Diagonally Implicit Runge-Kutta Methods for Solving Linear ODEs ab 58.99 € als Taschenbuch: Numerical Methods for ODEs. Aus dem Bereich: Bücher, Wissenschaft, Mathematik,
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.
The numerical approximation of solutions of ordinarydifferential equations played an important role inNumerical Analysis and still continues to be anactive field of research. In this book we are mainlyconcerned with the numerical solution of thefirst-order system of nonlinear two-point boundaryvalue problems. We will focus on the problem ofsolving singular perturbation problems since this hasfor many years been a source of difficulty to appliedmathematicians, engineers and numerical analystsalike. Firstly, we consider deferred correctionschemes based on Mono-Implicit Runge-Kutta (MIRK) andLobatto formulae. As is to be expected, the schemebased on Lobatto formulae, which are implicit, ismore stable than the scheme based on MIRK formulaewhich are explicit. Secondly, we construct high orderinterpolants to provide the continuous extension ofthe discrete solution. It will consider theconstruction of both explicit and implicitinterpolants. The estimation of conditioning numbersis also discussed and used to develop mesh selectionalgorithms which will be appropriate for solvingstiff linear and nonlinear boundary value problems.
Majority of the current numerical simulation methods for time-dependent flows are limited to second order accuracy in time discretization. Moreover, usual simulation methods rely on explicit time discretization methods for which numerical stability of solution is generally guaranteed only with very small time step sizes. Major commercial CFD software programs provide options for implicit time advancing, but the accuracy is limited to second order in time. In this book a stable simulation method is proposed that can be used to achieve arbitrarily high order of accuracy in time advancement in simulation of time-dependent flows and heat transfer. The strategy is to combine the state-of-the-art mathematical tools with proven flow simulation algorithms to develop simulation techniques with higher-order accuracy. A special class of implicit Runge-Kutta methods is used in conjunction with SIMPLE algorithm. The proposed method is called SIMPLE DIRK method. This book will be helpful to university students and researchers who are involved in research and code development for simulation of time-dependent flows and heat transfer.
The 5-moment two-fluid plasma model uses Euler equations to describe the ion and electron fluids, and Maxwell's equations to describe the electric and magnetic fields. Two-fluid physics becomes significant when the characteristic spatial scales are on the order of the ion skin depth and characteristic time scales are on the order of the inverse ion cyclotron frequency. The two-fluid plasma model has disparate characteristic speeds ranging from the ion and electron speeds of sound to the speed of light. Explicit and implicit time-stepping schemes are explored for the two-fluid plasma model to study the accuracy and computational effectiveness with which they could capture two-fluid physics. The explicit schemes explored include the high resolution wave propagation method (a finite volume method) and the Runge-Kutta discontinuous Galerkin method (a finite element method). The two-fluid plasma model is compared to the more commonly used Hall-MHD model for accuracy and computational effort using an explicit time-stepping scheme. Simulations of two-fluid instabilities in the Z-pinch and the field-reversed configuration are presented in 3-dimensions.
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.
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.
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".
This research presents numerical solution of Whitham-Broer-Kaup (WBK) shallow water model using some finite difference methods such as the explicit, implicit, Crank-Nicolson, exponential, Richardson, and DuFort-Frankel methods. The local truncation errors and consistency are also studied for these methods. Furthermore, the radial basis function-pseudospectral method is successfully applied for solving WBK model numerically. In this method, the model is discretized in the space x which leads to a system of ordinary differential equations with independent variable time t. Then the forward Euler's and fourth-order Runge-Kutta methods used to solve this system of ordinary differential equations. Finally, two different problems are considered to illustrate the efficiency of the methods. A comparative study is made between the approximate solution obtained by the presented numerical methods and the exact solution. The MATLAB codes have been written for all methods as can be found in the appendix.