Combined nonstandard numerical methods for ODEs with polynomial right-hand sides
The paper focuses on the nonstandard one-step finite-difference methods for general first order autonomus ordinary differential equations (ODE): \(\frac{dx}{dt}=f(x),\; x(t_0)=x_0,\) where the function \(f:\mathbb{R}\to\mathbb{R}\) is differentiable and \(x_0\in\mathbb{R}.\) The authors state conditions that guarantee second-order accuracy of these methods and develop a new class of this type of numerical methods. The first part is an introduction concerning the references related to this subject. The main one is given by \textit{R. E. Mickens}' research [Nonstandard finite difference models of differential equations (World Scientific, Singapore) (1994; Zbl 0810.65083)]. The second part provides some definitions of the mathematical notions used through this material and presents two sets of finite-difference schemes, based on results obtained by \textit{H. V. Kojouharov} and \textit{B. M. Chen-Charpentier} [Appl. Numer. Math. 49, No. 2, 225--243 (2004; Zbl 1146.76625)]. The main results are stated in the third section. The authors derive necessary and sufficient conditions for second-order accuracy of the general nonstandard one-step finite-difference method \[ \frac{x_{k+1}-x_k}{\varphi(h)}=g(x_k,x_{k+1},h) \] where \(\varphi(h)=h+O(h^2)\) is a nonnegative function. They apply these results to develop a new class of second-order nonstandard finite-difference schemes for the equation \(\frac{dx}{dt}=\text{ polynomial}(x).\) The demonstrations of the main results presented in the previous section are given in the fourth one. The last part contains a set of numerical examples which validate the efficiency of the theoretical results and some future research directions.
scientific article; zbMATH DE number 5073220
scientific article