High-Order Compact Finite-Difference Scheme for Singularly-Perturbed Reaction-Diffusion Problems on a New Mesh of Shishkin Type

In this work, a high-order compact finite-difference (HOCFD) scheme has been proposed to solve 1-dimensional (1D) and 2-dimensional (2D) elliptic and parabolic singularly-perturbed reaction-diffusion problems. A new kind of piecewise uniform mesh of Shishkin type (Miller et al. in Fitted Numerical Methods for Singular Perturbation Problems, 1996) has also been proposed and using this mesh the HOCFD scheme gives better results as compared to the results using the Shishkin mesh. Moreover, the stated method gives ε-uniform convergence and improved orders of convergence which have also been provided in the results for some test problems.

[1]  S. Lele.  Compact finite difference schemes with spectral-like resolution , 1992 .

[2]  Mahidhar Tatineni,et al.  High-order non-uniform grid schemes for numerical simulation of hypersonic boundary-layer stability and transition , 2003 .

[3]  J. Mackenzie,et al.  On a uniformly accurate finite difference approximation of a singulary peturbed reaction-diffusion problem using grid equidistribution , 2001 .

[4]  SOLVING SINGULARLY PERTURBED REACTION DIFFUSION PROBLEMS USING WAVELET OPTIMIZED FINITE DIFFERENCE AND CUBIC SPLINE ADAPTIVE WAVELET SCHEME , 2008 .

[5]  José Luis Gracia Lozano,et al.  A compact finite difference scheme for 2D reaction-diffusion singularly perturbed problems , 2005 .

[6]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[7]  Xiaolin Zhong,et al.  Derivation of high-order compact finite difference schemes for non-uniform grid using polynomial interpolation , 2005 .

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

[9]  Carmelo Clavero,et al.  High order methods for elliptic and time dependent reaction-diffusion singularly perturbed problems , 2005, Appl. Math. Comput..

[10]  A I Tolstykh.  High Accuracy Non-Centered Compact Difference Schemes for Fluid Dynamics Applications , 1994 .

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

[12]  Carmelo Clavero,et al.  High Order varepsilon-Uniform Methods for Singularly Perturbed Reaction-Diffusion Problems , 2000, NAA.

[13]  Mats Holmström,et al.  Solving Hyperbolic PDEs Using Interpolating Wavelets , 1999, SIAM J. Sci. Comput..

[14]  Eugene O'Riordan,et al.  Singularly perturbed convection-diffusion problems with boundary and weak interior layers , 2004 .

[15]  I. M. Navon,et al.  Uniformly convergent finite element methods for singularly perturbed elliptic boundary value problems I: Reaction-diffusion Type , 1998, Computers & Mathematics with Applications.