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Runge-Kutta methods with real eigenvalues

其他書名	dissertation
出版	B.Orel, 1991
URL	http://books.google.com.hk/books?id=p3YQtwAACAAJ&hl=&source=gbs_api

註釋We study Runge-Kutta methods with real eigenvalues for the solution of stiff initial value problems on parallel machines. Rational approximations to the exponential function are discussed first. A new family of biorthogonal polynomials is constructed as a basis of rational approximations with distinct real nodes. This biorthogonal polynomials are a natural generalization of the classic Laguerre polynomials. It is proved tyhat their zeros are all real, positive and distinct. The concept of the "real pole sandwich" is generalized to the subdiagonal approximations and the conjecture on the least value of the error constant is proved. The results on the study of A-acceptability of real pole rational approximations extend the known A-acceptability barriers bellow the main diagonal of the Padé tableau. Runge-Kutta methods which are based on the real pole approximations are studied. Multi-implicit Runge-Kutta methods with the stability function from the main diagonal of the Padé tableau can be constructed by collocation. Methods with the stability function below the main diagonal are not collocation type and the problem of calculating their coefficients was solved with a proper modification of the collocation-type construction. The local error estimate is provided by the use of embedding technique: an $s$-stage method is embedded within an $s+1$-stage error estimating method with order of accuracy greater than the order of embedded method. It is shown, that any $s$-stage L-acceptable method of order $s$ can be embedded within an $s+1$-stage method of order $s+2$.