Solving first-order initial-value problems by using an explicit non-standard \(A\)-stable one-step method in variable step-size formulation. (English) Zbl 1410.65256
Summary: This paper presents the construction of a new family of explicit schemes for the numerical solution of initial-value problems of ordinary differential equations (ODEs). The one-parameter family is constructed by considering a suitable rational approximation to the theoretical solution, resulting a family with second-order convergence and \(A\)-stable. Imposing that the principal term in the local truncation error vanishes, we obtain an expression for the parameter value in terms of the point \((x_{n}, y_{n})\) on each step. With this approach, the resulting method has third order convergence maintaining the characteristic of \(A\)-stability. Finally, combining this last method with other of order two in order to get an estimation for the local truncation error, an implementation in variable step-size has been considered. The proposed method can be used in a wide range of problems, for solving numerically a scalar ODE or a system of first order ODEs. Several numerical examples are given to illustrate the efficiency and performance of the proposed method in comparison with some existing methods in the literature.
MSC:
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 65L20 | Stability and convergence of numerical methods for ordinary differential equations |
| 34A34 | Nonlinear ordinary differential equations and systems |
Keywords:
non-linear algorithms; ordinary differential equations; initial value problems; rational approximation; stabilityReferences:
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