An Adaptive Finite Difference Solver for Nonlinear Two-Point Boundary Problems with Mild Boundary Layers

Lentini, Marianela; Pereyra, Víctor

doi:10.1137/0714006

Cited by 193 publications

(53 citation statements)

References 19 publications

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“…COLSYS has been tested on a large variety of problems, and a representative selection of them to demonstrate the performance of the code is available in [2]. Some comparisons with the codes in [35], [25] are also made in [2] in order to gain a relative perspective. For brevity, we examine only three examples here.…”

Section: =1 Azimentioning

confidence: 99%

See 1 more Smart Citation

A Collocation Solver for Mixed Order Systems of Boundary Value Problems

Ascher¹,

Christiansen²,

Russell³

1979

Mathematics of Computation

115

139

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Abstract.Implementation of a spline collocation method for solving boundary value problems for mixed order systems of ordinary differential equations is discussed.The aspects of this method considered include error estimation, adaptive mesh selection, B-spline basis function evaluation, linear system solution and nonlinear problem solution.The resulting general purpose code, COLSYS, is tested on a number of examples to demonstrate its stability, efficiency and flexibility.

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Section: =1 Azimentioning

confidence: 99%

“…A second approach has been implemented by Lentini and Pereyra [24], [25], where a finite difference method with deferred corrections is used.…”

Section: Introductionmentioning

confidence: 99%

A Collocation Solver for Mixed Order Systems of Boundary Value Problems

Ascher¹,

Christiansen²,

Russell³

1979

Mathematics of Computation

115

139

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“…In Christiansen and Russell (1979) a careful analysis of deferred corrections that does not involve asymptotic expansions is done for a realistic algorithm similar to the implementation of Lentini and Pereyra (1977) of iterated deferred corrections for two-point boundary value…”

Section: Error Estimation and Iterative Improvement For The Numericalmentioning

confidence: 99%

Numerical Methods for Initial Value Problems.

Skeel¹

1980

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“…The use of these schemes within an iterated deferred correction scheme for the numerical solution of boundary value problems has been described by Cash [10,11], and Cash and Wright [15]. This work has led to the development of a code called HAGRON, which has been shown to compare favorably with COLSYS and D02GAF, a code from the NAG library based on PASVA3 (Lentini and Pereyra [23]). The entire class of MIRK schemes has been considered for use in the numerical solution of boundary value ODEs by Gupta [21] and Enright and Muir [18].…”

Section: Introductionmentioning

confidence: 99%

Order barriers and characterizations for continuous mono-implicit Runge-Kutta schemes

Muir¹,

Owren²

1993

Math. Comp.

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Abstract. The mono-implicit Runge-Kutta (MIRK) schemes, a subset of the family of implicit Runge-Kutta (IRK) schemes, were originally proposed for the numerical solution of initial value ODEs more than fifteen years ago. During the last decade, a considerable amount of attention has been given to the use of these schemes in the numerical solution of boundary value ODE problems, where their efficient implementation suggests that they may provide a worthwhile alternative to the widely used collocation schemes. Recent work in this area has seen the development of some software packages for boundary value ODEs based on these schemes. Unfortunately, these schemes lead to algorithms which provide only a discrete solution approximation at a set of mesh points over the problem interval, while the collocation schemes provide a natural continuous solution approximation. The availability of a continuous solution is important not only to the user of the software but also within the code itself, for example, in estimation of errors, defect control, mesh selection, and the provision of initial solution estimates for new meshes. An approach for the construction of a continuous solution approximation based on the MIRK schemes is suggested by recent work in the area of continuous extensions for explicit Runge-Kutta schemes for initial value ODEs. In this paper, we describe our work in the investigation of continuous versions of the MIRK schemes: (i) we give some lower bounds relating the stage order to the minimal number of stages for general continuous IRK schemes, (ii) we establish lower bounds on the number of stages needed to derive continuous MIRK schemes of orders 1 through 6, and (iii) we provide characterizations of these schemes having a minimal number of stages for each of these orders.

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An Adaptive Finite Difference Solver for Nonlinear Two-Point Boundary Problems with Mild Boundary Layers

Cited by 193 publications

References 19 publications

A Collocation Solver for Mixed Order Systems of Boundary Value Problems

A Collocation Solver for Mixed Order Systems of Boundary Value Problems

Numerical Methods for Initial Value Problems.

Order barriers and characterizations for continuous mono-implicit Runge-Kutta schemes

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