General Modified Split-Step Balanced Methods for Stiff Stochastic Differential Equations

A class of general modified split-step balanced methods proposed in the paper can be applied to solve stiff stochastic differential systems with m-dimensional multiplicative noise. Compared to some other already reported split-step balanced methods, the drift increment function of the methods can be taken from any chosen ane-step ordinary differential equations （ODEs） solver. The schemes is proved to be strong convergent with order one. For the mean-square stability analysis, the investigation is confined to two cases. Some numerical experiments are reported to testify the performance and the effectiveness of the methods.

Modeling Fast Diffusion Processes in Time Integration of Stiff Stochastic Differential Equations

Numerical algorithms for stiff stochastic differential equations are developed using lin-ear approximations of the fast diffusion processes,under the assumption of decoupling between fast and slow processes.Three numerical schemes are proposed,all of which are based on the linearized formulation albeit with different degrees of approximation.The schemes are of comparable complexity to the classical explicit Euler-Maruyama scheme but can achieve better accuracy at larger time steps in stiff systems.Convergence analysis is conducted for one of the schemes,that shows it to have a strong convergence order of 1/2 and a weak convergence order of 1.Approximations arriving at the other two schemes are discussed.Numerical experiments are carried out to examine the convergence of the schemes proposed on model problems.

GENERAL FULL IMPLICIT STRONG TAYLOR APPROXIMATIONS FOR STIFF STOCHASTIC DIFFERENTIAL EQUATIONS

In this paper,we present the backward stochastic Taylor expansions for a Ito process,including backward Ito-Taylor expansions and backward Stratonovich-Taylor expansions.We construct the general full implicit strong Taylor approximations(including Ito-Taylor and Stratonovich-Taylor schemes)with implicitness in both the deterministic and the stochastic terms for the stiff stochastic differential equations(SSDE)by employing truncations of backward stochastic Taylor expansions.We demonstrate that these schemes will converge strongly with corresponding order 1,2,3,....Mean-square stability has been investigated for full implicit strong Stratonovich-Taylor scheme with order 2,and it has larger meansquare stability region than the explicit and the semi-implicit strong Stratonovich-Taylor schemes with order 2.We can improve the stability of simulations considerably without too much additional computational effort by using our full implicit schemes.The full implicit strong Taylor schemes allow a larger range of time step sizes than other schemes and are suitable for SSDE with stiffness on both the drift and the diffusion terms.Our numerical experiment show these points.

MODIFIED PARALLEL ROSENBROCK METHODS FOR STIFF DIFFERENTIAL EQUATIONS

Presents a class of modified parallel Rosenbrock methods (MPROW) which possesses more free parameters to improve further the various properties of the methods and will be similarly written as MPROW. Information on parallel Rosenbrock methods; Convergence and stability analysis; Discussion on two-stage third-order methods.

Comparison of some atmospheric chemical modeling schemes

The comparison of five atmospheric chemical modeling schemes was presented from accuracy,efficiency and progamming in term of Carbon bond Mechanism IV of atmospheric chemical reactions. The major results were as follows. The classical Gear scheme can provide an accurate solution and it is easy for programming by a computer automatically. The sparse matrix Gear type scheme can also provide an accurate solution but much faster than classical Gear scheme. QSSA,EBI and hybrid schemes can run with much longer time step without sacrificing of accuracy, therefore, much efficiently. If analytical solution is obtained by EBI scheme the accuracy and efficiency are much better.

Compensated split-step balanced methods for nonlinear stiff SDEs with jump-diffusion and piecewise continuous arguments

This paper deals with numerical solutions of nonlinear stiff stochastic differential equations with jump-diffusion and piecewise continuous arguments.By combining compensated split-step methods and balanced methods,a class of compensated split-step balanced(CSSB)methods are suggested for solving the equations.Based on the one-sided Lipschitz condition and local Lipschitz condition,a strong convergence criterion of CSSB methods is derived.It is proved under some suitable conditions that the numerical solutions produced by CSSB methods can preserve the mean-square exponential stability of the corresponding analytical solutions.Several numerical examples are presented to illustrate the obtained theoretical results and the effectiveness of CSSB methods.Moreover,in order to show the computational advantage of CSSB methods,we also give a numerical comparison with the adapted split-step backward Euler methods with or without compensation and tamed explicit methods.

CHEBYSHEV METHODS WITH DISCRETE NOISE:THE T-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.