A Superconsistent Chebyshev Collocation Method for Second-Order Differential Operators

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 _n 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.