A higher order accurate solution decomposition scheme for a singularly perturbed parabolic reaction-diffusion equation.

*(English. Russian original)*Zbl 1318.65056
Comput. Math. Math. Phys. 55, No. 3, 386-409 (2015); translation from Zh. Vychisl. Mat. Mat. Fiz. 55, No. 3, 393-416 (2015).

Summary: An initial-boundary value problem is considered for a singularly perturbed parabolic reaction-diffusion equation. For this problem, a technique is developed for constructing higher order accurate difference schemes that converge \(\varepsilon\)-uniformly in the maximum norm (where \(\varepsilon\) is the perturbation parameter multiplying the highest order derivative, \(\varepsilon\in(0, 1]\)). A solution decomposition scheme is described in which the grid subproblems for the regular and singular solution components are considered on uniform meshes. The Richardson technique is used to construct a higher order accurate solution decomposition scheme whose solution converges \(\varepsilon\)-uniformly in the maximum norm at a rate of \(\mathcal O(N^{-4}\ln^4N+N_0^{-2})\), where \(N+1\) and \(N_0+1\) are the numbers of nodes in uniform meshes in \(x\) and \(t\), respectively. Also, a new numerical-analytical Richardson scheme for the solution decomposition method is developed. Relying on the approach proposed, improved difference schemes can be constructed by applying the solution decomposition method and the Richardson extrapolation method when the number of embedded grids is more than two. These schemes converge \(\varepsilon\)-uniformly with an order close to the sixth in \(x\) and equal to the third in \(t\).

##### MSC:

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

35B25 | Singular perturbations in context of PDEs |

35K57 | Reaction-diffusion equations |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

##### Keywords:

singularly perturbed initial-boundary value problem; parabolic reaction-diffusion equation; perturbation parameter \(\varepsilon\); solution decomposition method; numerical-analytical scheme; improved Richardson difference scheme; \(\varepsilon\)-uniform convergence; maximum norm
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\textit{G. I. Shishkin} and \textit{L. P. Shishkina}, Comput. Math. Math. Phys. 55, No. 3, 386--409 (2015; Zbl 1318.65056); translation from Zh. Vychisl. Mat. Mat. Fiz. 55, No. 3, 393--416 (2015)

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##### References:

[1] | G. I. Marchuk and V. V. Shaidurov, Improvement of the Accuracy of Solutions to Difference Schemes (Nauka, Moscow, 1979) [in Russian]. · Zbl 0496.65046 |

[2] | Hemker, P W; Shishkin, G I; Shishkina, L P, High-order accurate decomposition of richardson’s method for a singularly perturbed elliptic reaction-diffusion equation, Comput. Math. Math. Phys., 44, 309-316, (2004) · Zbl 1114.65128 |

[3] | Shishkina, L P, The Richardson method of high-order accuracy in \(t\) for a semilinear singularly perturbed parabolic reaction-diffusion equation on a strip, 927-931, (2004), Novosibirsk |

[4] | Shishkin, G I; Shishkina, L P, A higher-order Richardson method for a quasilinear singularly perturbed elliptic reaction-diffusion equation, Differ. Equations, 41, 1030-1039, (2005) · Zbl 1096.65107 |

[5] | Shishkin, G I, Robust novel high-order accurate numerical methods for singularly perturbed convection-diffusion problems, Math. Model. Anal., 10, 393-412, (2005) · Zbl 1095.65089 |

[6] | Shishkin, G I, Richardson’s method for increasing the accuracy of difference solutions of singularly perturbed elliptic convection-diffusion equations, Russ. Math., 50, 57-71, (2006) |

[7] | Shishkin, G I, The Richardson scheme for the singularly perturbed parabolic reaction-diffusion equation in the case of a discontinuous initial condition, Comput. Math. Math. Phys., 49, 1348-1368, (2009) · Zbl 1199.65284 |

[8] | Shishkin, G I; Shishkina, L P, A higher order Richardson scheme for a singularly perturbed semilinear elliptic convection-diffusion equation, Comput. Math. Math. Phys., 50, 437-456, (2010) · Zbl 1224.76135 |

[9] | G. I. Shishkin and L. P. Shishkina, Difference Methods for Singular Perturbation Problems (Chapman and Hall/CRC, Boca Raton, 2009). · Zbl 1163.65062 |

[10] | Shishkin, G I; Shishkina, L P, Improved finite difference scheme of the solution decomposition method for a singularly perturbed reaction-diffusion equation, Proc. Steklov Inst. Math., 272, 197-214, (2010) · Zbl 1227.65067 |

[11] | Il’in, A M, Differencing scheme for a differential equation with a small parameter affecting the highest derivative, Math. Notes, 6, 596-602, (1969) · Zbl 0191.16904 |

[12] | Allen, D N; Southwell, R V, Relaxation methods applied to determine the motion, in two dimensions, of viscous fluid past a fixed cylinder, Quart. J. Mech. Appl. Math., 8, 129-145, (1955) · Zbl 0064.19802 |

[13] | Bakhvalov, N S, The optimization of methods of solving boundary value problems with a boundary layer, USSR Comput. Math. Math. Phys., 9, 139-166, (1969) · Zbl 0228.65072 |

[14] | Miller, J J H; O’Riordan, E, Necessity of fitted operators and shishkin meshes for resolving thin layer phenomena, CWI Quarterly, 10, 207-213, (1997) · Zbl 0909.65051 |

[15] | Shishkin, G I, Approximation of solutions of singularly perturbed boundary value problems with a parabolic boundary layer, USSR Comput. Math. Math. Phys., 29, 1-10, (1989) · Zbl 0709.65073 |

[16] | Shishkin, G I, Difference scheme of the solution decomposition method for a singularly perturbed parabolic reaction-diffusion equation, Russ. J. Numer. Anal. Math. Model., 25, 261-278, (2010) · Zbl 1192.65139 |

[17] | Shishkin, G I; Shishkina, L P, A Richardson scheme of the decomposition method for solving singularly perturbed parabolic reaction-diffusion equation, Comput. Math. Math. Phys., 50, 2003-2022, (2010) · Zbl 1224.35206 |

[18] | H. Schlichting, Boundary Layer Theory (McGraw-Hill, New York, 1979). · Zbl 0434.76027 |

[19] | Shishkin, G I; Shishkina, L P, Improved approximations of the solution and derivatives to a singularly perturbed reaction-diffusion equation based on the solution decomposition method, Comput. Math. Math. Phys., 51, 1020-1049, (2011) · Zbl 1249.35164 |

[20] | A. A. Samarskii, The Theory of Finite Difference Schemes (Nauka, Moscow, 1989; Marcel Dekker, New York, 2001). |

[21] | A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, Handbook of First-Order Partial Differential Equations (Fizmatlit, Moscow, 2001; Taylor and Francis, London, 2002). · Zbl 1031.35001 |

[22] | N. S. Bakhvalov, N. P. Zhidkov, and G. M. Kobel’kov, Numerical Methods (Laboratoriya Bazovykh Znanii, Moscow, 2001) [in Russian]. |

[23] | G. I. Shishkin, Grid Approximations of Singularly Perturbed Elliptic and Parabolic Equations (Ural Otd. Ross. Akad. Nauk, Yekaterinburg, 1992) [in Russian]. · Zbl 1397.65005 |

[24] | O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type (Nauka, Moscow, 1967; Am. Math. Soc., Providence, R.I., 1968). · Zbl 0164.12302 |

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