A collocation method for systems of nonlinear ordinary differential equations

Houstis, Elias N.

doi:10.1016/0022-247x(78)90215-9

Cited by 18 publications

(9 citation statements)

References 5 publications

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“…Finally, let us mention important papers on superconvergence for two-point problems using other than the Galerkin method. de Hoog and Weiss [97], Houstis [99], Nakao [137], Pereyra and Sewell [149], Wheeler [188]. de Hoog and Weiss [97], Houstis [99], Nakao [137], Pereyra and Sewell [149], Wheeler [188].…”

Section: Two-point Boundary Value Problemsmentioning

confidence: 99%

On superconvergence techniques

1987

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Section: Two-point Boundary Value Problemsmentioning

confidence: 99%

On superconvergence techniques

1987

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“…The use of collocation methods for the solution of partial differential equations (PDE) has become a subject of intensive interest in the past [Lanczos 1938;Ronto 1971;Russell and Shampine 1972;Finlayson 1972;De Boor and Swartz 1973;Diaz 1977;Houstis 1978;Botha and Pinder 1983]. While most research has focused on the local interpolation of a variable, not many works have appeared concerning the global collocation in two-dimensional problems [Frind and Pinder 1979;Hayes 1980;Van Blerk and Botha 1993].…”

Section: Introductionmentioning

confidence: 99%

A global collocation method for two-dimensional rectangular domains

Provatidis

2008

JOMMS

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This paper proposes the use of a global collocation procedure in conjunction with a previously developed functional set suitable for the numerical solution of Poisson's equation in rectangular domains. We propose to expand the unknown variable in a bivariate series of monomials x i y j that exist in Pascal's triangle. We also propose the use of the bivariate Gordon-Coons interpolation, apart from previous intuitive choices of the aforementioned monomials. The theory is sustained by two numerical examples of Dirichlet boundary conditions, in which we find that the approximate solution monotonically converges towards the exact solution.

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“…The past decade witnessed much research on spline collocation for ordinary differential equations, both nodal and orthogonal for quite general problems, splines of arbitrary order, etc. [22], [17], [3], [5], [40], [18], [23], [34], [38].…”

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confidence: 99%

On the Asymptotic Convergence of Spline Collocation Methods for Partial Differential Equations

Arnold¹,

Saranen²

1984

SIAM J. Numer. Anal.

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We examine the asymptotic accuracy of the method of collocation for the approximate solution of linear elliptic partial differential equations. Specifically we consider the nodal collocation of a second order equation in the plane with biperiodicity conditions using tensor product smooth splines of odd degree as trial functions. We prove optimal rates of convergence in L for partial derivatives of the approximate solution which are of order at least two in one variable, while the solution itself and its gradient converge in L at rates less than the optimal approximation theoretic results.

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A collocation method for systems of nonlinear ordinary differential equations

Cited by 18 publications

References 5 publications

On superconvergence techniques

On superconvergence techniques

A global collocation method for two-dimensional rectangular domains

On the Asymptotic Convergence of Spline Collocation Methods for Partial Differential Equations

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