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**A parameter-uniform second order numerical method for a weakly coupled system of singularly perturbed convection-diffusion equations with discontinuous convection coefficients and source terms.**
*(English)*
Zbl 1380.65132

The authors propose a uniformly convergent numerical scheme for singularly perturbed systems of weakly coupled convection-diffusion two-point boundary value problems (BVPs) with discontinuous convection coefficients and source terms. The solution of this system of BVPs exhibit interior layers. To discretize the domain piecewise-uniform Shishkin meshes are used. The differential equations are approximated by the cubic spline scheme in the interior layer regions, where the meshes are fine, and by the classical finite difference scheme in the outer regions, where the meshes are coarse. At the discontinuity point a one-sided second-order difference scheme is used. The proposed hybrid scheme is almost second-order convergent with a logarithmic term. Theoretical error estimates are obtained, and numerical experiments are carried out.

Reviewer: Srinivasan Natesan (Assam)

### MSC:

65L10 | Numerical solution of boundary value problems involving ordinary differential equations |

65L11 | Numerical solution of singularly perturbed problems involving ordinary differential equations |

65L12 | Finite difference and finite volume methods for ordinary differential equations |

34E15 | Singular perturbations for ordinary differential equations |

65L50 | Mesh generation, refinement, and adaptive methods for ordinary differential equations |

65L70 | Error bounds for numerical methods for ordinary differential equations |

65L20 | Stability and convergence of numerical methods for ordinary differential equations |

34B15 | Nonlinear boundary value problems for ordinary differential equations |

### Keywords:

singular perturbation; convection-diffusion problems; interior layers; Shishkin meshes; finite difference scheme; cubic spline; error estimates; convergence; numerical experiments
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\textit{M. B. Pathan} and \textit{S. Vembu}, Calcolo 54, No. 3, 1027--1053 (2017; Zbl 1380.65132)

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### References:

[1] | Naidu, D.S.: Singular Perturbation Methodology in Control Systems. Peter Peregrinus Ltd. on behalf of Institution of Electrical Engineers, London (1988) · Zbl 0697.93006 |

[2] | Kokotovic, P.V.: Applications of singular perturbation techniques to control problems. SIAM Rev. 26(4), 501-550 (1984) · Zbl 0548.93001 |

[3] | Linss, T., Stynes, M.: Numerical solution of systems of singularly perturbed differential equations. Comput. Methods Appl. Math. 9(2), 165-191 (2009) · Zbl 1193.65189 |

[4] | Farrell, P.A., Hegarty, A.F., Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Robust Computational Techniques for Boundary Layers. Chapman & Hall/CRC Press, Boca Raton (2000) · Zbl 0964.65083 |

[5] | Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Fitted Numerical Methods for Singular Perturbation Problems, revised edn. World-Scientific, Singapore (2012) · Zbl 1243.65002 |

[6] | Roos, H.G., Stynes, M., Tobiska, L.: Numerical Methods for Singularly Perturbed Differential Equations, 2nd edn. Springer, Berlin (2008) · Zbl 1155.65087 |

[7] | Linss, T.: Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems. Springer, Berlin (2010) · Zbl 1202.65120 |

[8] | Farrell, P.A., Hegarty, A.F., Miller, J.J.H., Riordan, E.O., Shishkin, G.I.: Singularly perturbed convection-diffusion problems with boundary and weak interior layers. Comput. Appl. Math. 166, 133-151 (2004) · Zbl 1041.65059 |

[9] | Farrell, P.A., Hegarty, A.F., Miller, J.J.H., Riordan, E.O., Shishkin, G.I.: Global maximum norm parameter-uniform numerical method for a singularly perturbed convection-diffusion problem with discontinuous convection coefficient. Math. Comput. Model. 40, 1375-1392 (2004) · Zbl 1075.65100 |

[10] | Cen, Z.: A hybrid difference scheme for a singularly perturbed convection-diffusion problem with discontinuous convection coefficient. Appl. Math. Comput. 169, 689-699 (2005) · Zbl 1087.65071 |

[11] | Bellew, S., O’Riordan, E.: A parameter robust numerical method for a system of two singularly perturbed convection-diffusion equations. Appl. Numer. Math. 51, 171-186 (2004) · Zbl 1059.65063 |

[12] | Linss, T.: Analysis of an upwind finite-difference scheme for a system of coupled singularly perturbed convection-diffusion equations. Computing 79, 23-32 (2007) · Zbl 1115.65084 |

[13] | Cen, Z.: Parameter-uniform finite dfference scheme for a system of coupled singularly perturbed convection-diffusion equations. Int. J. Comput. Math. 82, 177-192 (2005) · Zbl 1068.65101 |

[14] | Das, P., Natesan, S.: Numerical solution of a system of singularly perturbed convection-diffusion boundary value problems using mesh equidistribution technique. Aust. J. Math. Anal. Appl. 10(1), 1-7 (2007) |

[15] | Das, P., Natesan, S.: A uniformly convergent hybrid scheme for singularly perturbed system of reaction-diffusion robin type boundary value problems. J. Appl. Math. Comput. 41(1-2), 447-471 (2013) · Zbl 1299.65153 |

[16] | Priyadharshini, R.M., Ramanujam, N.: Uniformly convergent numerical methods for a system of coupled singularly perturbed convection-diffusion equations with mixed type boundary conditions. Math. Model. Anal. 18(5), 577-598 (2013) · Zbl 1293.65108 |

[17] | Tamilselvan, A., Ramanujam, N.: A numerical method for singularly perturbed system of second order ordinary differential equations of convection-diffusion type with a discontinuous source term. J. Appl. Math. Inform. 27(5-6), 1279-1292 (2009) |

[18] | Tamilselvan, A., Ramanujam, N.: A parameter uniform numerical method for a system of singularly perturbed convection-diffusion equations with discontinuous convection coefficients. Int. J. Comput. Math. 87(6), 1374-1388 (2010) · Zbl 1195.65098 |

[19] | Tamilselvan, A., Ramanujam, N.: An almost second order method for a system of singularly perturbed convection-diffusion equations with nonsmooth convection coefficients and source terms. Int. J. Comput. Math. 7(2), 261-277 (2010) · Zbl 1267.76077 |

[20] | Valanarasu, T., Priyadharshini, R.M., Ramanujam, N., Tamilselvan, A.: An \[\varepsilon\] ε-uniform numerical method for a system of convection-diffusion equations with discontinuous convection coefficients and source terms. Appl. Appl. Math. Int. J. 8(1), 191-213 (2013) · Zbl 1270.65040 |

[21] | Chawla, S., Chandra Sekhara Rao, S.: A uniformly convergent numerical method for a weakly coupled system of singularly perturbed convection-diffusion problems with boundary and weak interior layers. J. Appl. Math. Inform. 33(5-6), 635-648 (2015) · Zbl 1329.65157 |

[22] | Basha, P.M., Shanthi, V.: A parameter-uniform non-standard finite difference method for a weakly coupled system of singularly perturbed convection-diffusion equations with discontinuous source term. Int. J. Adv. Appl. Math. Mech. 3(2), 5-15 (2015) · Zbl 1359.65124 |

[23] | Basha, P.M., Shanthi, V.: A numerical method for singularly perturbed second order coupled system of convection-diffusion \[{R}\] Robin type boundary value problems with discontinuous source term. Int. J. Appl. Comput. Math. 1(3), 381-397 (2015) · Zbl 1398.65183 |

[24] | Tamilselvan, A., Ramanujam, N., Shanthi, V.: A numerical method for singularly perturbed weakly coupled system of two second order ordinary differential equations with discontinuous source term. J. Comput. Appl. Math. 202, 203-216 (2007) · Zbl 1115.65086 |

[25] | Basha, P.M., Shanthi, V.: A uniformly convergent scheme for a system of two coupled singularly perturbed reaction-diffusion \[{R}\] Robin type boundary value problems with discontinuous source term. Am. J. Numer. Anal. 3(2), 39-48 (2015) |

[26] | Basha, P.M., Shanthi, V.: Robust computational methods for a coupled system of singularly perturbed reaction-diffusion equations with discontinuous source term. J. Mod. Methods Numer. Math. 6(2), 64-85 (2015) · Zbl 1331.65104 |

[27] | Heinkenschloss, M., Leykekhman, D.: Local error estimates for supg solutions of advection-dominated elliptic linear-quadratic optimal control problems. SIAM J. Numer. Anal. 47(6), 4607-4638 (2010) · Zbl 1218.49036 |

[28] | Roos, H.G., Reibiger, C.: Numerical analysis of a system of singularly perturbed convection-diffusion equations related to optimal control. Numer. Math. Theory Methods Appl. 4(4), 562-575 (2011) · Zbl 1265.65123 |

[29] | Allendes, A., Hernandez, E., Otarola, E.: A robust numerical method for a control problem involving singularly perturbed equations. Comput. Math. Appl. 72(4), 974-991 (2016) · Zbl 1362.49018 |

[30] | O’Riordan, E., Stynes, J., Stynes, M.: A parameter-uniform finite difference method for a coupled system of convection-diffusion two-point boundary value problems. Numer. Math. Theory Methods Appl. 1(2), 176-197 (2008) · Zbl 1174.65441 |

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