Stability of implicit Runge-Kutta methods for nonlinear stiff differential equations

Under the assumption that an implicit Runge-Kutta method satisfies a certain stability estimate for linear systems with constant coefficientsl ₂-stability for nonlinear systems is proved. This assumption is weaker than algebraic stability since it is satisfied for many methods which are not evenA-stable. Some local smoothness in the right hand side of the differential equation is needed, but it may have a Jacobian and higher derivatives with large norms. The result is applied to a system derived from a strongly nonlinear parabolic equation by the method of lines.     

A class of split-step balanced methods for stiff stochastic differential equations

In this paper we design a class of general split-step balanced methods for solving It? stochastic differential systems with m-dimensional multiplicative noise, in which the drift or deterministic increment function can be taken from any chosen one-step ODE solver. We then give an analysis of their order of strong convergence in a general setting, but for the mean-square stability analysis, we confine our investigation to a special case in which the drift increment function of the methods is replaced by the one from the well known Rosenbrock method. The resulting class of stochastic differential equation (SDE) solvers will have more appropriate and useful mean-square stability properties for SDEs with stiffness in their drift and diffusion parts, compared to some other already reported split-step balanced methods. Finally, numerical results show the effectiveness of these methods.     

A special family of Runge-Kutta methods for solving stiff differential equations

An efficient way of implementing Implicit Runge-Kutta Methods was proposed by Butcher [3]. He showed that the most efficient methods when using this implementation are those whose characteristic polynomial of the Runge-Kutta matrix has a single reals-fold zero. In this paper we will construct such a family of methods and give some results concerning their maximum attainable order and stability properties. Some consideration is also given to showing how these methods can be efficiently implemented and, in particular, how local error estimates can be obtained by the use of embedding techniques.     

Regularity analysis for stochastic partial differential equations with nonlinear multiplicative trace class noise

In this article spatial and temporal regularity of the solution process of a stochastic partial differential equation (SPDE) of evolutionary type with nonlinear multiplicative trace class noise is analyzed.     

Efficient higher order implicit one-step methods for integration of stiff differential equations

A new implicit integration method is presented which can efficiently be applied in the solution of (stiff) differential equations. The given formulas are of a modified implicit Runge-Kutta type and areA-stable. They may containA-stable embedded methods for error estimation and step-size control.     

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

In this paper, we present the composite Milstein methods for the strong solution of Ito stochastic differential equations. These methods are a combination of semi-implicit and implicit Milstein methods. We give a criterion for choosing either the implicit or the semi-implicit scheme at each step of our numerical solution. The stability and convergence properties are investigated and discussed for the linear test equation. The convergence properties for the nonlinear case are shown numerically to be the same as the linear case. The stability properties of the composite Milstein methods are found to be more superior compared to those of the Milstein, the Euler and even better than the composite Euler method. This superiority in stability makes the methods a better candidate for the solution of stiff SDEs.     

Convergence analysis of trigonometric methods for stiff second-order stochastic differential equations

We study a class of numerical methods for a system of second-order SDE driven by a linear fast force generating high frequency oscillatory solutions. The proposed schemes permit the use of large step sizes, have uniform global error bounds in the position (i.e. independent of the large frequencies present in the SDE) and offer various additional properties. This new family of numerical integrators for SDE can be viewed as a stochastic generalisation of the trigonometric integrators for highly oscillatory deterministic problems.     

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.     

Optimal control for infinite dimensional stochastic differential equations with infinite Markov jumps and multiplicative noise

In this paper we solve an infinite-horizon linear quadratic control problem for a class of differential equations with countably infinite Markov jumps and multiplicative noise. The global solvability of the associated differential Riccati-type equations is studied under detectability hypotheses. A nonstochastic, operatorial approach is used. Some properties of the linear stochastic systems, such as stability, stabilizability and detectability, are also discussed on the basis of a new solution representation result. A generalized Ito's formula which applies to infinite dimensional stochastic differential equations with countably infinite Markov jumps is also provided.     

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.     

Predictor-corrector pseudospectral methods for stochastic partial differential equations with additive white noise

Commonly used finite-difference numerical schemes show some deficiencies in the integration of certain types of stochastic partial differential equations with additive white noise. In this paper efficient predictor-corrector spectral schemes to integrate these equations are discussed. They are all based on the discretization of the system in Fourier space. The nonlinear terms are treated using a pseudospectral approach so as to speed up the computations without a significant loss of accuracy. The proposed schemes are applied to solve, both in one and two spatial dimensions, two paradigmatic continuum models arising in the context of nonequilibrium dynamics of growing interfaces: the Kardar-Parisi-Zhang and Lai-Das Sarma-Villain equations. Numerical results about the Lai-Das Sarma-Villain equation in two spatial dimensions have not been previously reported in the literature.     

线性随机时滞微分方程裂步Milstein方法的收敛性(英文)

给出一个新的求解线性随机时滞微分方程的显式分裂步长Milstein格式.运用ItoTaylor展开式证明该格式相对于已有的求解随机时滞微分方程的分裂步长方法而言具有更好的收敛性.数值实验验证了理论分析的正确性.     

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.     

A family of implicit Chebyshev methods for the numerical integration of second-order differential equations

A family of implicit methods based on intra-step Chebyshev interpolation is developed for the solution of initial-value problems whose differential equations are of the special second-order form y″ = f(y(x); x). The general procedure allows stepsizes which are considerably larger than commonly used in conventional methods. Computation overhead is comparable to that required by high-order single or multistep procedures. In addition, the iterative nature of the method substantially reduces local errors while maintaining a low rate of global error growth.     

Convergence and stability of implicit runge-kutta methods for systems with multiplicative noise

A class ofimplicit Runge-Kutta schemes for stochastic differential equations affected bymultiplicative Gaussian white noise is shown to be optimal with respect to global order of convergence in quadratic mean. A test equation is proposed in order to investigate the stability of discretization methods for systems of this kind. Herestability is intended in a truly probabilistic sense, as opposed to the recently introduced extension of A-stability to the stochastic context, given for systems with additive noise. Stability regions for the optimal class are also given.Partially supported by the Italian Consiglio Nazionale delle Ricerche.     

Implicit Euler method for numerical solution of nonlinear stochastic partial differential equations with multiplicative trace class noise

In this paper, we consider the numerical approximation of stochastic partial differential equations with nonlinear multiplicative trace class noise. Discretization is obtained by spectral collocation method in space, and semi‐implicit Euler method is used for the temporal approximation. Our purpose is to investigate the convergence of the proposed method. The rate of convergence is obtained, and some numerical examples are included to illustrate the estimated convergence rate.     

Heterogeneous multiscale methods for stiff ordinary differential equations

The heterogeneous multiscale methods (HMM) is a general framework for the numerical approximation of multiscale problems. It is here developed for ordinary differential equations containing different time scales. Stability and convergence results for the proposed HMM methods are presented together with numerical tests. The analysis covers some existing methods and the new algorithms that are based on higher-order estimates of the effective force by kernels satisfying certain moment conditions and regularity properties. These new methods have superior computational complexity compared to traditional methods for stiff problems with oscillatory solutions.

     

Hitting probabilities for systems of non-linear stochastic heat equations with multiplicative noise

We consider a system of d non-linear stochastic heat equations in spatial dimension 1 driven by d-dimensional space-time white noise. The non-linearities appear both as additive drift terms and as multipliers of the noise. Using techniques of Malliavin calculus, we establish upper and lower bounds on the one-point density of the solution u(t, x), and upper bounds of Gaussian-type on the two-point density of (u(s, y),u(t, x)). In particular, this estimate quantifies how this density degenerates as (s, y) → (t, x). From these results, we deduce upper and lower bounds on hitting probabilities of the process ${\{u(t,x)\}_{t \in \mathbb{R}_+, x\in [0,1]}}$ , in terms of respectively Hausdorff measure and Newtonian capacity. These estimates make it possible to show that points are polar when d ≥ 7 and are not polar when d ≤ 5. We also show that the Hausdorff dimension of the range of the process is 6 when d > 6, and give analogous results for the processes ${t \mapsto u(t,x)}$ and ${x \mapsto u(t,x)}$ . Finally, we obtain the values of the Hausdorff dimensions of the level sets of these processes.