Optimal Collocation Nodes for Fractional Derivative Operators

Spectral discretizations of fractional derivative operators are examined, where the approximation basis is related to the set of Jacobi polynomials. The pseudo-spectral method is implemented by assuming that the grid, used to represent the function to be differentiated, may not be coincident with the collocation grid. The new option opens the way to the analysis of alternative techniques and the search of optimal distributions of collocation nodes, based on the operator to be approximated. Once the initial representation grid has been chosen, indications on how to recover the collocation grid are provided, with the aim of enlarging the dimension of the approximation space. As a results of this process, performances are improved. Applications to fractional type advection-diffusion equations, and comparisons in terms of accuracy and efficiency are made. As shown in the analysis, special choices of the nodes can also suggest tricks to speed up computations

A superconsistent Chebyshev collocation method for second-order differential operators

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