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May 2020, 13(5): 1567-1587. doi: 10.3934/dcdss.2020089
Stability and errors analysis of two iterative schemes of fractional steps type associated to a nonlinear reaction-diffusion equation
Costică Moroşanu
University "Al. I. Cuza" of Iaşi, 700506, Iaşi, Romania
Received January 2018 Revised August 2018 Published June 2019
We present the error analysis of two time-stepping schemes of fractional steps type, used in the discretization of a nonlinear reaction-diffusion equation with Neumann boundary conditions, relevant in phase transition and interface problems. We start by investigating the solvability of a such boundary value problems in the class $ W^{1,2}_p(Q) $. One proves the existence, the regularity and the uniqueness of solutions, in the presence of the cubic nonlinearity type. The convergence and error estimate results (using energy methods) for the iterative schemes of fractional steps type, associated to the nonlinear parabolic equation, are also established. The advantage of such method consists in simplifying the numerical computation. On the basis of this approach, a conceptual algorithm is formulated in the end. Numerical experiments are presented in order to validates the theoretical results (conditions of numerical stability) and to compare the accuracy of the methods.
Keywords: Nonlinear PDE of parabolic type, reaction-diffusion equations, finite difference methods, fractional steps method, stability and convergence of numerical methods, performance of numerical algorithms, thermodynamics, phase-changes.
Mathematics Subject Classification: Primary: 35K55; Secondary: 35K57, 65M06, 65M12, 65Y20, 80Axx.
Citation: Costică Moroşanu. Stability and errors analysis of two iterative schemes of fractional steps type associated to a nonlinear reaction-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2020, 13 (5) : 1567-1587. doi: 10.3934/dcdss.2020089
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Figure 1. Numerical stability: $ V^i $ at different levels of time
Figure 2. Errors $ \|v_e-V_j^M\|_\infty $ of the Newton, the linearized and the fractional steps methods: (10)-(11)
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