Integration of Newton Linearization into the Time Discretization of Initial-Boundary-Value Problems

We construct a one-step recursive scheme for the integration of the Cauchy problem for large systems of ordinary differential equations appearing after the space semidiscretization of the initial-boundary-value problems for systems of nonlinear parabolic equations. The main specific feature of the construction of this scheme is connected with balancing of the orders of the error of piecewise linear approximation with respect to time and the error of Newton linearization. The indicated feature enables us to construct a numerical predictor-corrector-type scheme with weight parameter. With the help of the principle of contracting mappings, we establish sufficient conditions for the correctness of discretized problems. It is shown that, in the case of sufficiently high regularity of the desired solution of the Cauchy problem, the proposed one-step recursive scheme can attain the quadratic rate of convergence of approximations to this solution. We present the results of numerical experiments characterizing the proposed scheme by comparing with the Runge–Kutta schemes of different orders and its application in modeling the reaction of oxidation of carbon monoxide on the platinum surface.

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Correspondence to О. V. Vovk.

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Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 56, No. 2, pp. 60–71, April–June, 2013.

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Vovk, О.V., Shynkarenko, H.A. Integration of Newton Linearization into the Time Discretization of Initial-Boundary-Value Problems. J Math Sci 203, 70–86 (2014). https://doi.org/10.1007/s10958-014-2091-9

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Keywords

  • Cauchy Problem
  • Contracting Mapping
  • Linear Algebraic Equation
  • Discretized Problem
  • Nonlinear Parabolic Equation