Adaptive multiquadric collocation for boundary layer problems

An adaptive collocation method based upon radial basis functions is presented for the solution of singularly perturbed two-point boundary value problems. Using a multiquadric integral formulation, the second derivative of the solution is approximated by multiquadric radial basis functions. This approach is combined with a coordinate stretching technique. The required variable transformation is accomplished by a conformal mapping, an iterated sine-transformation. A new error indicator function accurately captures the regions of the interval with insufficient resolution. This indicator is used to adaptively add data centres and collocation points. The method resolves extremely thin layers accurately with fairly few basis functions. The proposed adaptive scheme is very robust, and reaches high accuracy even when parameters in our coordinate stretching technique are not chosen optimally. The effectiveness of our new method is demonstrated on two examples with boundary layers, and one example featuring an interior layer. It is shown in detail how the adaptive method refines the resolution.

[1]  I. Babuska,et al.  The Partition of Unity Method , 1997 .

[2]  Zongmin Wu,et al.  Local error estimates for radial basis function interpolation of scattered data , 1993 .

[3]  W. Madych,et al.  Multivariate interpolation and condi-tionally positive definite functions , 1988 .

[4]  E. Kansa.  MULTIQUADRICS--A SCATTERED DATA APPROXIMATION SCHEME WITH APPLICATIONS TO COMPUTATIONAL FLUID-DYNAMICS-- II SOLUTIONS TO PARABOLIC, HYPERBOLIC AND ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS , 1990 .

[5]  J. Oden,et al.  H‐p clouds—an h‐p meshless method , 1996 .

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

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

[8]  T. Driscoll,et al.  Regular Article: A Fast Spectral Algorithm for Nonlinear Wave Equations with Linear Dispersion , 1999 .

[9]  Jungho Yoon,et al.  Spectral Approximation Orders of Radial Basis Function Interpolation on the Sobolev Space , 2001, SIAM J. Math. Anal..

[10]  R. Franke.  Scattered data interpolation: tests of some methods , 1982 .

[11]  Leevan Ling,et al.  A volumetric integral radial basis function method for time-dependent partial differential equations. I. Formulation , 2004 .

[12]  R. L. Hardy.  Theory and applications of the multiquadric-biharmonic method : 20 years of discovery 1968-1988 , 1990 .

[13]  Nam Mai-Duy,et al.  Numerical solution of differential equations using multiquadric radial basis function networks , 2001, Neural Networks.

[14]  Nam Mai-Duy,et al.  Numerical solution of Navier–Stokes equations using multiquadric radial basis function networks , 2001 .

[15]  E. Kansa,et al.  Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary , 2002 .

[16]  C. Canuto.  Spectral methods in fluid dynamics , 1991 .

[17]  S. Sarra.  Chebyshev super spectral viscosity solution of a two‐dimensional fluidized‐bed model , 2003 .

[18]  Eugene C. Gartland,et al.  Graded-mesh difference schemes for singularly perturbed two-point boundary value problems , 1988 .

[19]  Leslie Greengard,et al.  A fast algorithm for particle simulations , 1987 .

[20]  T. Belytschko,et al.  Element‐free Galerkin methods , 1994 .

[21]  E. J. Kansa,et al.  Multi-quadrics-a scattered data approximation scheme with applications to computational fluid dynamics-II , 1990 .

[22]  L. Ling,et al.  Multiquadric collocation method with integralformulation for boundary layer problems , 2004 .

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

[24]  Eugene O'Riordan,et al.  On piecewise-uniform meshes for upwind- and central-difference operators for solving singularly perturbed problems , 1995 .

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

[26]  R. L. Hardy.  Multiquadric equations of topography and other irregular surfaces , 1971 .