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.     

Convergence on error correction methods for solving initial value problems

Higher-order semi-explicit one-step error correction methods(ECM) for solving initial value problems are developed. ECM provides the excellent convergence O(h^2p+2)$O\left(h^{2p+2}\right)$ one wants to get without any iteration processes required by most implicit type methods. This is possible if one constructs a local approximation having a residual error O(h^p)$O\left(h^p\right)$ on each time step. As a practical example, we construct a local quadratic approximation. Further, it is shown that special choices of parameters for the local quadratic polynomial lead to the known explicit second-order methods which can be improved into a semi-explicit type ECM of the order of accuracy 6$6$. The stability function is also derived and numerical evidences are presented to support theoretical results with several stiff and non-stiff problems. It should be remarked that the ECM approach developed here does not yield explicit methods, but semi-implicit methods of the Rosenbrock type. Both ECM and Rosenbrock’s methods require to solve a few linear systems at each integration step, but the ECM approach involves 2p+2$2p+2$ evaluations of the Jacobian matrix per integration step whereas the Rosenbrock method demands one evaluation only. However, it is much easier to get high order methods by using the ECM approach.     

On the optimization of the initial boundary value problem for a conservation law

In the present note, the theory of shift differentiability for the Cauchy problem is extended to the case of an initial boundary value problem for a conservation law. This result allows to exhibit an Euler-Lagrange equation to be satisfied by the extrema of integral functionals defined on the solutions of initial boundary value problems of this kind.     

Numerical comparison of methods for solving second-order ordinary initial value problems

In this paper, we apply Adomian decomposition method (shortly, ADM) to develop a fast and accurate algorithm of a special second-order ordinary initial value problems. The ADM does not require discretization and consequently of massive computations. This paper is particularly concerned with the ADM and the results obtained are compared with previously known results using the Quintic C²-spline integration methods. The numerical results demonstrate that the ADM is relatively accurate and easily implemented.     

An automatic multistep method for solving stiff initial value problems

A multistep method with matricial coefficients is developed. It can be used to solve stiff initial value problems of the form y′ = Ay + g(x, y). This method bears the nature of the classical Adams—Bashforth—Moulton PC formula and can be shown to be consistent, convergent and A-stable. A careful reformulation of this method legitimatizes the implementation of this algorithm in a variable-step variable-order fashion. Numerical test results from a PECE mode of this method show its possible advantages.     

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 highly unstable initial value boundary value problem

In this paper we consider a two-dimensional diffusion equation on the closed right halfspace satisfying a boundary condition for which the minimum principle fails. As a consequence, the associated Cauchy initial value problem fails to be well-posed. In particular, solutions need not exist and, when they do exist, they may do so for only a finite length of time. Among other things, we provide a necessary and sufficient condition on the initial data in order that the solution exist for all time and remain non-negative.     

Symmetric multistep Obrechkoff methods with zero phase-lag for periodic initial value problems of second order differential equations

In this paper, symmetric multistep Obrechkoff methods of orders 8 and 12, involving a parameter p to solve a special class of second order initial value problems in which the first order derivative does not appear explicitly, are discussed. It is shown that the methods have zero phase-lag when p is chosen as 2π times the frequency of the given initial value problem.     

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.     

A Nyström method for solving 2sth order boundary value problems

A Nyström method is proposed for solving Fredholm integral equations equivalent to special boundary value problems of order 2s. The stability and the convergence of the proposed procedure is proved. Some numerical examples are provided in order to illustrate the accuracy of the method.     

On the stability of two new two-step explicit methods for the numerical integration of second order initial value problem on a variable mesh

We present two novel two-step explicit methods for the numerical solution of the second order initial value problem on a variable mesh. In the case of a constant mesh the method is superstable in the sense of Chawla (1985). Numerical experimentation is provided to verify the stability analysis.     

On the initial value problem and scattering of solutions for the generalized Davey-Stewartson systems

We study the initial value problem of the Davey-Stewartson systems for the elliptic-elliptic and hyperbolic-elliptic cases. The local and global existence and uniqueness of solutions in H^s is shown. Also, we prove that the scattering operator carries a band in H^s into H^s.     

Numerical treatment of sth order boundary value problems by solving second kind Fredholm integral equations

A Nyström method is proposed for solving Fredholm integral equations equivalent to boundary value problems of order s with complete differential equations. The stability and the convergence of the proposed procedure are proved. Some numerical examples are provided in order to illustrate the accuracy of the method and to compare the procedure with some other ones given in the literature.     

Solving a partially singularly perturbed initial value problem on Shishkin meshes

A coupled first order system of one singularly perturbed and one non-perturbed ordinary differential equation with prescribed initial conditions is considered. A Shishkin piecewise uniform mesh is constructed and used, in conjunction with a classical finite difference operator, to form a new numerical method for solving this problem. It is proved that the numerical approximations generated by this method are essentially first order convergent in the maximum norm at all points of the domain, uniformly with respect to the singular perturbation parameter. Numerical results are presented in support of the theory.     

Initial boundary value problem and asymptotic stabilization of the Camassa-Holm equation on an interval

We investigate the nonhomogeneous initial boundary value problem for the Camassa-Holm equation on an interval. We provide a local in time existence theorem and a weak-strong uniqueness result. Next we establish a result on the global asymptotic stabilization problem by means of a boundary feedback law.     

Numerical methods for solving the eigenvalue problem for a positive branch confocal unstable resonator

The resonator problem for a positive branch confocal unstable resonator reduces to a Fredholm homogeneous integral equation of the second kind, whose numerical solution here is based on a sequence of algebraic eigenvalue problems. We compare two algorithms for the solution of an optical resonator problem. These are obtained by (i) successive degenerate kernel approximation by Taylor polynomials of the Fredholm kernel and (ii) Nyström’s method with Simpson’s rule as the subordinate numerical integration method. The numerical results arising from these routines compare well with other published results, and have the added advantage of simplicity and easy adaptability to other resonator problems.     

An initial boundary value problem in ideal Magneto-Hydrodynamics

We consider the initial-boundary value problem for the system of equations of ideal Magneto-Hydrodynamics with a perfectly conducting wall boundary condition. Because of the characteristic boundary, where a loss of regularity in the normal direction to the boundary may occur, the natural functional setting is provided by anisotropic Sobolev spaces, which take account of such singular behavior. We show the existence of the regular solution in the anisotropic Sobolev space for . The result improves previous results of the author and of Yanagisawa - Matsumura. RID="h1" ID="h1"Research of the project Cofin. MURST 1998 "Problems and Methods in the Theory of Hyperbolic Equations".