Third order iterative method for nonlinear difference schemes. (English) Zbl 1454.65102

Iumanova, Irina; Solodushkin, Svyatoslav

Pinelas, Sandra (ed.) et al., Differential and difference equations with applications. Selected papers based on the presentations at the fourth international conference, ICDDEA 2019, Lisbon, Portugal, July 1–5, 2019. Cham: Springer. Springer Proc. Math. Stat. 333, 373-387 (2020).

Summary: A partial differential equation with fractional Riesz derivative and non-linearity in differentiation operator is studied. We considered an implicit method which is a fractional analogue of Crank-Nicolson method and, therefore, implies the necessity of iterative solving of non-linear system on each time layer. To accelerate the convergence we elaborate a two stage iterative method which does not use derivatives and could be considered as an analog of Stefensen’s method. The theorem of third order convergence is proved. Results of numerical examples coincides with theoretical ones.
For the entire collection see [Zbl 1445.34003].

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
65L06	Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations

Keywords:

partial differential equation; fractional derivative; non-linear operator; iterative solution

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