The Linear Rational Pseudospectral Method with Iteratively Optimized Poles for Two-Point Boundary Value Problems

An algorithm is proposed which improves upon the polynomial pseudospectral method for solving linear two-point boundary value problems. In the latter, the collocation points are the vertical projection onto the interval of points equidistant or nearly equidistant on the circle, and they therefore accumulate in the vicinity of the extremities of the interval. Thus the method is well-suited for solving problems whose solutions have boundary layers but not as good at approximating solutions with large gradients (shocks) away from the extremities of their domain of definition. Our idea is to modify the polynomial ansatz by attaching a denominator so as to make it a rational interpolant. The denominator is then successively optimized in an iterative procedure with each step consisting of the solution of two problems: an optimization of the denominator for given values of the approximation to the solution u at the interpolation points and a collocation in the linear space of the rational interpolants with the just obtained fixed denominator to obtain new approximate values of u. We show the efficiency of a Galerkin version of the method and discuss the power of the collocation version with several numerical examples.

[1]  E. H. Kaufman,et al.  Uniform rational approximation by differential correction and remes‐differential correlation , 1981 .

[2]  Matu-Tarow Noda,et al.  Hybrid computation of Cauchy-type singular integral equations , 1998, SIGS.

[3]  J. Weideman,et al.  Spectral Methods Based on Nonclassical Orthogonal Polynomials , 1999 .

[4]  Robert D. Russell,et al.  Numerical solution of boundary value problems for ordinary differential equations , 1995, Classics in applied mathematics.

[5]  Tao Tang,et al.  Boundary Layer Resolving Pseudospectral Methods for Singular Perturbation Problems , 1996, SIAM J. Sci. Comput..

[6]  U. Ascher,et al.  A collocation solver for mixed order systems of boundary value problems , 1979 .

[7]  H. Mittelmann,et al.  Lebesgue constant minimizing linear rational interpolation of continuous functions over the interval , 1997 .

[8]  Michael A. Driscoll,et al.  Rational function approximations in the numerical solution of Cauchy-type singular integral equations , 1985 .

[9]  K. Wright,et al.  Error estimation for collocation solution of linear ordinary differential equations , 1986 .

[10]  Peter Deuflhard,et al.  Numerische Mathematik. I , 2002 .

[11]  Leslie Greengard,et al.  Spectral integration and two-point boundary value problems , 1991 .

[12]  Leslie Greengard,et al.  A Fast Adaptive Numerical Method for Stiff Two-Point Boundary Value Problems , 1997, SIAM J. Sci. Comput..

[13]  Jean-Paul Berrut,et al.  The barycentric weights of rational interpolation with prescribed poles , 1997 .

[14]  K. Wright,et al.  Computable Error Bounds for Polynomial Collocation Methods , 1978 .

[15]  Sandro Ridella,et al.  Minimizing multimodal functions of continuous variables with the “simulated annealing” algorithmCorrigenda for this article is available here , 1987, TOMS.

[16]  U. Ascher,et al.  A New Basis Implementation for a Mixed Order Boundary Value ODE Solver , 1987 .

[17]  Daniele Funaro,et al.  Spectral Elements for Transport-Dominated Equations , 1997, Lecture Notes in Computational Science and Engineering.

[18]  Josef Stoer,et al.  Numerische Mathematik 2 , 1990 .

[19]  Jean-Paul Berrut,et al.  The errors in calculating the pseudospectral differentiation matrices for C̆ebys̆ev-Gauss-Lobatto points , 1999 .

[20]  P. Hemker,et al.  A numerical study of stiff two-point boundary problems , 1977 .

[21]  James F. Geer,et al.  Exponentially Accurate Approximations to Piece-Wise Smooth Periodic Functions , 1997 .

[22]  Weizhang Huang,et al.  Pseudospectral Solution of Near-Singular Problems using Numerical Coordinate Transformations Based on Adaptivity , 1998, SIAM J. Sci. Comput..

[23]  Jean-Paul Berrut,et al.  The linear rational collocation method , 2001 .

[24]  E. Grosse.  A catalogue of algorithms for approximation , 1990 .

[25]  R. Baltensperger.  Improving the accuracy of the matrix differentiation method for arbitrary collocation points , 2000 .

[26]  Jean-Paul Berrut,et al.  The Linear Rational Pseudospectral Method for Boundary Value Problems , 2001 .

[27]  H. Keller.  Numerical Methods for Two-Point Boundary-Value Problems , 1993 .

[28]  Tobin A. Driscoll,et al.  A Block Pseudospectral Method for Maxwell's Equations , 1998 .

[29]  W. Werner.  Polynomial interpolation: Lagrange versus Newton , 1984 .