Second derivative general linear methods with inherent Runge-Kutta stability. (English) Zbl 1351.65051
Summary: In this paper, we find some relationships among the coefficients matrices of second derivative general linear methods (SGLMs) which are sufficient conditions, but not necessary, to ensure the methods have Runge-Kutta stability (RKS) property. Considering these conditions, we construct some \(A\)- and \(L\)-stable SGLMs with inherent RKS of orders up to five. Also, some numerical experiments for the constructed methods in variable stepsize environment are given.
MSC:
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 65L20 | Stability and convergence of numerical methods for ordinary differential equations |
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 34A34 | Nonlinear ordinary differential equations and systems |
| 65L04 | Numerical methods for stiff equations |
| 65L50 | Mesh generation, refinement, and adaptive methods for ordinary differential equations |
Keywords:
stiff differential equations; second derivative methods; inherent Runge-Kutta stability; \(A\)-stability; \(L\)-stability; variable stepsize; numerical experimentsReferences:
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