A standard way to approximate the model problem -u = f, with u(+/-1) = 0, is to collocate the differential equation at the zeros of T-n': x(i), i = 1,..., n - 1, having denoted by T,, the nth Chebyshev polynomial. We introduce an alternative set of collocation nodes z(i), i = 1,..., n - 1, which will provide better numerical performances. The approximated solution is still computed at the nodes {x(i)}, but the equation is required to be satisfied at the new nodes {z(i)}, which are determined by asking an extra degree of consistency in the discretization of the differential operator.
A superconsistent Chebyshev collocation method for second-order differential operators / Funaro, Daniele. - In: NUMERICAL ALGORITHMS. - ISSN 1017-1398. - STAMPA. - 28:(2001), pp. 151-157. [10.1023/A:1014038615371]
A superconsistent Chebyshev collocation method for second-order differential operators
FUNARO, Daniele
2001
Abstract
A standard way to approximate the model problem -u = f, with u(+/-1) = 0, is to collocate the differential equation at the zeros of T-n': x(i), i = 1,..., n - 1, having denoted by T,, the nth Chebyshev polynomial. We introduce an alternative set of collocation nodes z(i), i = 1,..., n - 1, which will provide better numerical performances. The approximated solution is still computed at the nodes {x(i)}, but the equation is required to be satisfied at the new nodes {z(i)}, which are determined by asking an extra degree of consistency in the discretization of the differential operator.
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