Numerical Aspects of the Coeffcient Computation for LMMs

Keywords: Linear Multistep Methods, Boundary Value Methods, Conditioning, Vandermonde
matrix, Bernstein polynomials, Bjork-Pereyra algorithm

Abstract: The numerical solution of Boundary Value Problems usually requires the use
of an adaptive mesh selection strategy. For this reason, when a Linear Multistep Method
is considered, a dynamic computation of its coeffcients is necessary. This leads to solve
linear systems which can be expressed in dierent forms, depending on the polynomial
basis used to impose the order conditions. In this paper, we compare the accuracy of the
numerically computed coe cients for three dierent formulations. For all the considered
cases Vandermonde systems on general abscissae are involved and they are always solved
by the Bjork-Pereyra algorithm [3]. An adaptation of the forward error analysis given in
[8, 9] is proposed whose signi cance is con rmed by the numerical results.


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