A semi-analytic method for solving singularly perturbed twin-layer problems with a turning point
Abstract
This computational study investigates a class of singularly perturbed second-order boundary-value problems having dual (twin) boundary layers and simple turning points. It is well-known that the classical discretization methods fail to resolve sharp gradients arising in solving singularly perturbed differential equations as the perturbation (diffusion) parameter decreases, i.e., ε → 0+. To this end, this paper proposes a semi-analytic hybrid method consisting of a numerical procedure based on finite differences and an asymptotic method called the Successive Complementary Expansion Method (SCEM) to approximate the solution of such problems. Two numerical experiments are provided to demonstrate the method’s implementation and to evaluate its computational performance. Several comparisons with the numerical results existing in the literature are also made. The numerical observations reveal that the hybrid method leads to good solution profiles and achieves this in only a few iterations.
Keyword : asymptotic expansion, dual layers, finite differences, singular perturbation, turning point
How to Cite
Cengizci, S., Kumar, D., & Atay, M. T. (2023). A semi-analytic method for solving singularly perturbed twin-layer problems with a turning point. Mathematical Modelling and Analysis, 28(1), 102–117. https://doi.org/10.3846/mma.2023.14953
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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P. Cathalifaud, J. Mauss and J. Cousteix. Nonlinear aspects of high Reynolds number channel flows. European Journal of Mechanics - B/Fluids, 29(4):295– 304, 2010. https://doi.org/10.1016/j.euromechflu.2010.02.002
S. Cengizci. An asymptotic-numerical hybrid method for solving singularly perturbed linear delay differential equations. International Journal of Differential Equations, 2017:1–8, 2017. https://doi.org/10.1155/2017/7269450
L.-Y. Chen, N. Goldenfeld and Y. Oono. Renormalization group and singular perturbations: Multiple scales, boundary layers, and reductive perturbation theory. Physical Review E, 54(1):376–394, 1996. https://doi.org/10.1103/physreve.54.376
J. Cousteix and J. Mauss. Approximations of the Navier–Stokes equations for high Reynolds number flows past a solid wall. Journal of Computational and Applied Mathematics, 166(1):101–122, 2004. https://doi.org/10.1016/j.cam.2003.09.035
J. Cousteix and J. Mauss. Asymptotic analysis and boundary layers. Springer Science & Business Media, 2007. https://doi.org/10.1007/978-3-540-46489-1
J. Cousteix and J. Mauss. Interactive boundary layers in turbulent flow. Comptes Rendus M´ecanique, 335(9-10):590–605, 2007. https://doi.org/10.1016/j.crme.2007.08.014
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F.Z. Geng and S.P. Qian. Reproducing kernel method for singularly perturbed turning point problems having twin boundary layers. Applied Mathematics Letters, 26(10):998–1004, 2013. https://doi.org/10.1016/j.aml.2013.05.006
E. Hairer and G. Wanner. Solving Ordinary Differential Equations II. Springer Berlin Heidelberg, 1996. https://doi.org/10.1007/978-3-642-05221-7
E.J. Hinch. Perturbation Methods. Cambridge University Press, 1991. https://doi.org/10.1017/cbo9781139172189
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M.K. Kadalbajoo, P. Arora and V. Gupta. Collocation method using artificial viscosity for solving stiff singularly perturbed turning point problem having twin boundary layers. Computers & Mathematics with Applications, 61(6):1595–1607, 2011. https://doi.org/10.1016/j.camwa.2011.01.028
M.K. Kadalbajoo and V. Gupta. A parameter uniform B-spline collocation method for solving singularly perturbed turning point problem having twin boundary layers. International Journal of Computer Mathematics, 87(14):3218– 3235, 2010. https://doi.org/10.1080/00207160902980492
M.K. Kadalbajoo and K.C. Patidar. Variable mesh spline approximation method for solving singularly perturbed turning point problems having boundary layer(s). Computers & Mathematics with Applications, 42(10-11):1439–1453, 2001. https://doi.org/10.1016/s0898-1221(01)00253-x
M.K. Kadalbajoo and Y.N. Reddy. Asymptotic and numerical analysis of singular perturbation problems: A survey. Applied Mathematics and Computation, 30(3):223–259, 1989. https://doi.org/10.1016/0096-3003(89)90054-4
R.B. Kellogg and A. Tsan. Analysis of some difference approximations for a singular perturbation problem without turning points. Mathematics of Computation, 32(144):1025–1039, 1978. https://doi.org/10.1090/s0025-5718-1978-0483484-9
J. Kevorkian and J.D. Cole. Multiple Scale and Singular Perturbation Methods. Springer New York, 1996. https://doi.org/10.1007/978-1-4612-3968-0
J.A. Kierzenka and L.F. Shampine. A BVP solver that controls residual and error. Journal of Numerical Analysis, Industrial and Applied Mathematics, 3(12):27–41, 2008.
D. Kumar. A parameter-uniform method for singularly perturbed turning point problems exhibiting interior or twin boundary layers. International Journal of Computer Mathematics, 96(5):865–882, 2018. https://doi.org/10.1080/00207160.2018.1458098
M. Kumar and Parul. Methods for solving singular perturbation problems arising in science and engineering. Mathematical and Computer Modelling, 54(1-2):556– 575, 2011. https://doi.org/10.1016/j.mcm.2011.02.045
P.A. Lagerstrom. Matched Asymptotic Expansions: Ideas and Techniques. Springer, New York, 1988. https://doi.org/10.1007/978-1-4757-1990-1
J. Mauss and J. Cousteix. Uniformly valid approximation for singular perturbation problems and matching principle. Comptes Rendus M´ecanique, 330(10):697–702, 2002. https://doi.org/10.1016/s1631-0721(02)01522-x
J.B. Munyakazi and K.C. Patidar. Performance of Richardson extrapolation on some numerical methods for a singularly perturbed turning point problem whose solution has boundary layers. Journal of the Korean Mathematical Society, 51(4):679–702, 2014. https://doi.org/10.1016/j.cam.2015.05.022
D.S. Naidu and A.J. Calise. Singular perturbations and time scales in guidance and control of aerospace systems: A survey. Journal of Guidance, Control, and Dynamics, 24(6):1057–1078, 2001. https://doi.org/10.2514/2.4830
S. Natesan, J. Jayakumar and J. Vigo-Aguiar. Parameter uniform numerical method for singularly perturbed turning point problems exhibiting boundary layers. Journal of Computational and Applied Mathematics, 158(1):121–134, 2003. https://doi.org/10.1016/s0377-0427(03)00476-x
S. Natesan and M. Ramanujam. Initial-value technique for singularlyperturbed turning-point problems exhibiting twin boundary layers. Journal of Optimization Theory and Applications, 99(1):37–52, 1998. https://doi.org/10.1023/a:1021744025980
S. Natesan and N. Ramanujam. A computational method for solving singularly perturbed turning point problems exhibiting twin boundary layers. Applied Mathematics and Computation, 93(2-3):259–275, 1998. https://doi.org/10.1016/s0096-3003(97)10056-x
A.H. Nayfeh. Introduction to Perturbation Techniques. John Wiley & Sons, 2011.
K. Phaneendra, S. Rakmaiah and M.C.K. Reddy. Numerical treatment of singular perturbation problems exhibiting dual boundary layers. Ain Shams Engineering Journal, 6(3):1121–1127, 2015. https://doi.org/10.1016/j.asej.2015.02.012
K.K. Sharma, P. Rai and K.C. Patidar. A review on singularly perturbed differential equations with turning points and interior layers. Applied Mathematics and Computation, 219(22):10575–10609, 2013. https://doi.org/10.1016/j.amc.2013.04.049
F. Shen, P. Ju, M. Shahidehpour, Z. Li, C. Wang and X. Shi. Singular perturbation for the dynamic modeling of integrated energy systems. IEEE Transactions on Power Systems, 35(3):1718–1728, 2020. https://doi.org/10.1109/tpwrs.2019.2953672
G. Sun and M. Stynes. Finite element methods on piecewise equidistant meshes for interior turning point problems. Numerical Algorithms, 8(1):111–129, 1994. https://doi.org/10.1007/bf02145699
M. Uzunca, B. Karasözen and M. Manguoğlu. Adaptive discontinuous Galerkin methods for non-linear diffusion–convection–reaction equations. Computers & Chemical Engineering, 68:24–37, 2014. https://doi.org/10.1016/j.compchemeng.2014.05.002
F. Verhulst. Methods and Applications of Singular Perturbations. Springer New York, 2005. https://doi.org/10.1007/0-387-28313-7
R. Vulanović and P.A. Farrell. Continuous and numerical analysis of a multiple boundary turning point problem. SIAM Journal on Numerical Analysis, 30(5):1400–1418, 1993. https://doi.org/10.1137/0730073
S. Becher and H.-G. Roos. Richardson extrapolation for a singularly perturbed turning point problem with exponential boundary layers. Journal of Computational and Applied Mathematics, 290:334–351, 2015. https://doi.org/10.1016/j.cam.2015.05.022
A.E. Berger, H.D. Han and R.B. Kellogg. A priori estimates and analysis of a numerical method for a turning point problem. Mathematics of Computation, 42(166):465–492, 1984. https://doi.org/10.1090/s0025-5718-1984-0736447-2
J.P. Boyd. The devil’s invention: Asymptotic, superasymptotic and hyperasymptotic series. Acta Applicandae Mathematicae, 56(1):1–98, 1999. https://doi.org/10.1023/a:1006145903624
T. Braun, J. Reuter and J. Rudolph. A singular perturbation approach to nonlinear observer design with an application to electromagnetic actuators. International Journal of Control, 93(9):2015–2028, 2018. https://doi.org/10.1080/00207179.2018.1539873
P. Cathalifaud, J. Mauss and J. Cousteix. Nonlinear aspects of high Reynolds number channel flows. European Journal of Mechanics - B/Fluids, 29(4):295– 304, 2010. https://doi.org/10.1016/j.euromechflu.2010.02.002
S. Cengizci. An asymptotic-numerical hybrid method for solving singularly perturbed linear delay differential equations. International Journal of Differential Equations, 2017:1–8, 2017. https://doi.org/10.1155/2017/7269450
L.-Y. Chen, N. Goldenfeld and Y. Oono. Renormalization group and singular perturbations: Multiple scales, boundary layers, and reductive perturbation theory. Physical Review E, 54(1):376–394, 1996. https://doi.org/10.1103/physreve.54.376
J. Cousteix and J. Mauss. Approximations of the Navier–Stokes equations for high Reynolds number flows past a solid wall. Journal of Computational and Applied Mathematics, 166(1):101–122, 2004. https://doi.org/10.1016/j.cam.2003.09.035
J. Cousteix and J. Mauss. Asymptotic analysis and boundary layers. Springer Science & Business Media, 2007. https://doi.org/10.1007/978-3-540-46489-1
J. Cousteix and J. Mauss. Interactive boundary layers in turbulent flow. Comptes Rendus M´ecanique, 335(9-10):590–605, 2007. https://doi.org/10.1016/j.crme.2007.08.014
J.P. Fouque, G. Papanicolaou, R. Sircar and K. Solna. Singular perturbations in option pricing. SIAM Journal on Applied Mathematics, 63(5):1648–1665, 2003. https://doi.org/10.1137/s0036139902401550
F.Z. Geng and S.P. Qian. Reproducing kernel method for singularly perturbed turning point problems having twin boundary layers. Applied Mathematics Letters, 26(10):998–1004, 2013. https://doi.org/10.1016/j.aml.2013.05.006
E. Hairer and G. Wanner. Solving Ordinary Differential Equations II. Springer Berlin Heidelberg, 1996. https://doi.org/10.1007/978-3-642-05221-7
E.J. Hinch. Perturbation Methods. Cambridge University Press, 1991. https://doi.org/10.1017/cbo9781139172189
A. Iserles. A First Course in the Numerical Analysis of Differential Equations. Cambridge Texts in Applied Mathematics. Cambridge University Press, New York, 2nd edition, 2008.
M.K. Kadalbajoo, P. Arora and V. Gupta. Collocation method using artificial viscosity for solving stiff singularly perturbed turning point problem having twin boundary layers. Computers & Mathematics with Applications, 61(6):1595–1607, 2011. https://doi.org/10.1016/j.camwa.2011.01.028
M.K. Kadalbajoo and V. Gupta. A parameter uniform B-spline collocation method for solving singularly perturbed turning point problem having twin boundary layers. International Journal of Computer Mathematics, 87(14):3218– 3235, 2010. https://doi.org/10.1080/00207160902980492
M.K. Kadalbajoo and K.C. Patidar. Variable mesh spline approximation method for solving singularly perturbed turning point problems having boundary layer(s). Computers & Mathematics with Applications, 42(10-11):1439–1453, 2001. https://doi.org/10.1016/s0898-1221(01)00253-x
M.K. Kadalbajoo and Y.N. Reddy. Asymptotic and numerical analysis of singular perturbation problems: A survey. Applied Mathematics and Computation, 30(3):223–259, 1989. https://doi.org/10.1016/0096-3003(89)90054-4
R.B. Kellogg and A. Tsan. Analysis of some difference approximations for a singular perturbation problem without turning points. Mathematics of Computation, 32(144):1025–1039, 1978. https://doi.org/10.1090/s0025-5718-1978-0483484-9
J. Kevorkian and J.D. Cole. Multiple Scale and Singular Perturbation Methods. Springer New York, 1996. https://doi.org/10.1007/978-1-4612-3968-0
J.A. Kierzenka and L.F. Shampine. A BVP solver that controls residual and error. Journal of Numerical Analysis, Industrial and Applied Mathematics, 3(12):27–41, 2008.
D. Kumar. A parameter-uniform method for singularly perturbed turning point problems exhibiting interior or twin boundary layers. International Journal of Computer Mathematics, 96(5):865–882, 2018. https://doi.org/10.1080/00207160.2018.1458098
M. Kumar and Parul. Methods for solving singular perturbation problems arising in science and engineering. Mathematical and Computer Modelling, 54(1-2):556– 575, 2011. https://doi.org/10.1016/j.mcm.2011.02.045
P.A. Lagerstrom. Matched Asymptotic Expansions: Ideas and Techniques. Springer, New York, 1988. https://doi.org/10.1007/978-1-4757-1990-1
J. Mauss and J. Cousteix. Uniformly valid approximation for singular perturbation problems and matching principle. Comptes Rendus M´ecanique, 330(10):697–702, 2002. https://doi.org/10.1016/s1631-0721(02)01522-x
J.B. Munyakazi and K.C. Patidar. Performance of Richardson extrapolation on some numerical methods for a singularly perturbed turning point problem whose solution has boundary layers. Journal of the Korean Mathematical Society, 51(4):679–702, 2014. https://doi.org/10.1016/j.cam.2015.05.022
D.S. Naidu and A.J. Calise. Singular perturbations and time scales in guidance and control of aerospace systems: A survey. Journal of Guidance, Control, and Dynamics, 24(6):1057–1078, 2001. https://doi.org/10.2514/2.4830
S. Natesan, J. Jayakumar and J. Vigo-Aguiar. Parameter uniform numerical method for singularly perturbed turning point problems exhibiting boundary layers. Journal of Computational and Applied Mathematics, 158(1):121–134, 2003. https://doi.org/10.1016/s0377-0427(03)00476-x
S. Natesan and M. Ramanujam. Initial-value technique for singularlyperturbed turning-point problems exhibiting twin boundary layers. Journal of Optimization Theory and Applications, 99(1):37–52, 1998. https://doi.org/10.1023/a:1021744025980
S. Natesan and N. Ramanujam. A computational method for solving singularly perturbed turning point problems exhibiting twin boundary layers. Applied Mathematics and Computation, 93(2-3):259–275, 1998. https://doi.org/10.1016/s0096-3003(97)10056-x
A.H. Nayfeh. Introduction to Perturbation Techniques. John Wiley & Sons, 2011.
K. Phaneendra, S. Rakmaiah and M.C.K. Reddy. Numerical treatment of singular perturbation problems exhibiting dual boundary layers. Ain Shams Engineering Journal, 6(3):1121–1127, 2015. https://doi.org/10.1016/j.asej.2015.02.012
K.K. Sharma, P. Rai and K.C. Patidar. A review on singularly perturbed differential equations with turning points and interior layers. Applied Mathematics and Computation, 219(22):10575–10609, 2013. https://doi.org/10.1016/j.amc.2013.04.049
F. Shen, P. Ju, M. Shahidehpour, Z. Li, C. Wang and X. Shi. Singular perturbation for the dynamic modeling of integrated energy systems. IEEE Transactions on Power Systems, 35(3):1718–1728, 2020. https://doi.org/10.1109/tpwrs.2019.2953672
G. Sun and M. Stynes. Finite element methods on piecewise equidistant meshes for interior turning point problems. Numerical Algorithms, 8(1):111–129, 1994. https://doi.org/10.1007/bf02145699
M. Uzunca, B. Karasözen and M. Manguoğlu. Adaptive discontinuous Galerkin methods for non-linear diffusion–convection–reaction equations. Computers & Chemical Engineering, 68:24–37, 2014. https://doi.org/10.1016/j.compchemeng.2014.05.002
F. Verhulst. Methods and Applications of Singular Perturbations. Springer New York, 2005. https://doi.org/10.1007/0-387-28313-7
R. Vulanović and P.A. Farrell. Continuous and numerical analysis of a multiple boundary turning point problem. SIAM Journal on Numerical Analysis, 30(5):1400–1418, 1993. https://doi.org/10.1137/0730073