A Collocation Solver for Mixed Order Systems of Boundary Value Problems

Ascher, U.; Christiansen, J.; Russell, R. D.

doi:10.2307/2006301

Cited by 115 publications

(140 citation statements)

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“…The simplest type of method is probably collocation [3] where an approximate solution is sought in some finite space subject to the constraint that it satisfy the differential equation at certain specified points. This type of method has been applied very successfully in the case of mixed order systems of boundary value problems; see, for example, [1]. Other methods can be described based on the standard L2-Galerkin approximation [9], or on a combination of this and the collocation approach [7], [13], [25].…”

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confidence: 99%

“…[16] -eu" -xu' = sir2cos TTx + -nxsimrx, a: G (-1,1), u(-l) = -2, w(l) = 0, where e is a parameter. The solution is u(x) -costrx + crí(x/]¡2e )/erf(l/y2e J, which has a spike at x = 0; see [1]. The values of the parameters used in this study are given in Table 2, and we will refer to the corresponding problems as, for example, 1-3, meaning problem I with parameter choice 3, a = 100, x = 0.36388.…”

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confidence: 99%

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A computational study of finite element methods for second order linear two-point boundary value problems

Keast¹,

Fairweather²,

Díaz³

1983

Math. Comp.

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Abstract. A computational study of five finite element methods for the solution of a single second order linear ordinary differential equation subject to general linear, separated boundary conditions is described. In each method, the approximate solution is a piecewise polynomial expressed in terms of a B-spline basis, and is determined by solving a system of linear algebraic equations with an almost block diagonal structure. The aim of the investigation is twofold: to determine if the theoretical orders of convergence of the methods are realized in practice, and to compare the methods on the basis of cost for a given accuracy. In this study three parametrized families of test problems, containing problems of varying degrees of difficulty, are used. The conclusions drawn are rather straightforward. Collocation is the cheapest method for a given accuracy, and the easiest to implement. Also, for solving the linear algebraic equations, the use of a special purpose solver which takes advantage of the structure of the equations is advisable.

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confidence: 99%

mentioning

confidence: 99%

A computational study of finite element methods for second order linear two-point boundary value problems

Keast¹,

Fairweather²,

Díaz³

1983

Math. Comp.

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“…The constraint determines the multiplier by integrating (14) over [0,1]; eliminating λ we obtain the minimizer (12).…”

Section: Adaptivity As a Variational Problemmentioning

confidence: 99%

“…which imposes the condition that the step sizes exactly cover the entire interval [0,1]. It is important to note that the discrete normalization (8) differs from the continuous normalization (6).…”

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Automatic grid control in adaptive BVP solvers

2010

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Grid adaptation in two-point boundary value problems is usually based on mapping a uniform auxiliary grid to the desired nonuniform grid. Here we combine this approach with a new control system for constructing a grid density function φ(x). The local mesh width ∆x j+1/2 = x j+1 − x j with 0 = x 0 < x 1 < · · · < x N = 1 is computed as ∆x j+1/2 = ǫ N /ϕ j+1/2 , where {ϕ j+1/2 } N −1 0 is a discrete approximation to the continuous density function φ(x), representing mesh width variation. The parameter ǫ N = 1/N controls accuracy via the choice of N . For any given grid, a solver provides an error estimate. Taking this as its input, the feedback control law then adjusts the grid, and the interaction continues until the error has been equidistributed. Digital filters may be employed to process the error estimate as well as the density to ensure the regularity of the grid. Once φ(x) is determined, another control law determines N based on the prescribed tolerance tol. The paper focuses on the interaction between control system and solver, and the controller's ability to produce a nearoptimal grid in a stable manner as well as correctly predict how many grid points are needed. Numerical tests demonstrate the advantages of the new control system within the bvpsuite solver, ceteris paribus, for a selection of problems and over a wide range of tolerances. The control system is modular and can be adapted to other solvers and error criteria.

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“…Quasilinearization is applied to reduce the nonlinear problem to a sequence of linear problems (Ascher, Christiansen and Russell [4], deBoor and Swartz [10], Russell and Shampine [40]). This is achieved in our scheme by interpolating f(x, y) by a piecewise linear function of y for fixed x.…”

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confidence: 99%

A Hybrid Asymptotic-Finite Element Method for Stiff Two-Point Boundary Value Problems

Chin¹,

Krasny²

1983

SIAM J. Sci. and Stat. Comput.

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Abstract. An accurate and efficient numerical method has been developed for a nonlinear stiff second order two-point boundary value problem. The scheme combines asymptotic methods with the usual solution techniques for two-point boundary value problems. A new modification of Newton's method or quasilinearization is used to reduce the nonlinear problem to a sequence of linear problems. The resultant linear problem is solved by patching local solutions at the knots or equivalently by projecting onto an affine subset constructed from asymptotic expansions. In this way, boundary layers are naturally incorporated into the approximation. An adaptive mesh is employed to achieve an error of O(1/N2)+O(/e). Here, N is the number of intervals and e << is the singular perturbation parameter. Numerical computations are presented.

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A Collocation Solver for Mixed Order Systems of Boundary Value Problems

Cited by 115 publications

References 19 publications

A computational study of finite element methods for second order linear two-point boundary value problems

A computational study of finite element methods for second order linear two-point boundary value problems

Automatic grid control in adaptive BVP solvers

A Hybrid Asymptotic-Finite Element Method for Stiff Two-Point Boundary Value Problems

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