An overlapping Schwarz method for singularly perturbed fourth-order convection-diffusion type
Abstract
In this paper, we have constructed an iterative numerical method based on an overlapping Schwarz procedure with uniform mesh for singularly perturbed fourth-order of convection-diffusion type. The method splits the original domain into two overlapping subdomains. A hybrid difference scheme is proposed in which on the boundary layer region we use the central finite difference scheme on a uniform mesh while on the non-layer region we use the mid-point difference scheme on a uniform mesh. It is shown that the method produces numerical approximations which converge in the maximum norm to the exact solution. We prove that, when appropriate subdomains are used the method produces convergence of almost second-order. Furthermore, it is shown that, two iterations are sufficient to achieve the expected accuracy. Numerical examples are presented to support the theoretical results.
Keyword : singularly perturbed problems, convection-diffusion equations, Schwarz method, hybrid difference scheme
How to Cite
Roja, J. C., & Tamilselvan, A. (2020). An overlapping Schwarz method for singularly perturbed fourth-order convection-diffusion type. Mathematical Modelling and Analysis, 25(4), 661-679. https://doi.org/10.3846/mma.2020.10517
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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J. Christy Roja and A. Tamilselvan. Numerical method for singularly perturbed fourth-order ordinary differential equations of convection-diffusion type. Journal of Mathematical Modeling, 4(1):79–102, 2016.
J. Christy Roja and A. Tamilselvan. Schwarz method for singularly perturbed second-order convection-diffusion equations. J. Appl. Math. and Informatics, 36(3–4):181–203, 2018.
M.K. Kadalbajoo and Y.N. Reddy. A brief survey on numerical methods for solving singularly perturbed problems. Appl. Math. Comput., 2010. https://doi.org/10.1016/j.amc.2010.09.059
M. Feckan. Singularly perturbed higher-order boundary value problems. J. Differ. Equ., 3:79–102, 1994. https://doi.org/10.1006/jdeq.1994.1076
H. MacMullan, J.J.H. Miller, E. O’Riordan and G.I. Shishkin. A second-order parameter-uniform overlapping Schwarz method for reaction-diffusion problems with boundary layers. J. Comput. Appl. Math., 130(1-2):231–244, 2001. https://doi.org/10.1016/S0377-0427(99)00380-5
H. MacMullan, J.J.H. Miller, E. O’Riordan, G.I. Shishkin and S. Wang. A parameter-uniform Schwarz method for a singularly perturbed reactiondiffusion problem with an interrior layer. Appl. Num. Math., 35:323–337, 2000. https://doi.org/10.1016/S0168-9274(99)00140-3
H. MacMullen, J.J.H. Miller, E. O’Riordan and G.I. Shishkin. Schwarz iterative method for convection-diffusion problems with boundary layers. In L.G. Vulkov, J.J.H. Miller and G.I. Shishkin(Eds.), Analytical and numerical methods for convection-dominated and singularly perturbed problems, pp. 213–218. Nova Science Publishers, New York, 2000.
H. MacMullen, E. O’Riordan and G.I. Shishkin. The convergence of classical Schwarz methods applied to convection-diffusion problems with regular boundary layers. Appl. Num. Math., 43:297–313, 2002. https://doi.org/10.1016/S0168-9274(01)00177-5
J.J.H. Miller, E. O’Riordan and G.I. Shishkin. Fitted numerical methods for singular perturbation problems, Error estimates in the maximum norm for linear problems in one and two dimensions. World Scientific, Singapore, 1996. https://doi.org/10.1142/2933
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R. Mythili Priyadharshini, N. Ramanujam and A. Tamilselvan. Hybrid difference schemes for a system of singularly perturbed convection-diffusion equations. J. Appl. Math. and Informatics, 27(5-6):1001–1015, 2009.
S.C.S. Rao and S. Kumar. An almost fourth-order uniformly convergent domain decomposition method for a coupled system of singularly perturbed reaction diffusion equations. J. Comput. and Appl. Math., 235:3342–3354, 2011. https://doi.org/10.1016/j.cam.2011.01.047
H.G. Roos, M. Stynes and L. Tobiska. Numerical methods for singularly perturbed differential equations, Convection-diffusion and flow problems. Springer-Verlag, 1996. https://doi.org/10.1007/978-3-662-03206-0
V. Shanthi and N. Ramanujam. Asymptotic numerical method for boundary value problems for singularly perturbed fourth-order ordinary differential equations with a weak interior layer. Appl. Math. Comput., 133:559–579, 2002. https://doi.org/10.1016/S0096-3003(01)00257-0
V. Shanthi and N. Ramanujam. A boundary value technique for boundary value problems for singularly perturbed fourth-order ordinary differential equations. Comput. math. appl., 47:1673–1688, 2004. https://doi.org/10.1016/j.camwa.2004.06.015
V. Shanthi and N. Ramanujam. Computational methods for reaction-diffusion problems for fourth-order ordinary differentional equations with a small parameter at the highest derivative. Appl. Math. Comput., 147:97–113, 2004. https://doi.org/10.1016/S0096-3003(02)00654-9
M. Stephens and N. Madden. A parameter-uniform Schwarz method for a coupled system of reaction-diffusion equations. J. Comput. Appl. Math., 230:360– 370, 2009. https://doi.org/10.1016/j.cam.2008.12.009
M. Stynes and H.G. Roos. The midpoint upwind scheme. Appl. Numer. Math., 23(3):361–374, 1997. https://doi.org/10.1016/S0168-9274(96)00071-2
Sunil Kumar and Mukesh Kumar. An analysis of overlapping domain decomposition methods for singularly perturbed reaction-diffusion problems. J. Comput. Appl. Math., 281:250–262, 2015. https://doi.org/10.1016/j.cam.2014.12.018
J. Christy Roja and A. Tamilselvan. Numerical method for singularly perturbed fourth-order ordinary differential equations of convection-diffusion type. Journal of Mathematical Modeling, 4(1):79–102, 2016.
J. Christy Roja and A. Tamilselvan. Schwarz method for singularly perturbed second-order convection-diffusion equations. J. Appl. Math. and Informatics, 36(3–4):181–203, 2018.
M.K. Kadalbajoo and Y.N. Reddy. A brief survey on numerical methods for solving singularly perturbed problems. Appl. Math. Comput., 2010. https://doi.org/10.1016/j.amc.2010.09.059
M. Feckan. Singularly perturbed higher-order boundary value problems. J. Differ. Equ., 3:79–102, 1994. https://doi.org/10.1006/jdeq.1994.1076
H. MacMullan, J.J.H. Miller, E. O’Riordan and G.I. Shishkin. A second-order parameter-uniform overlapping Schwarz method for reaction-diffusion problems with boundary layers. J. Comput. Appl. Math., 130(1-2):231–244, 2001. https://doi.org/10.1016/S0377-0427(99)00380-5
H. MacMullan, J.J.H. Miller, E. O’Riordan, G.I. Shishkin and S. Wang. A parameter-uniform Schwarz method for a singularly perturbed reactiondiffusion problem with an interrior layer. Appl. Num. Math., 35:323–337, 2000. https://doi.org/10.1016/S0168-9274(99)00140-3
H. MacMullen, J.J.H. Miller, E. O’Riordan and G.I. Shishkin. Schwarz iterative method for convection-diffusion problems with boundary layers. In L.G. Vulkov, J.J.H. Miller and G.I. Shishkin(Eds.), Analytical and numerical methods for convection-dominated and singularly perturbed problems, pp. 213–218. Nova Science Publishers, New York, 2000.
H. MacMullen, E. O’Riordan and G.I. Shishkin. The convergence of classical Schwarz methods applied to convection-diffusion problems with regular boundary layers. Appl. Num. Math., 43:297–313, 2002. https://doi.org/10.1016/S0168-9274(01)00177-5
J.J.H. Miller, E. O’Riordan and G.I. Shishkin. Fitted numerical methods for singular perturbation problems, Error estimates in the maximum norm for linear problems in one and two dimensions. World Scientific, Singapore, 1996. https://doi.org/10.1142/2933
R. Mythili Priyadharshini, N. Ramanujam and V. Shanthi. Approximation of derivative in a system of singularly perturbed convectiondiffusion equations. J. Appl. Math. Comput., 30:369–383, 2009. https://doi.org/10.1007/s12190-008-0178-5
R. Mythili Priyadharshini, N. Ramanujam and A. Tamilselvan. Hybrid difference schemes for a system of singularly perturbed convection-diffusion equations. J. Appl. Math. and Informatics, 27(5-6):1001–1015, 2009.
S.C.S. Rao and S. Kumar. An almost fourth-order uniformly convergent domain decomposition method for a coupled system of singularly perturbed reaction diffusion equations. J. Comput. and Appl. Math., 235:3342–3354, 2011. https://doi.org/10.1016/j.cam.2011.01.047
H.G. Roos, M. Stynes and L. Tobiska. Numerical methods for singularly perturbed differential equations, Convection-diffusion and flow problems. Springer-Verlag, 1996. https://doi.org/10.1007/978-3-662-03206-0
V. Shanthi and N. Ramanujam. Asymptotic numerical method for boundary value problems for singularly perturbed fourth-order ordinary differential equations with a weak interior layer. Appl. Math. Comput., 133:559–579, 2002. https://doi.org/10.1016/S0096-3003(01)00257-0
V. Shanthi and N. Ramanujam. A boundary value technique for boundary value problems for singularly perturbed fourth-order ordinary differential equations. Comput. math. appl., 47:1673–1688, 2004. https://doi.org/10.1016/j.camwa.2004.06.015
V. Shanthi and N. Ramanujam. Computational methods for reaction-diffusion problems for fourth-order ordinary differentional equations with a small parameter at the highest derivative. Appl. Math. Comput., 147:97–113, 2004. https://doi.org/10.1016/S0096-3003(02)00654-9
M. Stephens and N. Madden. A parameter-uniform Schwarz method for a coupled system of reaction-diffusion equations. J. Comput. Appl. Math., 230:360– 370, 2009. https://doi.org/10.1016/j.cam.2008.12.009
M. Stynes and H.G. Roos. The midpoint upwind scheme. Appl. Numer. Math., 23(3):361–374, 1997. https://doi.org/10.1016/S0168-9274(96)00071-2
Sunil Kumar and Mukesh Kumar. An analysis of overlapping domain decomposition methods for singularly perturbed reaction-diffusion problems. J. Comput. Appl. Math., 281:250–262, 2015. https://doi.org/10.1016/j.cam.2014.12.018