwinding principle. The resulting schemes provide additional energy dissipation and better control of numerical oscillations near derivative singularities. Stability and convergence of the schemes are proved theoretically, and numerical simulation results are provided to compare with the scheme in [21].     

RELATIONS BETWEEN SOLUTIONS TO STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY SEMIMARTINGALE WITH NON-LIPSCHITZ COEFFICIENTS

In this paper, a class of stochastic differential equations (SDEs) driven by semi-martingale with non-Lipschitz coefficients is studied. We investigate the dependence of solutions to SDEs on the initial value. To obtain a continuous version, we impose the conditions on the local characteristic of semimartingale. In this case, it gives rise to a flow of homeomorphisms if the local characteristic is compactly supported.     

HIGH-ACCURACY EXPLICIT TWO-STEP METHODS WITH MINIMAL PHASE-LAG FOR y″= f（t,y）

In this paper, two families of high accuracy explicit two-step methods with minimal phase-lag are developed for the numerical integration of special second-order periodic initial-value problems. In comparison with some methods in [1-4,6], the advantage of these methods has a higher accuracy and minimal phase-lag. The methods proposed in this paper can be considered as a generalization of some methods in [1,3,4]. Numerical examples indicate that these new methods are generally more accurate than the methods used in [3,6]. second order periodic initial-value problems, phase-lag, local truncation error     

ON SOLVABILITY AND WAVEFORM RELAXATION METHODS FOR LINEAR VARIABLE-COEFFICIENT DIFFERENTIAL-ALGEBRAIC EQUATIONS

This paper is concerned with the solvability and waveform relaxation methods of linear variable-coefficient differential-algebraic equations （DAEs）. Most of the previous works have been focused on linear variable-coefficient DAEs with smooth coefficients and data, yet no results related to the convergence rate of the corresponding waveform relaxation methods has been obtained. In this paper, we develope the solvability theory for the linear variable-coefficient DAEs on Legesgue square-integrable function space in both traditional and least squares senses, and determine the convergence rate of the waveform relaxation methods for solving linear variable-coefficient DAEs.     

A CLASS OF TWO-STEP CONTINUITY RUNGE-KUTTA METHODS FOR SOLVING SINGULAR DELAY DIFFERENTIAL EQUATIONS AND ITS STABILITY ANALYSIS

In this paper, a class of two-step continuity Runge-Kutta（TSCRK） methods for solving singular delay differential equations（DDEs） is presented. Analysis of numerical stability of this methods is given. We consider the two distinct cases： （i）τ≥ h, （ii）τ 〈 h, where the delay τ and step size h of the two-step continuity Runge-Kutta methods are both constant. The absolute stability regions of some methods are plotted and numerical examples show the efficiency of the method.     

CHEBYSHEV METHODS WITH DISCRETE NOISE： THE τ-ROCK METHODS

Stabilized or Chebyshev explicit methods have been widely used in the past to solve stiff ordinary differential equations. Making use of special properties of Chebyshev-like polynomials, these methods have favorable stability properties compared to standard explicit methods while remaining explicit. A new class of such methods, called ROCK, introduced in [Numer. Math., 90, 1-18, 2001] has recently been extended to stiff stochastic differential equations under the name S-ROCK [C. R. Acad. Sci. Paris, 345（10）, 2007 and Commun. Math. Sci, 6（4）, 2008]. In this paper we discuss the extension of the S-ROCK methods to systems with discrete noise and propose a new class of methods for such problems, the τ-ROCK methods. One motivation for such methods is the simulation of multi-scale or stiff chemical kinetic systems and such systems are the focus of this paper, but our new methods could potentially be interesting for other stiff systems with discrete noise. Two versions of the τ-ROCK methods are discussed and their stability behavior is analyzed on a test problem. Compared to the τ-leaping method, a significant speed-up can be achieved for some stiff kinetic systems. The behavior of the proposed methods are tested on several numerical experiments.     

A NUMERICAL STUDY OF THE GAUSSIAN BEAM METHODS FOR SCHRODINGER-POISSON EQUATIONS

As an important model in quantum semiconductor devices, the SchrSdinger-Poisson equations have generated widespread interests in both analysis and numerical simulations in recent years. In this paper, we present Gaussian beam methods for the numerical simulation of the one-dimensional Schrodinger-Poisson equations. The Gaussian beam methods for high frequency waves outperform the geometrical optics method in that the former are accurate even around caustics. The purposes of the paper are first to develop the Gaussian beam methods, based on our previous methods for the linear SchrSdinger equation, for the Schrodinger-Poisson equations, and then check their validity for this weakly-nonlinear system.     

A DISCRETE-TIME ITÔ'S FORMULA

Abstract

The numerical methods on stochastic differential equations (SDEs) have been well established. There are several papers that study the numerical stability of SDEs with respect to sample paths or moments. In this paper, we study the stability in distribution of numerical solution of SDEs.     

Implicit Milstein method for stochastic differential equations via the Wong-Zakai approximation

The aim of this paper is to derive a numerical scheme for solving stochastic differential equations (SDEs) via Wong-Zakai approximation. One of the most important methods for solving SDEs is Milstein method, but this method is not so popular because the cost of simulating the double stochastic integrals is high. For overcoming this complexity, we present an implicit Milstein scheme based on Wong-Zakai approximation by approximating the Brownian motion with its truncated Haar expansion. The main advantages of this method lie in the fact that it preserves the convergence order and also stability region of the Milstein method while its simulation is much easier than Milstein scheme. We show the convergence rate of the method by some numerical examples.     

Split-step forward methods for stochastic differential equations

In this paper we discuss split-step forward methods for solving Itô stochastic differential equations (SDEs). Eight fully explicit methods, the drifting split-step Euler (DRSSE) method, the diffused split-step Euler (DISSE) method and the three-stage Milstein (TSM 1a-TSM 1f) methods, are constructed based on Euler-Maruyama method and Milstein method, respectively, in this paper. Their order of strong convergence is proved. The analysis of stability shows that the mean-square stability properties of the methods derived in this paper are improved on the original methods. The numerical results show the effectiveness of these methods in the pathwise approximation of Itô SDEs.     

Mean square polynomial stability of numerical solutions to a class of stochastic differential equations

The exponential stability of numerical methods to stochastic differential equations (SDEs) has been widely studied. In contrast, there are relatively few works on polynomial stability of numerical methods. In this letter, we address the question of reproducing the polynomial decay of a class of SDEs using the Euler–Maruyama method and the backward Euler–Maruyama method. The key technical contribution is based on various estimates involving the gamma function.     

A family of fully implicit Milstein methods for stiff stochastic differential equations with multiplicative noise

In this paper a family of fully implicit Milstein methods are introduced for solving stiff stochastic differential equations (SDEs). It is proved that the methods are convergent with strong order 1.0 for a class of SDEs. For a linear scalar test equation with multiplicative noise terms, mean-square and almost sure asymptotic stability of the methods are also investigated. We combine analytical and numerical techniques to get insights into the stability properties. The fully implicit methods are shown to be superior to those of the corresponding semi-implicit methods in term of stability property. Finally, numerical results are reported to illustrate the convergence and stability results.     

Pathwise accuracy and ergodicity of metropolized integrators for SDEs

Metropolized integrators for ergodic stochastic differential equations (SDEs) are proposed that (1) are ergodic with respect to the (known) equilibrium distribution of the SDEs and (2) approximate pathwise the solutions of the SDEs on finite‐time intervals. Both these properties are demonstrated in the paper, and precise strong error estimates are obtained. It is also shown that the Metropolized integrator retains these properties even in situations where the drift in the SDE is nonglobally Lipschitz, and vanilla explicit integrators for SDEs typically become unstable and fail to be ergodic. © 2009 Wiley Periodicals, Inc.     

A class of new Magnus-type methods for semi-linear non-commutative Itô stochastic differential equations

Numerical Algorithms - In this paper, a class of new Magnus-type methods is proposed for non-commutative Itô stochastic differential equations (SDEs) with semi-linear drift term and...     

Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise

The ergodic properties of SDEs, and various time discretizations for SDEs, are studied. The ergodicity of SDEs is established by using techniques from the theory of Markov chains on general state spaces, such as that expounded by Meyn–Tweedie. Application of these Markov chain results leads to straightforward proofs of geometric ergodicity for a variety of SDEs, including problems with degenerate noise and for problems with locally Lipschitz vector fields. Applications where this theory can be usefully applied include damped-driven Hamiltonian problems (the Langevin equation), the Lorenz equation with degenerate noise and gradient systems.The same Markov chain theory is then used to study time-discrete approximations of these SDEs. The two primary ingredients for ergodicity are a minorization condition and a Lyapunov condition. It is shown that the minorization condition is robust under approximation. For globally Lipschitz vector fields this is also true of the Lyapunov condition. However in the locally Lipschitz case the Lyapunov condition fails for explicit methods such as Euler–Maruyama; for pathwise approximations it is, in general, only inherited by specially constructed implicit discretizations. Examples of such discretization based on backward Euler methods are given, and approximation of the Langevin equation studied in some detail.     

High strong order stochastic Runge-Kutta methods for Stratonovich stochastic differential equations with scalar noise

This paper concerns the stochastic Runge-Kutta (SRK) methods with high strong order for solving the Stratonovich stochastic differential equations (SDEs) with scalar noise. Firstly, the new SRK methods with strong order 1.5 or 2.0 for the Stratonovich SDEs with scalar noise are constructed by applying colored rooted tree analysis and the theorem of order conditions for SRK methods proposed by Rößler (SIAM J. Numer. Anal. 48(3), 922–952, 2010). Secondly, a specific SRK method with strong order 2.0 for the Stratonovich SDEs whose drift term vanishes is proposed. And another specific SRK method with strong order 1.5 for the Stratonovich SDEs whose drift and diffusion terms satisfy the commutativity condition is proposed. The two specific SRK methods need only to use one random variable and do not need to simulate the multiple Stratonovich stochastic integrals. Finally, the numerical results show that performance of our methods is better than those of well-known SRK methods with strong order 1.0 or 1.5.