SERK2v2: A new second‐order stabilized explicit Runge‐Kutta method for stiff problems

Traditionally, explicit numerical algorithms have not been used with stiff ordinary differential equations (ODEs) due to their stability. Implicit schemes are usually very expensive when used to solve systems of ODEs with very large dimension. Stabilized Runge‐Kutta methods (also called Runge–Kutta–Chebyshev methods) were proposed to try to avoid these difficulties. The Runge–Kutta methods are explicit methods with extended stability domains, usually along the negative real axis. They can easily be applied to large problem classes with low memory demand, they do not require algebra routines or the solution of large and complicated systems of nonlinear equations, and they are especially suited for discretizations using the method of lines of two and three dimensional parabolic partial differential equations. In Martín‐Vaquero and Janssen [Comput Phys Commun 180 (2009), 1802–1810], we showed that previous codes based on stabilized Runge–Kutta algorithms have some difficulties in solving problems with very large eigenvalues and we derived a new code, SERK2, based on sixth‐order polynomials. Here, we develop a new method based on second‐order polynomials with up to 250 stages and good stability properties. These methods are efficient numerical integrators of very stiff ODEs. Numerical experiments with both smooth and nonsmooth data support the efficiency and accuracy of the new algorithms when compared to other well‐known second‐order methods such as RKC and ROCK2. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Nonlinear iterative operator splitting methods and applications for nonlinear differential equations

In this paper we discuss decomposition methods that are used for solving nonlinear differential equations. The motivation arose from the need to decouple nonlinear operator equations into simpler, computable operator equations [1]; [2]. We consider iterative operator-splitting methods for the decoupling of the differential equations and we apply iterative steps to achieve linearisation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Variable-Order ESIRK Methods for Stiff Differential Equations

ESIRK methods (Effective order Singly-Implicit Runge–Kutta methods) have been shown to be efficient for the numerical solution of stiff differential equations. In this paper, we consider a new implementation of these methods with a variable order strategy. We show that the efficiency of the ESIRK method for stiff problems is improved by using the proposed variable order schemes.     

Rational Krylov methods in exponential integrators for European option pricing

The aim of this paper is to analyze efficient numerical methods for time integration of European option pricing models. When spatial discretization is adopted, the resulting problem consists of an ordinary differential equation that can be approximated by means of exponential Runge–Kutta integrators, where the matrix‐valued functions are computed by the so‐called shift‐and‐invert Krylov method. To our knowledge, the use of this numerical approach is innovative in the framework of option pricing, and it reveals to be very attractive and efficient to solve the problem at hand. In this respect, we propose some a posteriori estimates for the error in the shift‐and‐invert approximation of the core‐functions arising in exponential integrators. The effectiveness of these error bounds is tested on several examples of interest. They can be adopted as a convenient stopping criterion for implementing the exponential Runge–Kutta algorithm in order to perform time integration. Copyright © 2013 John Wiley & Sons, Ltd.     

Runge–Kutta methods adapted to the numerical integration of oscillatory problems

New Runge–Kutta methods specially adapted to the numerical integration of IVPs with oscillatory solutions are obtained. The coefficients of these methods are frequency-dependent such that certain particular oscillatory solutions are computed exactly (without truncation errors). Based on the B-series theory and on the rooted trees we derive the necessary and sufficient order conditions for this class of RK methods. With the help of these order conditions we construct explicit methods (up to order 4) as well as pairs of embedded RK methods of orders 4 and 3. Some numerical examples show the excellent behaviour when they compete with classical RK methods.     

The Runge–Kutta method for hybrid fuzzy differential equations

In this paper we study numerical methods for addressing hybrid fuzzy differential equations by an application of the Runge–Kutta method for fuzzy differential equations using the Seikkala derivative. We state a convergence result and give a numerical example to illustrate the theory.     

飞机辅助动力装置引气特性计算方法

作为飞机环控系统与主发动机起动的气源,以目前广泛应用的带负载压气机结构APU(Auxiliary Power Unit)为研究对象,进行引气特性计算模型与计算方法研究。首先介绍了APU结构与引气工作特点,然后分析了建模时喘振控制阀SCV(Surge Control Valve)控制方法与APU共同工作机理,最后采用部件法建立了该类型APU引气计算数学模型。以某型APU为对象进行数值仿真并与实际试车数据比较,计算误差小于3%,表明所采用的建模方法是正确的,所建立的模型能够满足工程需求。      

Second-order accurate projective integrators for multiscale problems

We introduce new projective versions of second-order accurate Runge–Kutta and Adams–Bashforth methods, and demonstrate their use as outer integrators in solving stiff differential systems. An important outcome is that the new outer integrators, when combined with an inner telescopic projective integrator, can result in fully explicit methods with adaptive outer step size selection and solution accuracy comparable to those obtained by implicit integrators. If the stiff differential equations are not directly available, our formulations and stability analysis are general enough to allow the combined outer–inner projective integrators to be applied to legacy codes or perform a coarse-grained time integration of microscopic systems to evolve macroscopic behavior, for example.     

Second derivative general linear methods

A class of second derivative Runge Kutta methods (SD-RKM) which has a simple transformation to the general linear methods (GLM) are employed for the numerical integration of stiff initial value problems (IVPs) in ordinary differential equations (ODEs). The new GLM which incorporates second derivative (SD) terms have the advantage of higher order L-stable methods for a given number of stages compared with the classical GLM.     

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.

     

Adaptive stepsize based on control theory for stochastic differential equations

The numerical solution of stochastic differential equations (SDEs) has been focussed recently on the development of numerical methods with good stability and order properties. These numerical implementations have been made with fixed stepsize, but there are many situations when a fixed stepsize is not appropriate. In the numerical solution of ordinary differential equations, much work has been carried out on developing robust implementation techniques using variable stepsize. It has been necessary, in the deterministic case, to consider the “best” choice for an initial stepsize, as well as developing effective strategies for stepsize control—the same, of course, must be carried out in the stochastic case.

In this paper, proportional integral (PI) control is applied to a variable stepsize implementation of an embedded pair of stochastic Runge–Kutta methods used to obtain numerical solutions of nonstiff SDEs. For stiff SDEs, the embedded pair of the balanced Milstein and balanced implicit method is implemented in variable stepsize mode using a predictive controller for the stepsize change. The extension of these stepsize controllers from a digital filter theory point of view via PI with derivative (PID) control will also be implemented. The implementations show the improvement in efficiency that can be attained when using these control theory approaches compared with the regular stepsize change strategy.     

An improved class of generalized Runge–Kutta methods for stiff problems. Part I: The scalar case

A new family of p-stage methods for the numerical integration of some scalar equations and systems of ODEs is proposed. These methods can be seen as a generalization of the explicit p-stage Runge–Kutta ones, while providing better order and stability results. We will show in this first part that, at the cost of losing linearity in the formulas, it is possible to obtain explicit A-stable and L-stable methods for the numerical integration of scalar autonomous ODEs. Scalar autonomous ODEs are of very little interest in current applications. However, be begin studying this kind of problems because most of the work can be easily extended to a more general situation. In fact, we will show in a second part (entitled ‘The separated system case'), that it is possible to generalize our methods so that they can be applied to some non-autonomous scalar ODEs and systems. We will obtain linearly implicit L-stable methods which do not require Jacobian evaluations. In both parts, some numerical examples are discussed in order to show the good performance of the new schemes.     

B-Theory of Runge-Kutta methods for stiff Volterra functional differential equations

B-stability and B-convergence theories of Runge-Kutta methods for nonlinear stiff Volterra func-tional differential equations(VFDEs)are established which provide unified theoretical foundation for the studyof Runge-Kutta methods when applied to nonlinear stiff initial value problems(IVPs)in ordinary differentialequations(ODEs),delay differential equations(DDEs),integro-differential equatioons(IDEs)and VFDEs of     

Stein implicit Runge-Kutta methods with high stage order for large-scale ordinary differential equations

The Runge-Kutta method is one of the most popular implicit methods for the solution of stiff ordinary differential equations. For large problems, the main drawback of such methods is the cost required at each integration step for computing the solution of a nonlinear system of equations. In this paper, we propose to reduce the cost of the computation by transforming the linear systems arising in the application of Newton's method to Stein matrix equations. We propose an iterative projection method onto block Krylov subspaces for solving numerically such Stein matrix equations. Numerical examples are given to illustrate the performance of our proposed method.     

WENO schemes with Lax–Wendroff type time discretizations for Hamilton–Jacobi equations

In this paper, a class of weighted essentially non-oscillatory (WENO) schemes with a Lax–Wendroff time discretization procedure, termed WENO-LW schemes, for solving Hamilton–Jacobi equations is presented. This is an alternative method for time discretization to the popular total variation diminishing (TVD) Runge–Kutta time discretizations. We explore the possibility in avoiding the nonlinear weights for part of the procedure, hence reducing the cost but still maintaining non-oscillatory properties for problems with strong discontinuous derivative. As a result, comparing with the original WENO with Runge–Kutta time discretizations schemes (WENO-RK) of Jiang and Peng [G. Jiang, D. Peng, Weighted ENO schemes for Hamilton–Jacobi equations, SIAM J. Sci. Comput. 21 (2000) 2126–2143] for Hamilton–Jacobi equations, the major advantages of WENO-LW schemes are more cost effective for certain problems and their compactness in the reconstruction. Extensive numerical experiments are performed to illustrate the capability of the method.     

Numerical transformation methods: a constructive approach

Within group invariance theory we consider a constructive approach in order to define numerical transformation methods. These are initial-value methods for the solution of boundary value problems governed by ordinary differential equations. Here we consider the class of free boundary value problems governed by the most general second-order equation in normal form. For this class of problems the main theorem is concerned with the definition of an iterative transformation method. The definition of a noniterative method, applicable to a subclass of the original class of problems, follows as a corollary. Therefore, the proposed constructive approach allows us to establish a unifying framework for noniterative and iterative transformation methods.     

Some new additive Runge–Kutta methods and their applications

We propose some new additive Runge–Kutta methods of orders ranging from 2 to 4 that may be used for solving some nonlinear system of ODEs, especially for the temporal discretization of some nonlinear systems of PDEs with constraints. Only linear ODEs or PDEs need to be solved at each time step with these new methods.     

Error analysis for full discretizations of quasilinear parabolic problems on evolving surfaces

Convergence results are shown for full discretizations of quasilinear parabolic partial differential equations on evolving surfaces. As a semidiscretization in space the evolving surface finite element method is considered, using a regularity result of a generalized Ritz map, optimal order error estimates for the spatial discretization is shown. Combining this with the stability results for Runge–Kutta and backward differentiation formulae time integrators, we obtain convergence results for the fully discrete problems. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1200–1231, 2016     

New splitting methods for two-dimensional evolutionary equations

New second- and third-order splitting methods are proposed for evolutionary-type partial differential equations in a two-dimensional space. These methods are derived on the basis of diagonally implicit methods applied to the numerical analysis of stiff ordinary differential equations. The splitting methods are found to be absolutely unconditionally stable. Test calculations are presented.