The uniform convergence of upwind schemes on layer-adapted meshes for a singularly perturbed Robin BVP
ABSTRACT
In this paper, we discuss the uniform convergence of the simple upwind scheme on the Shishkin mesh and the Bakhvalov-Shishkin mesh for solving a singularly perturbed Robin boundary value problem, and investigate the midpoint upwind scheme on the Shishkin mesh and the Bakhvalov-Shishkin mesh to achieve better uniform convergence. The elaborate ε-uniform pointwise estimates are proved by using the comparison principle and barrier functions. The numerical experiments support the theoretical results for the schemes on the meshes.
In this paper, we discuss the uniform convergence of the simple upwind scheme on the Shishkin mesh and the Bakhvalov-Shishkin mesh for solving a singularly perturbed Robin boundary value problem, and investigate the midpoint upwind scheme on the Shishkin mesh and the Bakhvalov-Shishkin mesh to achieve better uniform convergence. The elaborate ε-uniform pointwise estimates are proved by using the comparison principle and barrier functions. The numerical experiments support the theoretical results for the schemes on the meshes.
Cite this paper
nullZheng, Q. , Huang, F. , Feng, X. and Han, M. (2012) The uniform convergence of upwind schemes on layer-adapted meshes for a singularly perturbed Robin BVP. Open Journal of Applied Sciences, 2, 66-69. doi: 10.4236/ojapps.2012.24B016.
nullZheng, Q. , Huang, F. , Feng, X. and Han, M. (2012) The uniform convergence of upwind schemes on layer-adapted meshes for a singularly perturbed Robin BVP. Open Journal of Applied Sciences, 2, 66-69. doi: 10.4236/ojapps.2012.24B016.
References
[1] J.J.H. Miller, R.E. O'Riordan, G.I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems. World Scientific, Singapore, 1996. [2] H.-G. Roos, M. Stynes, L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations, Springer-Verlag, Berlin, Heidelberg, 2008. [3] S. Natesan, N. Ramanujam, An asymptotic-numerical method for singularly perturbed Robin problems-I, Appl. Math. Comput. 126 (2002) 97-107. [4] Z.-D. Cen, H.-Y. Gao, Uniform convergence analysis of a singularly perturbed Robin problem on a Bakhvalov-Shishkin mesh, J. Zhejiang Univ. (Science Edition) 31 (2004) 373-375 (in Chinese). [5] R.B. Kellogg, A. Tsan, Analysis of some difference approximations for a singular perturbation problem without turning points, Math. Comput. 32 (1978) 1025-1039. [6] A.S. Bakhvalov, On the optimization of methods for solving boundary value problems in the presence of boundary layers, Zh. Vychisl. Mat. Mat. Fiz. 9(1969) 841-859 (in Russian). [7] G.I. Shishkin, Grid approximation of singularly perturbed elliptic and parabolic equations, Second doctorial thesis, Keldysh Institute, Moscow, 1990 (in Russian). [8] T. Linβ, An upwind difference scheme on a novel Shishkin-type mesh for a linear convection-diffusion problem, J. Comput. Appl. Math. 110 (1999) 93-104. [9] M. Stynes, H.-G. Roos, The midpoint upwind scheme, Appl. Numer. Math. 23 (1997) 361-374.
[1] J.J.H. Miller, R.E. O'Riordan, G.I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems. World Scientific, Singapore, 1996. [2] H.-G. Roos, M. Stynes, L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations, Springer-Verlag, Berlin, Heidelberg, 2008. [3] S. Natesan, N. Ramanujam, An asymptotic-numerical method for singularly perturbed Robin problems-I, Appl. Math. Comput. 126 (2002) 97-107. [4] Z.-D. Cen, H.-Y. Gao, Uniform convergence analysis of a singularly perturbed Robin problem on a Bakhvalov-Shishkin mesh, J. Zhejiang Univ. (Science Edition) 31 (2004) 373-375 (in Chinese). [5] R.B. Kellogg, A. Tsan, Analysis of some difference approximations for a singular perturbation problem without turning points, Math. Comput. 32 (1978) 1025-1039. [6] A.S. Bakhvalov, On the optimization of methods for solving boundary value problems in the presence of boundary layers, Zh. Vychisl. Mat. Mat. Fiz. 9(1969) 841-859 (in Russian). [7] G.I. Shishkin, Grid approximation of singularly perturbed elliptic and parabolic equations, Second doctorial thesis, Keldysh Institute, Moscow, 1990 (in Russian). [8] T. Linβ, An upwind difference scheme on a novel Shishkin-type mesh for a linear convection-diffusion problem, J. Comput. Appl. Math. 110 (1999) 93-104. [9] M. Stynes, H.-G. Roos, The midpoint upwind scheme, Appl. Numer. Math. 23 (1997) 361-374.