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.
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References
C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral Methods in Fluid Dynamics (Springer, New York, 1988).
D. Funaro, A new scheme for the approximation of advection-diffusion equations by collocation, SIAM J. Numer. Anal. 19(4) (1982) 816–832.
D. Funaro, Improving the performances of implicit schemes for hyperbolic equations, J. Sci. Comput. 12(4) (1997) 385–394.
D. Funaro, Spectral Elements for Transport-Dominated Equations, LNCSE, Vol. 1 (Springer, Heidelberg, 1997).
D. Funaro, Some remarks about the collocation method on a modified Legendre grid, Comput. Math. Appl. 33(1/2) (1997) 95–103.
D. Funaro, A note on second-order finite-difference schemes on uniform meshes for advectiondiffusion equations, Numer. Methods PDEs 15 (1999) 581–588.
D. Funaro, Superconsistent discretization of integral type equations, to appear.
D. Funaro and G. Pontrelli, Spline approximation of advection-diffusion problems using upwind type collocation nodes, J. Comput. Appl. Math. 110 (1999) 141–153.
D. Gottlieb and S.A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications (SIAM, Philadelphia, 1977).
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Funaro, D. A Superconsistent Chebyshev Collocation Method for Second-Order Differential Operators. Numerical Algorithms 28, 151–157 (2001). https://doi.org/10.1023/A:1014038615371
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DOI: https://doi.org/10.1023/A:1014038615371