A stepsize control algorithm for SDEs with small noise based on stochastic Runge–Kutta Maruyama methods

A variable stepsize control algorithm for solution of stochastic differential equations (SDEs) with a small noise parameter ?? is presented. In order to determine the optimal stepsize for each stage of the algorithm, an estimate of the global error is introduced based on the local error of the Stochastic Runge?CKutta Maruyama (SRKM) methods. Based on the relation of the stepsize and the small noise parameter, the local mean-square stochastic convergence order can be different from stage to stage. Using this relation, a strategy for producing and controlling the stepsize in the numerical integration of SDEs is proposed. Numerical experiments on several standard SDEs with small noise are presented to illustrate the effectiveness of this approach.     

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.     

Optimized Runge–Kutta pairs for problems with oscillating solutions

Three types of methods for integrating periodic initial value problems are presented. These methods are (i) phase-fitted, (ii) zero dissipation (iii) both zero dissipative and phase fitted. Some particular modifications of well-known explicit Runge–Kutta pairs of orders five and four are constructed. Numerical experiments show the efficiency of the new pairs in a wide range of oscillatory problems.     

Invariants preserving schemes based on explicit Runge–Kutta methods

Numerical integration of ordinary differential equations with some invariants is considered. For such a purpose, certain projection methods have proved its high accuracy and efficiency. Unfortunately, however, sometimes they can exhibit instability. In this paper, a new, highly efficient projection method is proposed based on explicit Runge–Kutta methods. The key there is to employ the idea of the perturbed collocation method, which gives a unified way to incorporate scheme parameters for projection. Numerical experiments confirm the stability of the proposed method.     

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.     

High-order convergent deferred correction schemes based on parameterized Runge–Kutta–Nyström methods for second-order boundary value problems

Iterated deferred correction is a widely used approach to the numerical solution of first-order systems of nonlinear two-point boundary value problems. Normally, the orders of accuracy of the various methods used in a deferred correction scheme differ by 2 and, as a direct result, each time deferred correction is used the order of the overall scheme is increased by a maximum of 2. In [16], however, it has been shown that there exist schemes based on parameterized Runge–Kutta methods, which allow a higher increase of the overall order. A first example of such a high-order convergent scheme which allows an increase of 4 orders per deferred correction was based on two mono-implicit Runge–Kutta methods. In the present paper, we will investigate the possibility for high-order convergence of schemes for the numerical solution of second-order nonlinear two-point boundary value problems not containing the first derivative. Two examples of such high-order convergent schemes, based on parameterized Runge–Kutta-Nyström methods of orders 4 and 8, are analysed and discussed.     

Optimal discrete and continuous mono‐implicit Runge–Kutta schemes for BVODEs

Recent investigations of discretization schemes for the efficient numerical solution of boundary value ordinary differential equations (BVODEs) have focused on a subclass of the well‐known implicit Runge–Kutta (RK) schemes, called mono‐implicit RK (MIRK) schemes, which have been employed in two software packages for the numerical solution of BVODEs, called TWPBVP and MIRKDC. The latter package also employs continuous MIRK (CMIRK) schemes to provide C ¹ continuous approximate solutions. The particular schemes implemented in these codes come, in general, from multi‐parameter families and, in some cases, do not represent optimal choices from these families. In this paper, several optimization criteria are identified and applied in the derivation of optimal MIRK and CMIRK schemes for orders 1–6. In some cases the schemes obtained result from the analysis of existent multi‐parameter families; in other cases new families are derived from which specific optimal schemes are then obtained. New MIRK and CMIRK schemes are presented which are superior to those currently available. Numerical examples are provided to demonstrate the practical improvements that can be obtained by employing the optimal schemes. This revised version was published online in June 2006 with corrections to the Cover Date.     

Stable Runge–Kutta–Nyström methods for dissipative stiff problems

The definition of stability for Runge–Kutta–Nyström methods applied to stiff second-order in time problems has been recently revised, proving that it is necessary to add a new condition on the coefficients in order to guarantee the stability. In this paper, we study the case of second-order in time problems in the nonconservative case. For this, we construct an $RThe definition of stability for Runge–Kutta–Nystr?m methods applied to stiff second-order in time problems has been recently revised, proving that it is necessary to add a new condition on the coefficients in order to guarantee the stability. In this paper, we study the case of second-order in time problems in the nonconservative case. For this, we construct an -stable Runge–Kutta–Nystr?m method with two stages satisfying this condition of stability and we show numerically the advantages of this new method.This research was supported by MTM 2004-08012 and JCYL VA103/04.     

Evolutionary derivation of Runge–Kutta pairs for addressing inhomogeneous linear problems

Numerical Algorithms - Two new Runge–Kutta (RK) pairs of orders 6(4) and 7(5) are presented for solving numerically the inhomogeneous linear initial value problems with constant coefficients....     

An energy-preserving exponentially-fitted continuous stage Runge–Kutta method for Hamiltonian systems

Recently, the symplectic exponentially-fitted methods for Hamiltonian systems with periodic or oscillatory solutions have been attracting a lot of interest. As an alternative to them, in this paper, we propose a class of energy-preserving exponentially-fitted methods. For this aim, we show sufficient conditions for energy-preservation in terms of the coefficients of continuous stage Runge–Kutta (RK) methods, and extend the theory of exponentially-fitted RK methods in the context of continuous stage RK methods. Then by combining these two theories, we derive second and fourth order energy-preserving exponentially-fitted schemes.     

A new family of Runge–Kutta type methods for the numerical integration of perturbed oscillators

Summary. Our task in this paper is to present a new family of methods of the Runge–Kutta type for the numerical integration of perturbed oscillators. The key property is that those algorithms are able to integrate exactly, without truncation error, harmonic oscillators, and that, for perturbed problems the local error contains the perturbation parameter as a factor. Some numerical examples show the excellent behaviour when they compete with Runge–Kutta–Nystr?m type methods. Received June 12, 1997 / Revised version received July 9, 1998     

Global error estimation based on the tolerance proportionality for some adaptive Runge–Kutta codes

Modern codes for the numerical solution of Initial Value Problems (IVPs) in ODEs are based in adaptive methods that, for a user supplied tolerance δ$\delta$, attempt to advance the integration selecting the size of each step so that some measure of the local error is ?δ$?\delta$. Although this policy does not ensure that the global errors are under the prescribed tolerance, after the early studies of Stetter [Considerations concerning a theory for ODE-solvers, in: R. Burlisch, R.D. Grigorieff, J. Schröder (Eds.), Numerical Treatment of Differential Equations, Proceedings of Oberwolfach, 1976, Lecture Notes in Mathematics, vol. 631, Springer, Berlin, 1978, pp. 188–200; Tolerance proportionality in ODE codes, in: R. März (Ed.), Proceedings of the Second Conference on Numerical Treatment of Ordinary Differential Equations, Humbold University, Berlin, 1980, pp. 109–123] and the extensions of Higham [Global error versus tolerance for explicit Runge–Kutta methods, IMA J. Numer. Anal. 11 (1991) 457–480; The tolerance proportionality of adaptive ODE solvers, J. Comput. Appl. Math. 45 (1993) 227–236; The reliability of standard local error control algorithms for initial value ordinary differential equations, in: Proceedings: The Quality of Numerical Software: Assessment and Enhancement, IFIP Series, Springer, Berlin, 1997], it has been proved that in many existing explicit Runge–Kutta codes the global errors behave asymptotically as some rational power of δ$\delta$. This step-size policy, for a given IVP, determines at each grid point _tn$t_n$ a new step-size _hn+1=h(_tn;δ)$h_{n+1}=h(t_n;\delta )$ so that h(t;δ)$h(t;\delta )$ is a continuous function of t$t$.     

Two-derivative Runge–Kutta methods for PDEs using a novel discretization approach

We develop a novel and general approach to the discretization of partial differential equations. This approach overcomes the rigid restriction of the traditional method of lines (MOL) and provides flexibility in the treatment of spatial discretization. This method is essential for developing efficient numerical schemes for PDEs based on two-derivative Runge–Kutta (TDRK) methods, where the first and second derivatives must be discretized in an efficient way. This is unlikely to be achieved by using MOL. We then apply the explicit TDRK methods to the advection equations and analyze the numerical stability in the linear advection equation case. We conduct numerical experiments on the Burgers’ equation using the TDRK methods developed. We also apply a two-stage semi-implicit TDRK method of order-4 and stage-order-4 to the heat equation. A very significant improvement in the efficiency of this TDRK method is observed when compared to the popular Crank-Nicolson method. This paper is partially based on the work in Tsai’s PhD thesis (2011) [10].     

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.     

An efficient time-step-based self-adaptive algorithm for predictor–corrector methods of Runge–Kutta type

Finding an efficient implementation variant for the numerical solution of problems from computational science and engineering involves many implementation decisions that are strongly influenced by the specific hardware architecture. The complexity of these architectures makes it difficult to find the best implementation variant by manual tuning. For numerical solution methods from linear algebra, auto-tuning techniques based on a global search engine as they are used for ATLAS or FFTW can be used successfully. These techniques generate different implementation variants at installation time and select one of these implementation variants either at installation time or at runtime, before the computation starts. For some numerical methods, auto-tuning at installation time cannot be applied directly, since the best implementation variant may strongly depend on the specific numerical problem to be solved. An example is solution methods for initial value problems (IVPs) of ordinary differential equations (ODEs), where the coupling structure of the ODE system to be solved has a large influence on the efficient use of the memory hierarchy of the hardware architecture. In this context, it is important to use auto-tuning techniques at runtime, which is possible because of the time-stepping nature of ODE solvers.In this article, we present a sequential self-adaptive ODE solver that selects the best implementation variant from a candidate pool at runtime during the first time steps, i.e., the auto-tuning phase already contributes to the progress of the computation. The implementation variants differ in the loop structure and the data structures used to realize the numerical algorithm, a predictor–corrector (PC) iteration scheme with Runge–Kutta (RK) corrector considered here as an example. For those implementation variants in the candidate pool that use loop tiling to exploit the memory hierarchy of a given hardware platform we investigate the selection of tile sizes. The self-adaptive ODE solver combines empirical search with a model-based approach in order to reduce the search space of possible tile sizes. Runtime experiments demonstrate the efficiency of the self-adaptive solver for different IVPs across a range of problem sizes and on different hardware architectures.     

Parallel iterated methods based on variable step‐size multistep Runge—Kutta methods of Radau type for stiff problems

In our previous paper [3], the performance of a variable step‐size implementation of Parallel Iterated Methods based on Multistep Runge–Kutta methods (PIMRK) is far from satisfactory. This is due to the fact that the underlying parameters of the Multistep Runge–Kutta (MRK) method, and the splitting matrices W that are needed to solve the nonlinear system are designed on a fixed step‐size basis. Similar unsatisfactory results based on this method were also noted by Schneider [12], who showed that the method is only suitable when the step‐size does not vary too often. In this paper, we design the Variable step‐size Multistep Runge–Kutta (VMRK) method as the underlying formula for Parallel Iterated methods. The numerical results show that Parallel Iterated Variable step‐size MRK (PIVMRK) methods improve substantially on the PIMRK methods and are usually competitive with Parallel Iterated Runge–Kutta methods (PIRKs). This revised version was published online in June 2006 with corrections to the Cover Date.     

Energy-stable Runge–Kutta schemes for gradient flow models using the energy quadratization approach

In this letter, we present a novel class of arbitrarily high-order and unconditionally energy-stable algorithms for gradient flow models by combining the energy quadratization (EQ) technique and a specific class of Runge–Kutta (RK) methods, which is named the EQRK schemes. First of all, we introduce auxiliary variables to transform the original model into an equivalent system, with the transformed free energy a quadratic functional with respect to the new variables and the modified energy dissipative law is conserved. Then a special class of RK methods is employed for the reformulated system to arrive at structure-preserving time-discrete schemes. Along with rigorous proofs, numerical experiments are presented to demonstrate the accuracy and unconditionally energy-stability of the EQRK schemes.