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.     

Input-to-state stability of Runge–Kutta methods for nonlinear control systems

In the paper we introduce input-to-state stability (ISS) of Runge–Kutta methods for control systems. The ISS properties of Runge–Kutta methods are studied for linear control systems and nonlinear control systems, respectively. The previously reported results in literature are special cases of ISS of Runge–Kutta methods.     

A Runge–Kutta type modified Landweber method for nonlinear ill-posed operator equations

In this paper, we investigate the convergence behavior of a Runge–Kutta type modified Landweber method for nonlinear ill-posed operator equations. In order to improve the stability and convergence of the Landweber iteration, a 2-stage Gauss-type Runge–Kutta method is applied to the continuous analogy of the modified Landweber method, to give a new modified Landweber method, called R–K type modified Landweber method. Under some appropriate conditions, we prove the convergence of the proposed method. We conclude with a numerical example confirming the theoretical results, including comparisons to the modified Landweber iteration.     

Runge–Kutta methods for affinely controlled nonlinear systems

Rooted tree analysis is adapted from stochastic differential equations to derive systematically general Runge–Kutta methods for deterministic affinely controlled nonlinear systems. Order conditions are found and some specific coefficients for second- and third-order methods are determined, which are then used for simulations compared with the Taylor methods for affinely controlled nonlinear systems derived by Grüne and Kloeden.     

Enumeration results for Runge–Kutta methods for index 2 DAEs

Recursive formulas are presented that give the number of order conditions for single-step Runge–Kutta methods for index 2 DAEs.     

Structure preservation of exponentially fitted Runge–Kutta methods

The preservation of some structure properties of the flow of differential systems by numerical exponentially fitted Runge–Kutta (EFRK) methods is considered. A complete characterisation of EFRK methods that preserve linear or quadratic invariants is given and, following the approach of Bochev and Scovel [On quadratic invariants and symplectic structure, BIT 34 (1994) 337–345], the sufficient conditions on symplecticity of EFRK methods derived by Van de Vyver [A fourth-order symplectic exponentially fitted integrator, Comput. Phys. Comm. 174 (2006) 255–262] are obtained. Further, a family of symplectic EFRK two-stage methods with order four has been derived. It includes the symplectic EFRK method proposed by Van de Vyver as well as a collocation method at variable nodes that can be considered as the natural collocation extension of the classical RK Gauss method. Finally, the results of some numerical experiments are presented to compare the relative merits of several fitted and nonfitted fourth-order methods in the integration of oscillatory systems.     

Weak second order S-ROCK methods for Stratonovich stochastic differential equations

It is well known that the numerical solution of stiff stochastic ordinary differential equations leads to a step size reduction when explicit methods are used. This has led to a plethora of implicit or semi-implicit methods with a wide variety of stability properties. However, for stiff stochastic problems in which the eigenvalues of a drift term lie near the negative real axis, such as those arising from stochastic partial differential equations, explicit methods with extended stability regions can be very effective. In the present paper our aim is to derive explicit Runge–Kutta schemes for non-commutative Stratonovich stochastic differential equations, which are of weak order two and which have large stability regions. This will be achieved by the use of a technique in Chebyshev methods for ordinary differential equations.     

Runge–Kutta methods and viscous wave equations

We study the numerical time integration of a class of viscous wave equations by means of Runge–Kutta methods. The viscous wave equation is an extension of the standard second-order wave equation including advection–diffusion terms differentiated in time. The viscous wave equation can be very stiff so that for time integration traditional explicit methods are no longer efficient. A-Stable Runge–Kutta methods are then very good candidates for time integration, in particular diagonally implicit ones. Special attention is paid to the question how the A-Stability property can be translated to this non-standard class of viscous wave equations.      

Complexity and effective dimension of discrete Lévy areas

Discretisation methods to simulate stochastic differential equations belong to the main tools in mathematical finance. For Itô processes, there exist several Euler- or Runge–Kutta-like methods which are analogues of well-known approximation schemes in the nonstochastic case. In the multidimensional case, there appear several difficulties, caused by the mixed second order derivatives. These mixed terms (or more precisely their differences) correspond to special random variables called Lévy stochastic area terms. In the present paper, we compare three approximation methods for such random variables with respect to computational complexity and the so-called effective dimension.     

A note on iterated splitting schemes

Iterated splittings seem attractive in view of consistency and local accuracy. In this note it will be shown, however, that for stiff systems the stability properties are quite poor. Specific Runge–Kutta implementations can improve stability, but this leads to classes of methods that are better studied in their own right.     

Second order Runge–Kutta methods for Stratonovich stochastic differential equations

The weak approximation of the solution of a system of Stratonovich stochastic differential equations with a m–dimensional Wiener process is studied. Therefore, a new class of stochastic Runge–Kutta methods is introduced. As the main novelty, the number of stages does not depend on the dimension m of the driving Wiener process which reduces the computational effort significantly. The colored rooted tree analysis due to the author is applied to determine order conditions for the new stochastic Runge–Kutta methods assuring convergence with order two in the weak sense. Further, some coefficients for second order stochastic Runge–Kutta schemes are calculated explicitly. AMS subject classification (2000)  65C30, 65L06, 60H35, 60H10     

A fourth-order Runge–Kutta method based on BDF-type Chebyshev approximations

In this paper we consider a new fourth-order method of BDF-type for solving stiff initial-value problems, based on the interval approximation of the true solution by truncated Chebyshev series. It is shown that the method may be formulated in an equivalent way as a Runge–Kutta method having stage order four. The method thus obtained have good properties relatives to stability including an unbounded stability domain and large α$\alpha$-value concerning A(α)$A(\alpha )$-stability. A strategy for changing the step size, based on a pair of methods in a similar way to the embedding pair in the Runge–Kutta schemes, is presented. The numerical examples reveals that this method is very promising when it is used for solving stiff initial-value problems.     

Diagonally implicit Runge–Kutta methods for 3D shallow water applications

We construct A‐stable and L‐stable diagonally implicit Runge–Kutta methods of which the diagonal vector in the Butcher matrix has a minimal maximum norm. If the implicit Runge–Kutta relations are iteratively solved by means of the approximately factorized Newton process, then such iterated Runge–Kutta methods are suitable methods for integrating shallow water problems in the sense that the stability boundary is relatively large and that the usually quite fine vertical resolution of the discretized spatial domain is not involved in the stability condition. This revised version was published online in June 2006 with corrections to the Cover Date.     

Mean-square stability of second-order Runge–Kutta methods for multi-dimensional linear stochastic differential systems

In this paper, the mean-square stability of second-order Runge–Kutta schemes for multi-dimensional linear stochastic differential systems is studied. Motivated by the work of Tocino [Mean-square stability of second-order Runge–Kutta methods for stochastic differential equations, J. Comput. Appl. Math. 175 (2005) 355–367] and Saito and Mitsui [Mean-square stability of numerical schemes for stochastic differential systems, in: International Conference on SCIentific Computation and Differential Equations, July 29–August 3 2001, Vancouver, British Columbia, Canada] we investigate the mean-square stability of second-order Runge–Kutta schemes for multi-dimensional linear stochastic differential systems with one multiplicative noise. Stability criteria are established and numerical examples that confirm the theoretical results are also presented.     

Weak second-order stochastic Runge–Kutta methods for non-commutative stochastic differential equations

A new explicit stochastic Runge–Kutta scheme of weak order 2 is proposed for non-commutative stochastic differential equations (SDEs), which is derivative-free and which attains order 4 for ordinary differential equations. The scheme is directly applicable to Stratonovich SDEs and uses 2m-1$2m-1$ random variables for one step in the m-dimensional Wiener process case. It is compared with other derivative-free and weak second-order schemes in numerical experiments.     

Beyond conventional Runge–Kutta methods in numerical integration of ODEs and DAEs by use of structures and local models

There are two parts in this paper. In the first part we consider an overdetermined system of differential-algebraic equations (DAEs). We are particularly concerned with Hamiltonian and Lagrangian systems with holonomic constraints. The main motivation is in finding methods based on Gauss coefficients, preserving not only the constraints, symmetry, symplecticness, and variational nature of trajectories of holonomically constrained Hamiltonian and Lagrangian systems, but also having optimal order of convergence. The new class of (s,s)$(s,s)$-Gauss–Lobatto specialized partitioned additive Runge–Kutta (SPARK) methods uses greatly the structure of the DAEs and possesses all desired properties. In the second part we propose a unified approach for the solution of ordinary differential equations (ODEs) mixing analytical solutions and numerical approximations. The basic idea is to consider local models which can be solved efficiently, for example analytically, and to incorporate their solution into a global procedure based on standard numerical integration methods for the correction. In order to preserve also symmetry we define the new class of symmetrized Runge–Kutta methods with local model (SRKLM).     

Derivation of continuous explicit two-step Runge–Kutta methods of order three

We describe a construction of continuous extensions to a new representation of two-step Runge–Kutta methods for ordinary differential equations. This representation makes possible the accurate and reliable estimation of local discretization error, facilitates the efficient implementation of these methods in variable stepsize environment, and adapts readily to the numerical solution of a class of delay differential equations. A number of numerical tests carried out on the obtained methods of order 3 with quadratic interpolants show their efficiency and robust performance which allow them to compete with the state-of-the-art dde23 code from Matlab.     

Runge–Kutta–collocation methods for systems of functional–differential and functional equations

Systems of functional–differential and functional equations occur in many biological, control and physics problems. They also include functional–differential equations of neutral type as special cases. Based on the continuous extension of the Runge–Kutta method for delay differential equations and the collocation method for functional equations, numerical methods for solving the initial value problems of systems of functional–differential and functional equations are formulated. Comprehensive analysis of the order of approximation and the numerical stability are presented.     

Runge-Kutta methods without order reduction for linear initial boundary value problems

Summary. It is well-known the loss of accuracy when a Runge–Kutta method is used together with the method of lines for the full discretization of an initial boundary value problem. We show that this phenomenon, called order reduction, is caused by wrong boundary values in intermediate stages. With a right choice, the order reduction can be avoided and the optimal order of convergence in time is achieved. We prove this fact for time discretizations of abstract initial boundary value problems based on implicit Runge–Kutta methods. Moreover, we apply these results to the full discretization of parabolic problems by means of Galerkin finite element techniques. We present some numerical examples in order to confirm that the optimal order is actually achieved. Received July 10, 2000 / Revised version received March 13, 2001 / Published online October 17, 2001     

Stability of IMEX Runge–Kutta methods for delay differential equations

Stability of IMEX (implicit–explicit) Runge–Kutta methods applied to delay differential equations (DDEs) is studied on the basis of the scalar test equation du/dt=λu(t)+μu(t-τ)$\mathrm{d}u/\mathrm{d}t=\lambda u(t)+\mu u(t-\tau )$, where τ$\tau$ is a constant delay and λ,μ$\lambda ,\mu$ are complex parameters. More specifically, P-stability regions of the methods are defined and analyzed in the same way as in the case of the standard Runge–Kutta methods. A new IMEX method which possesses a superior stability property for DDEs is proposed. Some numerical examples which confirm the results of our analysis are presented.