Continuous extensions to Nyström methods for second order initial value problems

In this work we consider interpolants for Nyström methods, i.e., methods for solving second order initial value problems. We give a short introduction to the theory behind the discrete methods, and extend some of the work to continuous, explicit Nyström methods. Interpolants for continuous, explicit Runge-Kutta methods have been intensively studied by several authors, but there has not been much effort devoted to continuous Nyström methods. We therefore extend some of the work by Owren.     

Superconvergent Nyström and degenerate kernel methods for eigenvalue problems

The Nyström and degenerate kernel methods, based on projections at Gauss points onto the space of (discontinuous) piecewise polynomials of degree ?r-1, for the approximate solution of eigenvalue problems for an integral operator with a smooth kernel, exhibit order 2r. We propose new superconvergent Nyström and degenerate kernel methods that improve this convergence order to 4r for eigenvalue approximation and to 3r for spectral subspace approximation in the case where the kernel is sufficiently smooth. Moreover for a simple eigenvalue, we show that by using an iteration technique, an eigenvector approximation of order 4r can be obtained. The methods introduced here are similar to that studied by Kulkarni in [10] and exhibit the same convergence orders, so a comparison with these methods is worked out in detail. Also, the error terms are analyzed and the obtained methods are numerically tested. Finally, these methods are extended to the case of discontinuous kernel along the diagonal and superconvergence results are also obtained.     

Numerical methods for Fredholm integral equations on the square

In this paper we shall investigate the numerical solution of two-dimensional Fredholm integral equations by Nyström and collocation methods based on the zeros of Jacobi orthogonal polynomials. The convergence, stability and well conditioning of the method are proved in suitable weighted spaces of functions. Some numerical examples illustrate the efficiency of the methods.     

A family of explicit parallel Runge-Kutta-Nyström methods

A procedure for the construction of high-order explicit parallel Runge-Kutta-Nyström (RKN) methods for solving second-order nonstiff initial value problems (IVPs) is analyzed. The analysis reveals that starting the procedure with a reference symmetric RKN method it is possible to construct high-order RKN schemes which can be implemented in parallel on a small number of processors. These schemes are defined by means of a convex combination of k disjoint s_i-stage explicit RKN methods which are constructed by connecting s_i steps of a reference explicit symmetric method. Based on the reference second-order Störmer-Verlet methods we derive a family of high-order explicit parallel schemes which can be implemented in variable-step codes without additional cost. The numerical experiments carried out show that the new parallel schemes are more efficient than some sequential and parallel codes proposed in the scientific literature for solving second-order nonstiff IVPs.     

Analysis of partitioned methods for the Biot System

In this work, we present a comprehensive study of several partitioned methods for the coupling of flow and mechanics. We derive energy estimates for each method for the fully‐discrete problem. We write the obtained stability conditions in terms of a key control parameter defined as a ratio of the coupling strength and the speed of propagation. Depending on the parameters in the problem, give the choice of the partitioned method which allows the largest time step. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1769–1813, 2015     

Numerical analysis of two partitioned methods for uncoupling evolutionary MHD flows

Magnetohydrodynamics (MHD) studies the dynamics of electrically conducting fluids, involving Navier–Stokes (NSE) equations in fluid dynamics and Maxwell equations in eletromagnetism. The physical processes of fluid flows and electricity and magnetism are quite different and numerical simulations of each subprocess can require different meshes, time steps, and methods. In most terrestrial applications, MHD flows occur at low‐magnetic Reynold numbers. We introduce two partitioned methods to solve evolutionary MHD equations in such cases. The methods we study allow us at each time step to call NSE and Maxwell codes separately, each possibly optimized for the subproblem's respective physics. Complete error analysis and computational tests supporting the theory are given.Copyright © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1083–1102, 2014     

A-stable diagonally implicit Runge-Kutta-Nyström methods for parallel computers

In this paper, we study diagonally implicit Runge-Kutta-Nyström methods (DIRKN methods) for use on parallel computers. These methods are obtained by diagonally implicit iteration of fully implicit Runge-Kutta-Nyström methods (corrector methods). The number of iterations is chosen such that the method has the same order of accuracy as the corrector, and the iteration parameters serve to make the method at least A-stable. Since a large number of the stages can be computed in parallel, the methods are very efficient on parallel computers. We derive a number of A-stable, strongly A-stable and L-stable DIRKN methods of orderp withs ^* (p) sequential, singly diagonal-implicit stages wheres ^*(p)=[(p+1)/2] ors ^* (p)=[(p+1)/2]+1,[°] denoting the integer part function.These investigations were supported by the University of Amsterdam with a research grant to enable the author to spend a total of two years in Amsterdam.     

ORDER PROPERTIES AND CONSTRUCTION OF SYMPLECTIC RUNGE-KUTTA METHODS

1. IntroductionFOr a given s stage Runge-Kutta methodwith A = [ail], p = [pl, PZt... 5 P.]T and ac = [afl, ry23... ) %]T / 0, we introduce thefollowing simplifying conditions as in Butcher [1]and make the notational convensionwhere 1 5 m? pi(x), i ~ 1, 2, 3,' ? are arbitrarily given i--th polynomials with the property that pi(0) = 0,Note that B(P), C(P) and D(P) are equivalent to BI,. = 0, CI,P = 0 and DI,. = 0respectively. We shall always denote BI,., CI,., DI,. and VI,. by B, …     

Construction of starting algorithms for the RK-Gauss methods

In this paper starting algorithms for the numerical solution of stage equations in Runge-Kutta-Gauss formulae with 2, 3 and 4 stages are constructed. For each of these formulae, three types of starting algorithms are given according to their requirement of none, one or two additional function evaluations per step. Numerical experiments with Hamiltonian systems are presented to show the superior performance of the new starting algorithms of high order.     

On multistep interval methods for solving the initial value problem

In this paper we shortly complete our previous considerations on interval versions of Adams multistep methods [M. Jankowska, A. Marciniak, Implicit interval multistep methods for solving the initial value problem, Comput. Meth. Sci. Technol. 8(1) (2002) 17–30; M. Jankowska, A. Marciniak, On explicit interval methods of Adams–Bashforth type, Comput. Meth. Sci. Technol. 8(2) (2002) 46–57; A. Marciniak, Implicit interval methods for solving the initial value problem, Numerical Algorithms 37 (2004) 241–251]. It appears that there exist two families of implicit interval methods of this kind. More considerations are dealt with two new kinds of interval multistep methods based on conventional well-known Nyström and Milne–Simpson methods. For these new interval methods we prove that the exact solution of the initial value problem belongs to the intervals obtained. Moreover, we present some estimations of the widths of interval solutions. Some conclusions bring this paper to the end.     

Stochastic partitioned averaged vector field methods for stochastic differential equations with a conserved quantity

In this paper, stochastic differential equations in the Stratonovich sense with a conserved quantity are considered. A stochastic partitioned averaged vector field method is proposed and analyzed. We prove this numerical method is able to preserve the conserved quantity of the original system. Then the convergence analysis is carried out in detail and we derive the method is convergent with order $1$ in the mean-square sense. Finally some numerical examples are reported to verify the effectiveness and flexibility of the proposed method.     

Starting algorithms for low stage order RKN methods

When second order differential equations are solved with Runge-Kutta-Nyström methods, the computational effort is dominated by the cost of solving the nonlinear system. That is why it is important to have good starting values to begin the iterations. In this paper we consider a type of starting algorithms without additional computational cost. We study the general order conditions and the maximum order achieved when the Runge-Kutta-Nyström method satisfies some simplifying assumptions.     

ERKN methods for long-term integration of multidimensional orbital problems

This paper is devoted to introducing ERKN methods for long-term integration of multidimensional orbital problems. For the general multidimensional perturbed oscillators y^″+My=f(t,y)$y^{″}+\mathit{My}=f(t\text{,}y)$ with M∈R^m×m$M\in {\mathbb{R}}^{m\times m}$, the extended Runge–Kutta–Nyström (ERKN) methods are proposed by Wu et al. [X. Wu, X. You, W. Shi, B. Wang, ERKN integrators for systems of oscillatory second-order differential equations, Comput. Phys. Commun. 181 (2010) 1873–1887]. These methods exactly integrate the multidimensional unperturbed oscillators and are highly efficient when the perturbing forces are small. In this paper, we pay attention to the applications of ERKN methods to multidimensional orbital problems. Numerical experiments accompanied demonstrate that for long-term integration of multidimensional orbital problems the multidimensional ERKN methods are more efficient compared with high-quality codes proposed in the scientific literature. In particular, when an orbital problem under consideration is a Hamiltonian system, the symplectic ERKN methods preserve the Hamiltonian very well, and has better accuracy than the high-quality codes with the same computational cost.     

Absolute stability of explicit Runge-Kutta-Nyström methods for y″=ƒ(x,y, y′)

We examine absolute stability of s-stage explicit Runge-Kutta-Nyström (R-K-N) methods of order s for $\text{s=2, 3, 4}\text{for}\text{y″=?(x, y, y′)}$ by applying these methods to the test equation: y″+2λy′+λ²y=0, λ>0. We show the existence of R-K-N methods of orders two, three and four possessing intervals of absolute stability as large as that of explicit Runge-Kutta (r-K) methods of respective orders.     

Twostep-by-twostep continuous PIRKN-type PC methods for nonstiff IVPs

In this paper, we start with the consideration of direct collocation-based Runge-Kutta-Nyström (RKN) methods with continuous output formulas for solving nonstiff initial-value problems (IVPs) for systems of special second-order differential equations y″(t) = f(t, y(t)). At nth step, the continuous output formulas can be used for calculating the step values at (n + 2)th step and the integration processes can be proceeded twostep-by-twostep. In this case, we obtain twostep-by-twostep RKN methods with continuous output formulas (continuous TBTRKN methods). Furthermore, we consider a parallel predictor-corrector (PC) iteration scheme using the continuous TBTRKN methods as corrector methods with predictor methods defined by the continuous output formulas. The resulting twostep-by-twostep parallel-iterated RKN-type PC methods with continuous output formulas (twostep-by-twostep continuous PIRKN-type PC methods or TBTCPIRKN methods) give us a faster integration processes. Numerical comparisons based on the solution of a few widely-used test problems show that the new TBTCPIRKN methods are much more efficient than the well-known PIRKN methods, the famous nonstiff sequential ODEX2, DOP853 codes and comparable with the CPIRKN methods.     

Positive solutions for some non-autonomous Schrödinger-Poisson systems

In this paper we study the Schrödinger-Poisson system

(SP)     

Nyström methods and extrapolation for solving Steklov eigensolutions and its application in elasticity

Based on potential theory, Steklov eigensolutions of elastic problems can be converted into eigenvalue problems of boundary integral equations (BIEs). The kernels of these BIEs are characterized by logarithmic and Hilbert singularities. In this article, the Nyström methods are presented for obtaining eigensolutions (λ⁽ⁱ⁾,u⁽ⁱ⁾), which have to deal with the two kinds of singularities simultaneously. The solutions possess high accuracy orders O(h³) and an asymptotic error expansion with odd powers. Using h³ ‐Richardson extrapolation algorithms, we can greatly improve the accuracy orders to O(h⁵). Furthermore, a generalized Fourier series is constructed by the eigensolutions, and then solving the elasticity displacement and traction problems involves just calculating the coefficients of the series. A class of elasticity problems with boundary Γ is solved with high convergence rate O(h⁵). The efficiency is illustrated by a numerical example. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012     

A Nyström method for a class of Fredholm integral equations of the third kind on unbounded domains

The author proposes a numerical procedure in order to approximate the solution of a class of Fredholm integral equations of the third kind on unbounded domains. The given equation is transformed in a Fredholm integral equation of the second kind. Hence, according to the integration interval, the equation is regularized by means of a suitable one-to-one map or is transformed in a system of two Fredholm integral equations that are subsequently regularized. In both cases a Nyström method is applied, the convergence and the stability of which are proved in spaces of weighted continuous functions. Error estimates and numerical tests are also included.     

Parallelism across the steps in iterated Runge-Kutta methods for stiff initial value problems

For the parallel integration of stiff initial value problems (IVPs) three main approaches can be distinguished: approaches based on parallelism across the problem, on parallelism across the method and on parallelism across the steps. The first type of parallelism does not require special integration methods can be exploited within any available IVP solver. The methodparallel approach received some attention in the case of Runge-Kutta based methods. For these methods, the required number of processors is roughly half the order of the generating Runge-Kutta method and the speed-up with respect to a good sequential IVP solver is about a factor 2. The third type of parallelism (step-parallelism) can be achieved in any IVP solver based on predictor-corrector iteration. Most step-parallel methods proposed so far employ a large number of processors, but lack the property of robustness, due to a poor convergence behaviour in the iteration process. Hence, the effective speed-up is rather poor. The step-parallel iteraction process proposed in the present paper is less massively parallel, but turns out to be sufficiently robust to solve the four-stage Radau IIA corrector used in our experiments within a few effective iterations per step and to achieve speed-up factors up to 10 with respect to the best sequential codes.The research reported in this paper was partly supported by the Technology Foundation (STW) in the Netherlands.     

On higher-order semi-explicit symplectic partitioned Runge-Kutta methods for constrained Hamiltonian systems

Summary. In this paper we generalize the class of explicit partitioned Runge-Kutta (PRK) methods for separable Hamiltonian systems to systems with holonomic constraints. For a convenient analysis of such schemes, we first generalize the backward error analysis for systems in to systems on manifolds embedded in . By applying this analysis to constrained PRK methods, we prove that such methods will, in general, suffer from order reduction as well-known for higher-index differential-algebraic equations. However, this order reduction can be avoided by a proper modification of the standard PRK methods. This modification increases the number of projection steps onto the constraint manifold but leaves the number of force evaluations constant. We also give a numerical comparison of several second, fourth, and sixth order methods. Received May 5, 1995 / Revised version received February 7, 1996