The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods ab 26.49 € als Taschenbuch: Auflage 1989. Aus dem Bereich: Bücher, Wissenschaft, Mathematik,
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 book has three chapters and an Appendix on different methods to calculate approximate areas of plain regions enclosed by curves. Tutorials, Tests and Examination Papers of different Universities where the author taught the course at the faculties of Engineering and Business Studies are provided. A short bibliography of books on the subject and alphabetical index of the topics covered are given in the end.The first chapter accounts the different methods for numerical solutions of ordinary differential equations. Picard's, Taylor's, Euler's, Runge-Kutta, Milne's and Adams-Bashforth's methods are given. The problems of curve fittings and spline fittings are explained in the second chapter. The Gaussian method of least squares' for the curve of best fit' is included. The concepts of Linear Programming are studied in the third chapter. Solution(s) of linear relations obtained analytically are also discussed by graphical methods. General problems of Linear Programming (LPP), canonical and standard forms of LPP and Simplex methods are discussed. Trapezoidal rule and Simpson's rules for approximate areas of plain regions are included in the Appendix
A numerical analysis was performed as to investigate the heat transfer characteristics of an evaporating thin-film meniscus. A mathematical model was used in the formulation of a third order ordinary differential equation. This equation governs the evaporating thinfilm through use of continuity, momentum, energy equations and the Kelvin-Clapeyron model. This governing equation was treated as an initial value problem and was solved numerically using a Runge-Kutta technique. The numerical model uses varying thermophysical properties and boundary conditions such as channel width, applied superheat, accommodation coefficient and working fluid which can be tailored by the user. This work focused mainly on the effects of altering accommodation coefficient and applied superheat. A unified solution is also presented which models the meniscus to half channel width. The model was validated through comparison to literature values.
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. The finite element method (FEM) (its practical application often known as finite element analysis (FEA)) is a numerical technique for finding approximate solutions of partial differential equations (PDE) as well as of integral equations. The solution approach is based either on eliminating the differential equation completely (steady state problems), or rendering the PDE into an approximating system of ordinary differential equations, which are then numerically integrated using standard techniques such as Euler''s method, Runge-Kutta, etc.
This multi-author contributed proceedings volume contains recent advances in several areas of Computational and Applied Mathematics. Each review is written by well known leaders of Computational and Applied Mathematics.The book gives a comprehensive account of a variety of topics including - Efficient Global Methods for the Numerical Solution of Nonlinear Systems of Two point Boundary Value Problems, Advances on collocation based numerical methods for Ordinary Differential Equations and Volterra Integral Equations, Basic Methods for Computing Special Functions, Melt Spinning: Optimal Control and Stability Issues, Brief survey on the CP methods for the Schrödinger equation, Symplectic Partitioned Runge-Kutta methods for the numerical integration of periodic and oscillatory problems.Recent Advances in Computational and Applied Mathematics is aimed at advanced undergraduates and researchers who are working in these fast moving fields.
The study of flow-induced noise, known as aeroacoustics, is concerned with the sound generated by turbulent and/or unsteady vortical flows including the effects of any solid boundaries in the flow. Flow-induced noise is a serious problem in many engineering applications. The most notorious one is the jet noise. In this work a shock-capturing adaptive filter is designed. It is indispensable with non linear propagation of acoustic waves or in cases when discontinuities are presents. With the combination of Dispersion Relation Schemes, Runge-Kutta optimized algorithm, selective and nonlinear filter it is possible to calculate directly acoustic and aerodynamic field in a single computation, by solving the compressible unsteady Navier-Stokes equations. Actually, traditional theories using acoustic analogy are appropriate for predicting noise when a sufficiently accurate solution of turbulent flow is available. On the contrary, the approach proposed here can be applied to cases when aerodynamic sources are influenced by acoustic motion such as flames, jet or wake instabilities. This work has been done in collaboration with Dr. Christophe Bogey of École Centrale de Lyon, France.
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Linear multistep methods are used for the numerical solution of ordinary differential equations. Conceptually, a numerical method starts from an initial point and then takes a short step forward in time to find the next solution point. The process continues with subsequent steps to map out the solution. Single-step methods (such as Euler''s method) refer to only one previous point and its derivative to determine the current value. Methods such as Runge-Kutta take some intermediate steps (for example, a half-step) to obtain a higher order method, but then discard all previous information before taking a second step. Multistep methods attempt to gain efficiency by keeping and using the information from previous steps rather than discarding it. Consequently, multistep methods refer to several previous points and derivative values. In the case of linear multistep methods, a linear combination of the previous points and derivative values is used.