A family of multistep methods to integrate orbits on spheres

Summary A trajectory problem is an initial value problem where the interest lies in obtaining the curve traced by the solution, rather than in finding the actual correspondence between the values of the parameter and the points on that curve. This paper introduces a family of multi-stage, multi-step numerical methods to integrate trajectory problems whose solution is on a spherical surface. It has been shown that this kind of algorithms has good numerical properties: consistency, stability, convergence and others that are not standard. The latest ones make them a better choice for certain problems.     

On the Effect of Boundary Conditions in Collocation by Polynomial Splines for the Solution of Boundary Value Problems in Ordinary Differential Equations

Consider the numerical solution of a boundary-value problemfor a differential equation of order m using collocation ofa polynomial spline of degree n m on a uniform mesh of sizeh. We describe several collocation schemes which differ onlyin the boundary collocation conditions and which include a "natural"spline collocation scheme. Taking account of derived asymptoticerror bounds most of which are, roughly speaking, of O(hn^12m+1), we discuss the computational effectiveness of the variousschemes.     

Convergence of the Lambert-McLeod trajectory solver and of the celf method

Summary A trajectory problem is an initial value problemd y/dt=f(y),y(0)= where the interest lies in obtaining the curve traced by the solution (the trajectory), rather than in finding the actual correspondanc between values of the parametert and points on that curve. We prove the convergence of the Lambert-McLeod scheme for the numerical integration of trajectory problems. We also study the CELF method, an explicit procedure for the integration in time of semidiscretizations of PDEs which has some useful conservation properties. The proofs rely on the concept of restricted stability introduced by Stetter. In order to show the convergence of the methods, an idea of Strang is also employed, whereby the numerical solution is compared with a suitable perturbation of the theoretical solution, rather than with the theoretical solution itself.     

Method for constructing an external polyhedral estimate of the trajectory tube for a nonlinear dynamic system

A numerical method is proposed for constructing an external polyhedral estimate for the trajectory tube of a nonlinear dynamic system described by a differential inclusion. The method is based on the approximation of cross sections of the trajectory tube (reachable sets) for an auxiliary system described by the convex hull of the graph of the differential inclusion. It produces polyhedral estimates suitable for the direct study of tubes via computer visualization and for the solution of more general problems.

     

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.     

An integrable fourth-order nonlinear evolution equation applied to thermal grooving of metal surfaces

The fourth-order nonlinear partial differential equation forsurface diffusion is approximated by a new integrable nonlinearevolution equation. Exact solutions are obtained for thermalgrooving, subject to boundary conditions representing a sectionof a grain boundary. When the slope m of the groove centre islarge, the linear model grossly overestimates the groove depth.In the linear model dimensionless groove depth increases linearlywith m, but in the nonlinear model it approaches an upper limitA nontrivial similarity solution is found for the limiting caseof a thermal groove whose central slope is vertical.     

On Implicit Ordinary Differential Equations

In a recent paper Fox & Mayers discuss the numerical solutionof implicit ordinary differential equations of the form f(x,y(x), y'(x)) = 0. They find that numerical methods can be veryunreliable near the point where f_y' = 0. In this paper we givea theoretical analysis of the problem which enables us to explainwhen to expect numerical difficulties. We suggest a possibleline of approach for the solution of such problems, and discusssome numerical examples. Research supported by the National Science Foundation, the Officeof Naval Research, the Army Research, and the Air Force Officeof Scientific Research. Travel funding provided by the Universityof Toronto and the British Council.     

Asymptotic Properties of Collocation Matrix Norms 2: Piecewise Polynomial Approximation

This paper considers a matrix related to the solution by piecewisepolynomial collocation using n subintervals of an mth-orderordinary differential boundary-value problem. It is shown thatif the maximum subinterval size tends to zero as the matrix norm tends to the norm of an operator related tothe differential equation, under the assumption that the collocationpoints in each subinterval are assumed to be distributed identicallyand their associated interpolatory quadrature weights are positive.     

LAX CONSTRAINTS IN SEMISIMPLE LIE GROUPS

Instead of studying Lax equations as such, a solution Z of aLax equation is assumed to be given. Then Z is regarded as defininga constraint on a non-autonomous linear differential equationassociated with the Lax equation. In generic cases, quadratureand sometimes algebraic formulae in terms of Z are then provedfor solution x of the linear differential equation, and examplesare given where these formulae lead to new results in higher-ordervariational problems for curves in general semisimple Lie groupsG, extending results previously obtained by different methodsfor the case where G has dimension 3. The new construction isexplored in detail for G = SU(m).     

Pattern formation in a 2D simple chemical system with general orders of autocatalysis and decay

In this paper, we investigate pattern formation in a coupledsystem of reaction–diffusion equations in two spatialdimensions. These equations arise as a model of isothermal chemicalautocatalysis with termination in which the orders of autocatalysisand termination, m and n, respectively, are such that 1 <n < m. We build on the preliminary work by Leach & Wei(2003, Physica D, 180, 185–209) for this coupled systemin one spatial dimension, by presenting rigorous stability analysisand detailed numerical simulations for the coupled system intwo spatial dimensions. We demonstrate that spotty patternsare observed over a wide parameter range.