Implicit Stochastic Runge–Kutta Methods for Stochastic Differential Equations

In this paper we construct implicit stochastic Runge–Kutta (SRK) methods for solving stochastic differential equations of Stratonovich type. Instead of using the increment of a Wiener process, modified random variables are used. We give convergence conditions of the SRK methods with these modified random variables. In particular, the truncated random variable is used. We present a two-stage stiffly accurate diagonal implicit SRK (SADISRK2) method with strong order 1.0 which has better numerical behaviour than extant methods. We also construct a five-stage diagonal implicit SRK method and a six-stage stiffly accurate diagonal implicit SRK method with strong order 1.5. The mean-square and asymptotic stability properties of the trapezoidal method and the SADISRK2 method are analysed and compared with an explicit method and a semi-implicit method. Numerical results are reported for confirming convergence properties and for comparing the numerical behaviour of these methods. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

High order accurate semi-implicit WENO schemes for hyperbolic balance laws

In this paper we propose a family of well-balanced semi-implicit numerical schemes for hyperbolic conservation and balance laws. The basic idea of the proposed schemes lies in the combination of the finite volume WENO discretization with Roe’s solver and the strong stability preserving (SSP) time integration methods, which ensure the stability properties of the considered schemes [S. Gottlieb, C.-W. Shu, E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev. 43 (2001) 89-112]. While standard WENO schemes typically use explicit time integration methods, in this paper we are combining WENO spatial discretization with optimal SSP singly diagonally implicit (SDIRK) methods developed in [L. Ferracina, M.N. Spijker, Strong stability of singly diagonally implicit Runge-Kutta methods, Appl. Numer. Math. 58 (2008) 1675-1686]. In this way the implicit WENO numerical schemes are obtained. In order to reduce the computational effort, the implicit part of the numerical scheme is linearized in time by taking into account the complete WENO reconstruction procedure. With the proposed linearization the new semi-implicit finite volume WENO schemes are designed.A detailed numerical investigation of the proposed numerical schemes is presented in the paper. More precisely, schemes are tested on one-dimensional linear scalar equation and on non-linear conservation law systems. Furthermore, well-balanced semi-implicit WENO schemes for balance laws with geometrical source terms are defined. Such schemes are then applied to the open channel flow equations. We prove that the defined numerical schemes maintain steady state solution of still water. The application of the new schemes to different open channel flow examples is shown.     

Three-stage stochastic Runge–Kutta methods for stochastic differential equations

In this paper we discuss three-stage stochastic Runge–Kutta (SRK) methods with strong order 1.0 for a strong solution of Stratonovich stochastic differential equations (SDEs). Higher deterministic order is considered. Two methods, a three-stage explicit (E3) method and a three-stage semi-implicit (SI3) method, are constructed in this paper. The stability properties and numerical results show the effectiveness of these methods in the pathwise approximation of several standard test problems.     

Two-stage and Three-stage Diagonally Implicit Runge-Kutta Nystrom Methods of Orders Three and Four

We investigate the existence of two-stage and three-stage R-stable,P-stable, RL-stable, and dispersively enhanced diagonally implicitRunge-Kutta Nyström methods of orders three and four. Wefirst show that a one-parameter family of two-stage third-orderR-stable diagonally implicit methods exists, and that theirdispersive order is at most four. From this we show that two-stagefourth-order R-stable, and third-order P-stable and RL-stablediagonally implicit methods do not exist. Next we show a two-parameterfamily of three-stage fourth-order R-stable diagonally implicitmethods exists with dispersive order at most four, and thatthis family contains a one-parameter family of P-stable methodsand a unique RL-stable. We also show that a one-parameter familyof fourth-order diagonally implicit methods with dispersiveorder at least six exists, and that they are not R-stable. Wepresent third- and fourth-order R-stable and P-stable methodswith small principal truncation coefficients and discuss howthese methods might be implemented in an efficient integrator.     

Singly implicit diagonally extended Runge-Kutta methods of fourth order

Singly implicit diagonally extended Runge-Kutta methods make it possible to combine the merits of diagonally implicit methods (namely, the simplicity of implementation) and fully implicit ones (high stage order). Due to this combination, they can be very efficient at solving stiff and differential-algebraic problems. In this paper, fourth-order methods with an explicit first stage are examined. The methods have the third or fourth stage order. Consideration is given to an efficient implementation of these methods. The results of tests in which the proposed methods were compared with the fifth-order RADAU IIA method are presented.     

On implicit Runge-Kutta methods with high stage order

It is well known that high stage order is a desirable property for implicit Runge-Kutta methods. In this paper it is shown that it is always possible to construct ans-stage IRK method with a given stability function and stage orders−1 if the stability function is an approximation to the exponential function of at least orders. It is further indicated how to construct such methods as well as in which cases the constructed methods will be stiffly accurate.     

随机延迟微分方程的全隐式Euler方法

研究随机延迟微分方程数值解具有重要的意义,目前已有显式和半隐式两种数值方法,还没有全隐式的数值方法.本文构造了一种全隐式Euler方法,在该方法中用一些截断的随机变量代替维纳过程增量△W<,n>,接着证明了全隐式方法是1/2阶收敛的并通过数值实验验证了该方法的收敛性.最后,用数值实验表明在某些情况下全隐式方法的稳定性比半隐式方法好一些.     

Singly Diagonally Implicit Runge–Kutta Methods with an Explicit First Stage

The purpose of this paper is to construct methods for solving stiff ODEs, in particular singular perturbation problems. We consider embedded pairs of singly diagonally implicit Runge–Kutta methods with an explicit first stage (ESDIRKs). Stiffly accurate pairs of order 3/2, 4/3 and 5/4 are constructed.     

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.     

An exponential high‐order compact ADI method for 3D unsteady convection–diffusion problems

In this article, we develop an exponential high order compact alternating direction implicit (EHOC ADI) method for solving three dimensional (3D) unsteady convection–diffusion equations. The method, which requires only a regular seven‐point 3D stencil similar to that in the standard second‐order methods, is second order accurate in time and fourth‐order accurate in space and unconditionally stable. The resulting EHOC ADI scheme in each alternating direction implicit (ADI) solution step corresponding to a strictly diagonally dominant matrix equation can be solved by the application of the one‐dimensional tridiagonal Thomas algorithm with a considerable saving in computing time. Numerical experiments for three test problems are carried out to demonstrate the performance of the present method and to compare it with the classical Douglas–Gunn ADI method and the Karaa's high‐order compact ADI method. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Asymptotic preserving time‐discretization of optimal control problems for the Goldstein–Taylor model

We consider the development of implicit‐explicit time integration schemes for optimal control problems governed by the Goldstein–Taylor model. In the diffusive scaling, this model is a hyperbolic approximation to the heat equation. We investigate the relation of time integration schemes and the formal Chapman–Enskog‐type limiting procedure. For the class of stiffly accurate implicit–explicit Runge–Kutta methods, the discrete optimality system also provides a stable numerical method for optimal control problems governed by the heat equation. Numerical examples illustrate the expected behavior. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1770–1784, 2014     

Numerical integration of Hamiltonian problems by G-symplectic methods

It is the purpose of this paper to consider the employ of General Linear Methods (GLMs) as geometric numerical solvers for the treatment of Hamiltonian problems. Indeed, even if the numerical flow generated by a GLM cannot be symplectic, we exploit here a concept of near conservation for such methods which, properly combined with other desirable features (such as symmetry and boundedness of parasitic components), allows to achieve an accurate conservation of the Hamiltonian. In this paper we focus our attention on the connection between order of convergence and Hamiltonian deviation by multivalue methods. Moreover, we derive a semi-implicit GLM which results competitive to symplectic Runge-Kutta methods, which are notoriously implicit.     

On the construction of stiffly accurate and B-stable Runge-Kutta methods

In the solution of stiff ODEs and especially DAEs it is desirable that the method used is stiffly accurate and B-stable. In this paper guidelines for the construction of Runge-Kutta methods with these properties are presented.     

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.     

Second-order time-integration in inelastic dynamical systems

In structural dynamics the initial boundary value problem has to be solved in the space and time domain. The spatial discretization is done using finite elements. For the temporal discretization two classes of Runge-Kutta methods and the generalized-α method are compared. The representatives for the Runge-Kutta methods are diagonally implicit Runge-Kutta schemes (DIRK) and diagonally implicit Runge-Kutta-Nyström schemes (DIRKN). (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Singly diagonally implicit runge‐kutta method for time‐dependent reaction‐diffusion equation

In this article, an efficient fourth‐order accurate numerical method based on Padé approximation in space and singly diagonally implicit Runge‐Kutta method in time is proposed to solve the time‐dependent one‐dimensional reaction‐diffusion equation. In this scheme, we first approximate the spatial derivative using the second‐order central finite difference then improve it to fourth‐order by applying Padé approximation. A three stage fourth‐order singly diagonally implicit Runge‐Kutta method is then used to solve the resulting system of ordinary differential equations. It is also shown that the scheme is unconditionally stable, and is suitable for stiff problems. Several numerical examples are solved by the scheme and the efficiency and accuracy of the new scheme are compared with two widely used high‐order compact finite difference methods. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1423–1441, 2011     

A Reliable Error Estimation for Diagonally Implicit Multistage Integration Methods

A new approach to the estimation of the local discretization error for diagonally implicit multistage integration methods (DIMSIMs) is described. The error estimates that are obtained are very accurate and very reliable for both explicit and implicit methods for any stepsize pattern.This revised version was published online in October 2005 with corrections to the Cover Date.     

Stability analysis and classification of Runge–Kutta methods for index 1 stochastic differential-algebraic equations with scalar noise

The problem of solving stochastic differential-algebraic equations (SDAEs) of index 1 with a scalar driving Wiener process is considered. Recently, the authors have proposed a class of stiffly accurate stochastic Runge–Kutta (SRK) methods that do not involve any pseudo-inverses or projectors for the numerical solution of the problem. Based on this class of approximation methods, classifications for the coefficients of stiffly accurate SRK methods attaining strong order 0.5 as well as strong order 1.0 are calculated. Further, the mean-square stability of the considered class of SRK methods is analyzed. As the main result, families of A-stable efficient order 0.5 and 1.0 stiffly accurate SRK methods with a minimal number of stages for SDEs as well as for SDAEs are presented.