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- W1978798265 abstract "When the standard Runge-Kutta method is applied to a certain system of linear differential equations, the numerical solution converges to the origin but the true solution diverges to infinity. This phenomenon is named the abnormal convergence of numerical solutions. Abnormal convergence has not been studied fully; however, many concepts involving A-stability have been established to treat the phenomenon that the numerical solution diverges even though the true one converges, as is apt to occur in integration of a stiff system.An example will be illustrated here in which the standard Runge-Kutta method generates abnormal convergence. Further, it will be proved that the abnormality is caused by the existence of an intersection of the absolute stability region and the right-half plane. Even if the 1/8 formula method is used instead of the standard one, the same phenomenon can be seen because all four-stage, fourth-order, explicit Runge-Kutta methods have the unique absolute stability region.This paper seeks for Runge-Kutta methods that are A-stable and never generate abnormal convergence, even though this might be an excessive requirement. Concretely, two types of methods will be proposed whose absolute stability region coincides with the left-half plane. Furthermore, it will be reported that Gauss-Legendre methods are typical ones." @default.
- W1978798265 created "2016-06-24" @default.
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- W1978798265 date "1992-03-01" @default.
- W1978798265 modified "2023-09-25" @default.
- W1978798265 title "Abnormal convergence of numerical solutions and Runge-Kutta-type methods admitting no such phonemenon" @default.
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- W1978798265 doi "https://doi.org/10.1002/ecjc.4430750309" @default.
- W1978798265 hasPublicationYear "1992" @default.
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