A class of two-step P-stable methods for the accurate integration of second order periodic initial value problems

In this paper we consider a two parameter family of two-step methods for the accurate numerical integration of second order periodic initial value problems. By applying the methods to the test equation y″ + λ²y = 0, we determine the parameters α, β so that the phase-lag (frequency distortion) of the method is minimal. The resulting method is a P-stable method with a minimal phase-lag λ⁶h⁶/42000. The superiority of the method over the other P-stable methods is illustrated by a comparative study of the phase-lag errors and by illustrating with a numerical example.     

P-stable methods for periodic initial value problems of second order differential equations

A class ofP-stable finite difference methods is discussed for solving initial value problems of second order differential equations which have periodic solutions. The methods depend upon a parameterp>0, and reduce to the classical Störmer-Cowell methods forp=0. It is shown that whenp is chosen for linear problems as the square of the frequency of the periodic solution, the methods areP-stable and for some suitable choice ofp, they have extended finite interval of periodicity.     

High orderP-stable formulae for the numerical integration of periodic initial value problems

Summary Recently there has been considerable interest in the approximate numerical integration of the special initial value problemy=f(x, y) for cases where it is known in advance that the required solution is periodic. The well known class of Störmer-Cowell methods with stepnumber greater than 2 exhibit orbital instability and so are often unsuitable for the integration of such problems. An appropriate stability requirement for the numerical integration of periodic problems is that ofP-stability. However Lambert and Watson have shown that aP-stable linear multistep method cannot have an order of accuracy greater than 2. In the present paper a class of 2-step methods of Runge-Kutta type is discussed for the numerical solution of periodic initial value problems.P-stable formulae with orders up to 6 are derived and these are shown to compare favourably with existing methods.     

P-stable Obrechkoff methods of arbitrary order for second-order differential equations

Following the ideas of Ananthakrishnaiah we develop a family of P-stable Obrechkoff methods of arbitrary even order. The coefficients of these methods follow from a recursive algorithm. It is also shown that the stability functions of the thus obtained methods can be expressed as Padé approximants of the exponential function with a complex argument. A numerical example is given to illustrate the performance of the methods.     

Hybrid numerical methods for periodic initial value problems involving second-order differential equations

Computer simulation of problems in celestial mechanics often leads to the numerical solution of the system of second-order initial value problems with periodic solutions. When conventional methods are applied to obtain the solution, the time increment must be limited to a value of the order of the reciprocal of the frequency of the periodic solution.In this paper hybrid methods of orders four and six which are P-stable are developed. Further, the adaptive hybrid methods of polynomial order four and trigonometric order one have also been discussed. The numerical results for the undamped Duffing equation with a forced harmonic function are listed.     

EfficientP-stable methods for periodic initial value problems

Efficient families ofP-stable formulae are developed for the numerical integration of periodic initial value problems where the required solution has an unknown period. Formulae of orders 4 and 6 requiring respectively 2 and 4 function evaluations per step are derived and some numerical results are given.     

On the numerical integration of nonlinear initial value problems by linear multistep methods

We study the numerical solution of the nonlinear initial value problem $$\left\{ {\begin{array}{*{20}c} {{{du(t)} \mathord{\left/ {\vphantom {{du(t)} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} + Au(t) = f(t),t > 0} \\ {u(0) = c,} \\ \end{array} } \right.$$ whereA is a nonlinear operator in a real Hilbert space. The problem is discretized using linear multistep methods, and we assume that their stability regions have nonempty interiors. We give sharp bounds for the global error by relating the stability region of the method to the monotonicity properties ofA. In particular we study the case whereAu is the gradient of a convex functional φ(u).     

Blended extended linear multistep methods for the accurate numerical integration of stiff initial value problems

A class of blended extended linear multistep methods suitable for the approximate numerical integration of stiff systems of first order ordinary differential equations is described. These methods are formulated as a result of combining the second derivative extended backward differentiation formulae of Cash and the blended linear multistep methods of Skeel and Kong. The new methods combine a high order or accuracy with good stability properties and, as a direct consequence, they are often suitable for the numerical integration of stiff differential systems when high accuracy is requested. In the first part of the present paper we consider the derivation of these new blended methods and give the coefficients and stability regions for formulae of order up to and including 10. In the second half we consider their practical implementation. In particular we describe a variable order/variable step package based on these blended formulae and we evaluate the performance of this package on the well known DETEST test set. It is shown that the new code is reliable on this test set and is competitive with the well known second derivative method of Enright.     

Spurious solutions of numerical methods for initial value problems

It is well known that some numerical methods for initial valueproblems admit spurious limit sets. Here the existence and behaviourof spurious solutions of Runge-Kutta, linear multistep and predictor-correctormethods are studied in the limit as the step-size h0. In particular,it is shown that for ordinary differential equations definedby globally Lipschitz vector fields, spurious fixed points andperiod 2 solutions cannot exist for h arbitrarily small, whilstfor locally Lipschitz vector fields, spurious solutions mayexist for h arbitrarily small, but must become unbounded ash0. The existence of spurious solutions is also studied forvector fields merely assumed to be continuous, and an exampleis given, showing that in this case spurious solutions may remainbounded as h0. It is shown that if spurious fixed points orperiod 2 solutions of continuous problems exist for h arbitrarilysmall, then as h0 spurious solutions either converge to steadysolutions of the underlying differential equation or divergeto infinity. A necessary condition for the bifurcation spurioussolutions from h=0 is derived. To prove the above results forimplicit Runge-Kutta methods, an additional assumption on theiteration scheme used to solve the nonlinear equations definingthe method is needed; an example of a Runge-Kutta method whichgenerates a bounded spurious solution for a smooth problem withh arbitrarily small is given, showing that such an assumptionis necessary. It is also shown that an Adams-Bashforth/Adams-Moultonpredictor-corrector method in PC^m implementation can generatespurious fixed point solutions for any m.     

A generator of high-order embedded P-stable methods for the numerical solution of the Schrödinger equation

A generator of new embedded P-stable methods of order 2n+2, where n is the number of layers used by the embedded methods, for the approximate numerical integration of the one-dimensional Schrödinger equation is developed in this paper. These new methods are called embedded methods because of a simple natural error control mechanism. Numerical results obtained for one-dimensional differential equations of the Schrödinger type show the validity of the developed theory.     

An explicit six-step singularly P-stable Obrechkoff method for the numerical solution of second-order oscillatory initial value problems

Numerical Algorithms - In this paper, we present an explicit six-step singularly P-stable Obrechkoff method of tenth algebraic order for solving second-order linear periodic and oscillatory initial...     

P-stable exponentially-fitted Obrechkoff methods of arbitrary order for second-order differential equations

We consider the construction of P-stable exponentially-fitted symmetric two-step Obrechkoff methods for solving second order differential equations related to an initial value problem. Our approach is based on two ideas: for the exponential fitting, we follow the ideas of Ixaru and Vanden Berghe; for the P-stability we introduce exponentially-fitted Padé approximants to the exponential function. By combining these two ideas, we are able to construct P-stable methods of arbitrary (even) order.     

Parameter-uniform numerical methods for some singularly perturbed nonlinear initial value problems

Parameter-uniform numerical methods for singularly perturbed nonlinear scalar initial value problems are both constructed and analysed in this paper. The conditions on the initial condition for a stable initial layer to form are identified. The character of a stable initial layer in the vicinity of a double root of the reduced algebraic problem is different to the standard layer structures appearing in the neighbourhood of a single stable root of the reduced problem. Results for a problem where two reduced solutions intersect are also discussed. Numerical results are presented to illustrate the theoretical results obtained.     

Adaptive methods for periodic initial value problems of second order differential equations

In this paper numerical methods involving higher order derivatives for the solution of periodic initial value problems of second order differential equations are derived. The methods depend upon a parameter p > 0 and reduce to their classical counter parts as p → 0. The methods are periodically stable when the parameter p is chosen as the square of the frequency of the linear homogeneous equation. The numerical methods involving derivatives of order up to 2q are of polynomial order 2q and trigonometric order one. Numerical results are presented for both the linear and nonlinear problems. The applicability of implicit adaptive methods to linear systems is illustrated.     

A multi-point numerical integrator with trigonometric coefficients for initial value problems with periodic solutions

Based on the collocation technique, we introduced a unifying approach for deriving a family of multi-point numerical integrators with trigonometric coefficients for the numerical solution of periodic initial value problems. A practical 3-point numerical integrator was presented, whose coefficients are generalizations of classical linear multistep methods such that the coefficients are functions of an estimate of the angular frequency ω. The collocation technique yields a continuous method, from which the main and complementary methods are recovered and expressed as a block matrix finite difference formula that integrates a second-order differential equation over non-overlapping intervals without predictors. Some properties of the numerical integrator were investigated and presented. Numerical examples are given to illustrate the accuracy of the method.     

Four-stage symplectic and P-stable SDIRKN methods with dispersion of high order

New SDIRKN methods specially adapted to the numerical integration of second-order stiff ODE systems with periodic solutions are obtained. Our interest is focused on the dispersion (phase errors) of the dominant components in the numerical oscillations when these methods are applied to the homogeneous linear test model. Based on this homogeneous test model we derive the dispersion and P-stability conditions for SDIRKN methods which are assumed to be zero dissipative. Two four-stage symplectic and P-stable methods with algebraic order 4 and high order of dispersion are obtained. One of the methods is symmetric and sixth-order dispersive whereas the other method is nonsymmetric and eighth-order dispersive. These methods have been applied to a number of test problems (linear as well as nonlinear) and some numerical results are presented to show their efficiency when they are compared with other methods derived by Sharp et al. [IMA J. Numer. Anal. 10 (1990) 489–504].     

Discontinuous Galerkin methods for periodic boundary value problems

This article considers the extension of well‐known discontinuous Galerkin (DG) finite element formulations to elliptic problems with periodic boundary conditions. Such problems routinely appear in a number of applications, particularly in homogenization of composite materials. We propose an approach in which the periodicity constraint is incorporated weakly in the variational formulation of the problem. Both H¹ and L² error estimates are presented. A numerical example confirming theoretical estimates is shown. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

On the numerical integration of multi-dimensional,initial boundary value problems for the Euler equations in quasi-linear form

A matricial formalism to solve multi-dimensional initial boundary values problems for hyperbolic equations written in quasi-linear based on the λ scheme approach is presented. The derivation is carried out for nonorthogonal, moving systems of curvilinear coordinates. A uniform treatment of the integration at the boundaries, when the boundary conditions can be expressed in terms of combinations of time or space derivatives of the primitive variables, is also presented. The methodology is validated against a two-dimensional test case, the supercritical flow through the Hobson cascade n.2, and in three-dimensional test cases such as the supersonic flow about a sphere and the flow through a plug nozzle. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 781–814, 1998     

A numerical comparison of Chebyshev methods for solving fourth order semilinear initial boundary value problems

In solving semilinear initial boundary value problems with prescribed non-periodic boundary conditions using implicit-explicit and implicit time stepping schemes, both the function and derivatives of the function may need to be computed accurately at each time step. To determine the best Chebyshev collocation method to do this, the accuracy of the real space Chebyshev differentiation, spectral space preconditioned Chebyshev tau, real space Chebyshev integration and spectral space Chebyshev integration methods are compared in the L² and W^2,2 norms when solving linear fourth order boundary value problems; and in the L^∞([0,T];L²) and L^∞([0,T];W^2,2) norms when solving initial boundary value problems. We find that the best Chebyshev method to use for high resolution computations of solutions to initial boundary value problems is the spectral space Chebyshev integration method which uses sparse matrix operations and has a computational cost comparable to Fourier spectral discretization.     

Contractivity in the numerical solution of initial value problems

Numerische Mathematik - Consider a linear autonomous system of ordinary differential equations with the property that the norm |U(t)| of each solutionU(t) satisfies |U(t)|≦|U(0)| (t≧0)....