Numerical study of an adaptive domain decomposition algorithm based on Chebyshev tau method for solving singular perturbed problems

It is known that spectral methods offer exponential convergence for infinitely smooth solutions. However, they are not applicable for problems presenting singularities or thin layers, especially true for the ones with the location of singularity unknown. An adaptive domain decomposition method (DDM) integrated with Chebyshev tau method based on the highest derivative (CTMHD) is introduced to solve singular perturbed boundary value problems (SPBVPs). The proposed adaptive algorithm uses the refinement indicators based on Chebyshev coefficients to determine which subintervals need to be refined. Numerical experiments have been conducted to demonstrate the superior performance of the method for SPBVPs with a number of singularities including boundary layers, interior layers and dense oscillations. A fourth order nonlinear SPBVP is also concerned. The numerical results illustrate the efficiency and applicability of our adaptive algorithm to capture the locations of singularities, and the higher accuracy in comparison with some existing numerical methods in the literature. Adaptive domain decomposition method for singular perturbed problems is proposed.The proposed adaptive algorithm captures the locations of the layers effectively.Chebyshev tau method based on the highest derivative is applied.The refinement indicator is based on the tails of the Chebyshev coefficients.A numerical example with internal layers and oscillations is computed.

[1]  Lloyd N. Trefethen,et al.  A Rational Spectral Collocation Method with Adaptively Transformed Chebyshev Grid Points , 2006, SIAM J. Sci. Comput..

[2]  S. Chandra Sekhara Rao,et al.  Exponential B-spline collocation method for self-adjoint singularly perturbed boundary value problems , 2008 .

[3]  Wil H. A. Schilders,et al.  Uniform Numerical Methods for Problems with Initial and Boundary Layers , 1980 .

[4]  An efficient domain-decomposition pseudo-spectral method for solving elliptic differential equations , 2007 .

[5]  Gene H. Golub,et al.  Matrix computations , 1983 .

[6]  Jie Shen,et al.  Efficient Spectral-Galerkin Method I. Direct Solvers of Second- and Fourth-Order Equations Using Legendre Polynomials , 1994, SIAM J. Sci. Comput..

[7]  Nam Mai-Duy.  An effective spectral collocation method for the direct solution of high-order ODEs , 2005 .

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

[9]  N. Ramanujam,et al.  A boundary value technique for boundary value problems for singularly perturbed fourth-order ordinary differential equations , 2004 .

[10]  Wenting Shao,et al.  Chebyshev tau meshless method based on the highest derivative for fourth order equations , 2013 .

[11]  Jie Shen,et al.  Efficient Spectral-Galerkin Method II. Direct Solvers of Second- and Fourth-Order Equations Using Chebyshev Polynomials , 1995, SIAM J. Sci. Comput..

[12]  Suheil A. Khuri,et al.  The boundary layer problem: A fourth-order adaptive collocation approach , 2012, Comput. Math. Appl..

[13]  Yingwei Wang,et al.  A rational spectral collocation method for solving a class of parameterized singular perturbation problems , 2010, J. Comput. Appl. Math..

[14]  T. Tran-Cong,et al.  A multidomain integrated‐radial‐basis‐function collocation method for elliptic problems , 2008 .

[15]  Marc Garbey,et al.  Adaptive pseudo-spectral domain decomposition and the approximation of multiple layers , 1995 .

[16]  Leevan Ling,et al.  Adaptive multiquadric collocation for boundary layer problems , 2006 .

[17]  Natalia Kopteva,et al.  Shishkin meshes in the numerical solution of singularly perturbed differential equations , 2010 .

[18]  T. Tran-Cong,et al.  A second-order continuity domain decomposition technique based on integrated Chebyshev polynomials for two-dimensional elliptic problems , 2008 .

[19]  Catherine Mavriplis,et al.  Adaptive mesh strategies for the spectral element method , 1992 .

[20]  Sheehan Olver,et al.  A Fast and Well-Conditioned Spectral Method , 2012, SIAM Rev..

[21]  Z. Uzelac,et al.  Uniformly convergent difference schemes for a singularly perturbed third order boundary value problem , 2015 .

[22]  JohnM . Miller,et al.  Robust Computational Techniques for Boundary Layers , 2000 .

[23]  Vikas Gupta,et al.  Collocation method using artificial viscosity for solving stiff singularly perturbed turning point problem having twin boundary layers , 2011, Comput. Math. Appl..

[24]  T. Tran-Cong,et al.  Mesh-free radial basis function network methods with domain decomposition for approximation of functions and numerical solution of Poisson's equations , 2002 .

[25]  Xionghua Wu,et al.  Boundary reduction technique and rational Sinc domain decomposition method , 2012 .

[26]  D. F. Yun,et al.  Adaptive least squares finite integration method for higher-dimensional singular perturbation problems with multiple boundary layers , 2015, Appl. Math. Comput..