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This book deals with the derivation of diagonally implicit Runge-Kutta (DIRK) methods of order four and five which are specially designed for the integration of linear ordinary differential equations (LODEs). The restriction to LODEs with constant coefficients reduces the number of order equations which the coefficients of Runge-Kutta (RK) methods must satisfy. The coefficients of the RK methods are chosen such that the error norm is minimized, this resulted in methods which are almost one order higher than the actual order. The stability polynomials and stability regions of the methods are then obtained using MATHEMATICA package. Codes using C++ programming based on the methods are developed to test sets of problems on linear ordinary differential equations. Numerical results show that the new methods are more efficient than the existing methods.

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Erscheinungsdatum: 11/2011, Medium: Taschenbuch, Einband: Kartoniert / Broschiert, Titel: Diagonally Implicit Runge-Kutta Methods for Solving Linear ODEs, Titelzusatz: Numerical Methods for ODEs, Autor: Che Jawias, Nurizzati // Ismail, Fudziah, Verlag: LAP Lambert Acad. Publ., Sprache: Englisch, Rubrik: Mathematik // Sonstiges, Seiten: 140, Informationen: Paperback, Gewicht: 225 gr, Verkäufer: averdo

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The main contribution of this work is illustrated as follows: 1- The derivation for embedded singly diagonally implicit Runge-Kutta (SDIRK) method of fourth-order six stages in fifth-order seven stages is introduced to solve ordinary and delay differential equations. The stability region is presented and the numerical results are compared with the other existing methods. 2- Singly diagonally implicit Runge-Kutta Nystróm (SDIRKN) of third-order three stages embedded in fourth-order four stages is constructed. The stability region of the new method is presented and numerical results are compared with the same method of lower order. 3- A new singly diagonally implicit Runge-Kutta-Nystróm general (SDIRKNG) method of third-order embedded in fourth-order is derived to solve second order ordinary differential equations. Analysis the stability region of the new method is discussed and numerical results are presented.

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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In numerical analysis, the Runge Kutta methods are an important family of implicit and explicit iterative methods for the approximation of solutions of ordinary differential equations. These techniques were developed around 1900 by the German mathematicians C. Runge and M.W. Kutta. See the article on numerical ordinary differential equations for more background and other methods. See also List of Runge Kutta methods. One member of the family of Runge Kutta methods is so commonly used that it is often referred to as "RK4", "classical Runge-Kutta method" or simply as "the Runge Kutta method".

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Two implicit Hybrid block methods at step length k=2 and 3 were derived through collocation procedures. The two derived block methods also reconstructed to equivalent S stage Runge-Kutta type methods for the solution of y^'=f(x,y). Both methods were tested on the same numerical experiments. Runge-Kutta Type Methods (RKTM) gives better result over the equivalent linear multi-step methods of the same value of step length k.

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This work presents an efficient solution procedure for the elastohydrodynamic (EHD) contact problem considering structural dynamics. The contact bodies are modeled using reduced finite element models. Singly diagonal implicit Runge-Kutta (SDIRK) methods are used for adaptive time integration. The structural model is coupled with the nonlinear Reynolds Equation using a monolithic coupling approach. Finally, a reduced order model of the complete nonlinear coupled problem is constructed.

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