Abstract
The Cauchy problem for a semilinear parabolic equation is considered. Under the conditions u(x, t) = X(x)T1(t) + T2(t), ∂u/∂x ≠ = 0, it is shown that the problem is equivalent to the system of two ordinary differential equations for which exact difference scheme (EDS) with special Steklov averaging and difference schemes with arbitrary order of accuracy (ADS) are constructed on the moving mesh. The special attention is paid to investigating approximation, stability and convergence of the ADS. The convergence of the iteration method is also considered. The presented numerical examples illustrate theoretical results investigated in the paper.
Highlights
In which cases an exact difference scheme (EDS) or an ADS approximating nonlinear parabolic equation can be constructed? The paper deals with this question
In papers [1, 6] the EDSs and truncated difference schemes of an arbitrary rank were constructed for the nonlinear second order differential equation and for the systems of first-order nonlinear
0, we show that problem is equivalent to the system of two ordinary differential equations [7]
Summary
In which cases an EDS or an ADS approximating nonlinear parabolic equation can be constructed? The paper deals with this question. In which cases an EDS or an ADS approximating nonlinear parabolic equation can be constructed? The problem of constructing a difference scheme of high order of accuracy is topical. Difference schemes of arbitrary order of accuracy for semilinear parabolic equations ordinary differential equations, respectively. In [10] the investigations of the order of approximation, stability, and convergence of the high accuracy difference schemes for the nonlinear transfer equation. The EDS and the difference schemes of an arbitrary order of approximation for the parabolic equations with travelling wave solutions u(x, t) = U (x − at) were constructed in [7, 8]. The main aim of this paper is to investigate approximation, stability and convergence of the nonlinear scheme of an arbitrary order of accuracy.
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