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**Monotone method and convergence acceleration for finite-difference solutions of parabolic problems with time delays.**
*(English)*
Zbl 0839.65096

For multidimensional quasilinear systems of reaction-diffusion equations with time delays monotone finite difference iterative schemes are studied. By means of the method of upper and lower solutions existence, comparison, and uniqueness theorems are established. The author obtains the convergence in \(C\)-norm of the discrete solution to the continuous solution of differential problem with first order by time and second order by space. For a scheme obtained from the method mentioned above by means of the developed convergence acceleration method quadratic convergence of the monotone sequences of the finite difference systems to the solution is shown. Significant numerical results for some model problems are discussed.

Reviewer: P.Matus (Minsk)

### MSC:

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

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

35K57 | Reaction-diffusion equations |

### Keywords:

multidimensional quasilinear systems; reaction-diffusion equations with time delays; monotone finite difference iterative schemes; method of upper and lower solutions; convergence acceleration; quadratic convergence; numerical results
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\textit{X. Lu}, Numer. Methods Partial Differ. Equations 11, No. 6, 591--602 (1995; Zbl 0839.65096)

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