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Splitting Methods For Partial Differential Equations With Rough Solutions
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Book Synopsis Splitting Methods for Partial Differential Equations with Rough Solutions by : Helge Holden
Download or read book Splitting Methods for Partial Differential Equations with Rough Solutions written by Helge Holden and published by European Mathematical Society. This book was released on 2010 with total page 238 pages. Available in PDF, EPUB and Kindle. Book excerpt: Operator splitting (or the fractional steps method) is a very common tool to analyze nonlinear partial differential equations both numerically and analytically. By applying operator splitting to a complicated model one can often split it into simpler problems that can be analyzed separately. In this book one studies operator splitting for a family of nonlinear evolution equations, including hyperbolic conservation laws and degenerate convection-diffusion equations. Common for these equations is the prevalence of rough, or non-smooth, solutions, e.g., shocks. Rigorous analysis is presented, showing that both semi-discrete and fully discrete splitting methods converge. For conservation laws, sharp error estimates are provided and for convection-diffusion equations one discusses a priori and a posteriori correction of entropy errors introduced by the splitting. Numerical methods include finite difference and finite volume methods as well as front tracking. The theory is illustrated by numerous examples. There is a dedicated Web page that provides MATLABR codes for many of the examples. The book is suitable for graduate students and researchers in pure and applied mathematics, physics, and engineering.
Book Synopsis Iterative Splitting Methods for Differential Equations by : Juergen Geiser
Download or read book Iterative Splitting Methods for Differential Equations written by Juergen Geiser and published by CRC Press. This book was released on 2011-06-01 with total page 325 pages. Available in PDF, EPUB and Kindle. Book excerpt: Iterative Splitting Methods for Differential Equations explains how to solve evolution equations via novel iterative-based splitting methods that efficiently use computational and memory resources. It focuses on systems of parabolic and hyperbolic equations, including convection-diffusion-reaction equations, heat equations, and wave equations.In th
Book Synopsis Time-split Methods for Partial Differential Equations by : Randall J. LeVeque
Download or read book Time-split Methods for Partial Differential Equations written by Randall J. LeVeque and published by . This book was released on 1982 with total page 236 pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis concerns the use of time-split methods for the numerical solution of time-dependent partial differential equations. Frequently the differential operator splits additively into two or more pieces such that the corresponding subproblems are each easier to solve than the original equation, or are best handled by different techniques. In the time-split method the solution to the original equation is advanced by alternately solving the subproblems. In this thesis a unified approach to splitting methods is developed which simplifies their analysis. Particular emphasis is given to splittings of hyperbolic problems into subproblems with disparate wave speeds. Three main aspects of the method are considered. The first is the accuracy and efficiency of the time-split method relative to unsplit methods. The second topic is stability for split methods. The final topic is the proper specification of boundary data for the intermediate solutions, e.g., the solution obtained after solving only one of the subproblems. The main emphasis is on hyperbolic problems, and the one-dimensional shallow water equations are used as a specific example throughout. The final chapter is devoted to some other applications or the theory. Two-dimensional hyperbolic problems, convection-diffusion equations, and the Peaceman-Rachford ADI method for the heat equation are considered.
Book Synopsis Splitting Methods for Fourth Order Parabolic Partial Differential Equations by : E. Jan W. ter Maten
Download or read book Splitting Methods for Fourth Order Parabolic Partial Differential Equations written by E. Jan W. ter Maten and published by . This book was released on 1982 with total page 24 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Splitting Methods for the Numerical Solution of the Incompressible Navier-Stokes Equations by : R. Glowinski
Download or read book Splitting Methods for the Numerical Solution of the Incompressible Navier-Stokes Equations written by R. Glowinski and published by . This book was released on 1984 with total page 49 pages. Available in PDF, EPUB and Kindle. Book excerpt: Splitting methods provide efficient tools for solving linear and nonlinear time dependent problems modelled by partial differential equations. In this report we discuss the numerical solution of the Navier-Stokes equations for incompressible viscous fluids by such methods. The splitting permits decoupling the two main difficulties in the problem, namely the nonlinearity and the incompressibility. Actually these splitting methods have a broad range of applicability and can be applied for example, to the solution of eigenvalue problems. Originator supplied keywords include: operator splitting methods, nonlinear least squares, preconditioned conjugate gradient algorithms, finite element approximations, eigenvalue calculation, and variational methods.
Book Synopsis Group Explicit Methods for the Numerical Solution of Partial Differential Equations by : David J. Evans
Download or read book Group Explicit Methods for the Numerical Solution of Partial Differential Equations written by David J. Evans and published by CRC Press. This book was released on 1997-05-22 with total page 478 pages. Available in PDF, EPUB and Kindle. Book excerpt: A new class of methods, termed "group explicit methods," is introduced in this text. Their applications to solve parabolic, hyperbolic and elliptic equations are outlined, and the advantages for their implementation on parallel computers clearly portrayed. Also included are the introductory and fundamental concepts from which the new methods are derived, and on which they are dependent. With the increasing advent of parallel computing into all aspects of computational mathematics, there is no doubt that the new methods will be widely used.
Book Synopsis Splitting Methods for Second-order Initial Value Problems by : Pieter Jacobus Houwen
Download or read book Splitting Methods for Second-order Initial Value Problems written by Pieter Jacobus Houwen and published by . This book was released on 1998 with total page 18 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: "We consider stiff initial-value problems for second-order differential equations of the special form y"=f(y). Stiff initial-value problem solvers are necessarily implicit, hence, we are faced with the problem of solving systems of implicit relations. This paper focuses on the construction and analysis of iterative solution methods which are effective in cases where the Jacobian of the righthand side of the differential equation can be split into a sum of matrices with a simple structure. These iterative methods consist of the modified Newton method and an iterative linear solver to deal with the linear Newton systems. The linear solver is based on the approximate factorization of the system matrix associated with the linear Newton systems. A number of convergence results are derived for the linear solver in the case where the Jacobian matrix can be split into commuting matrices. Such often [sic] problems arise in the spatial discretization of time-dependent partial differential equations. Furthermore, the stability matrix and the order of accuracy of the integration process are derived in the case of a finite number of iterations."
Book Synopsis A general Formulation of linear splitting methods for ordinary and partial differential equations by : Piet J. van der Houwen
Download or read book A general Formulation of linear splitting methods for ordinary and partial differential equations written by Piet J. van der Houwen and published by . This book was released on 1977 with total page 17 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Additive Operator-Difference Schemes by : Petr N. Vabishchevich
Download or read book Additive Operator-Difference Schemes written by Petr N. Vabishchevich and published by Walter de Gruyter. This book was released on 2013-11-27 with total page 370 pages. Available in PDF, EPUB and Kindle. Book excerpt: Applied mathematical modeling is concerned with solving unsteady problems. Splitting schemes are attributed to the transition from a complex problem to a chain of simpler problems. This book shows how to construct additive difference schemes (splitting schemes) to solve approximately unsteady multi-dimensional problems for PDEs. Two classes of schemes are highlighted: methods of splitting with respect to spatial variables (alternating direction methods) and schemes of splitting into physical processes. Also regionally additive schemes (domain decomposition methods) and unconditionally stable additive schemes of multi-component splitting are considered for evolutionary equations of first and second order as well as for systems of equations. The book is written for specialists in computational mathematics and mathematical modeling. All topics are presented in a clear and accessible manner.
Book Synopsis Sixth-Order Parallel Splitting Method for Diffusion Equation by : Syed Ali Mardan Azmi
Download or read book Sixth-Order Parallel Splitting Method for Diffusion Equation written by Syed Ali Mardan Azmi and published by LAP Lambert Academic Publishing. This book was released on 2011-04 with total page 92 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this book a sixth-order parallel splitting method is developed to get more accurate numerical solution of Parabolic Partial Differential equations. Here sixth-order approximations for second-order parabolic partial differential equations and their algorithms are developed using six parallel processors. In addition to giving a more favorable truncation error, the use of higher order approximates to the matrix exponential functions improves the stability properties of finite difference schemes yielded, which is clear in examples and results are compared.
Book Synopsis Splitting Methods in Communication, Imaging, Science, and Engineering by : Roland Glowinski
Download or read book Splitting Methods in Communication, Imaging, Science, and Engineering written by Roland Glowinski and published by Springer. This book was released on 2017-01-05 with total page 822 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is about computational methods based on operator splitting. It consists of twenty-three chapters written by recognized splitting method contributors and practitioners, and covers a vast spectrum of topics and application areas, including computational mechanics, computational physics, image processing, wireless communication, nonlinear optics, and finance. Therefore, the book presents very versatile aspects of splitting methods and their applications, motivating the cross-fertilization of ideas.
Book Synopsis A Concise Introduction to Geometric Numerical Integration by : Sergio Blanes
Download or read book A Concise Introduction to Geometric Numerical Integration written by Sergio Blanes and published by CRC Press. This book was released on 2017-11-22 with total page 287 pages. Available in PDF, EPUB and Kindle. Book excerpt: Discover How Geometric Integrators Preserve the Main Qualitative Properties of Continuous Dynamical Systems A Concise Introduction to Geometric Numerical Integration presents the main themes, techniques, and applications of geometric integrators for researchers in mathematics, physics, astronomy, and chemistry who are already familiar with numerical tools for solving differential equations. It also offers a bridge from traditional training in the numerical analysis of differential equations to understanding recent, advanced research literature on numerical geometric integration. The book first examines high-order classical integration methods from the structure preservation point of view. It then illustrates how to construct high-order integrators via the composition of basic low-order methods and analyzes the idea of splitting. It next reviews symplectic integrators constructed directly from the theory of generating functions as well as the important category of variational integrators. The authors also explain the relationship between the preservation of the geometric properties of a numerical method and the observed favorable error propagation in long-time integration. The book concludes with an analysis of the applicability of splitting and composition methods to certain classes of partial differential equations, such as the Schrödinger equation and other evolution equations. The motivation of geometric numerical integration is not only to develop numerical methods with improved qualitative behavior but also to provide more accurate long-time integration results than those obtained by general-purpose algorithms. Accessible to researchers and post-graduate students from diverse backgrounds, this introductory book gets readers up to speed on the ideas, methods, and applications of this field. Readers can reproduce the figures and results given in the text using the MATLAB® programs and model files available online.
Book Synopsis Partial Differential Equations by : Walter A. Strauss
Download or read book Partial Differential Equations written by Walter A. Strauss and published by John Wiley & Sons. This book was released on 2007-12-21 with total page 467 pages. Available in PDF, EPUB and Kindle. Book excerpt: Our understanding of the fundamental processes of the natural world is based to a large extent on partial differential equations (PDEs). The second edition of Partial Differential Equations provides an introduction to the basic properties of PDEs and the ideas and techniques that have proven useful in analyzing them. It provides the student a broad perspective on the subject, illustrates the incredibly rich variety of phenomena encompassed by it, and imparts a working knowledge of the most important techniques of analysis of the solutions of the equations. In this book mathematical jargon is minimized. Our focus is on the three most classical PDEs: the wave, heat and Laplace equations. Advanced concepts are introduced frequently but with the least possible technicalities. The book is flexibly designed for juniors, seniors or beginning graduate students in science, engineering or mathematics.
Book Synopsis Front Tracking for Hyperbolic Conservation Laws by : Helge Holden
Download or read book Front Tracking for Hyperbolic Conservation Laws written by Helge Holden and published by Springer. This book was released on 2015-12-10 with total page 521 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the second edition of a well-received book providing the fundamentals of the theory hyperbolic conservation laws. Several chapters have been rewritten, new material has been added, in particular, a chapter on space dependent flux functions and the detailed solution of the Riemann problem for the Euler equations. Hyperbolic conservation laws are central in the theory of nonlinear partial differential equations and in science and technology. The reader is given a self-contained presentation using front tracking, which is also a numerical method. The multidimensional scalar case and the case of systems on the line are treated in detail. A chapter on finite differences is included. From the reviews of the first edition: "It is already one of the few best digests on this topic. The present book is an excellent compromise between theory and practice. Students will appreciate the lively and accurate style." D. Serre, MathSciNet "I have read the book with great pleasure, and I can recommend it to experts as well as students. It can also be used for reliable and very exciting basis for a one-semester graduate course." S. Noelle, Book review, German Math. Soc. "Making it an ideal first book for the theory of nonlinear partial differential equations...an excellent reference for a graduate course on nonlinear conservation laws." M. Laforest, Comp. Phys. Comm.
Book Synopsis Innovative Algorithms and Analysis by : Laurent Gosse
Download or read book Innovative Algorithms and Analysis written by Laurent Gosse and published by Springer. This book was released on 2016-05-26 with total page 362 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume gathers contributions reflecting topics presented during an INDAM workshop held in Rome in May 2016. The event brought together many prominent researchers in both Mathematical Analysis and Numerical Computing, the goal being to promote interdisciplinary collaborations. Accordingly, the following thematic areas were developed: 1. Lagrangian discretizations and wavefront tracking for synchronization models; 2. Astrophysics computations and post-Newtonian approximations; 3. Hyperbolic balance laws and corrugated isometric embeddings; 4. “Caseology” techniques for kinetic equations; 5. Tentative computations of compressible non-standard solutions; 6. Entropy dissipation, convergence rates and inverse design issues. Most of the articles are presented in a self-contained manner; some highlight new achievements, while others offer snapshots of the “state of the art” in certain fields. The book offers a unique resource, both for young researchers looking to quickly enter a given area of application, and for more experienced ones seeking comprehensive overviews and extensive bibliographic references.
Book Synopsis Computing Qualitatively Correct Approximations of Balance Laws by : Laurent Gosse
Download or read book Computing Qualitatively Correct Approximations of Balance Laws written by Laurent Gosse and published by Springer Science & Business Media. This book was released on 2013-03-30 with total page 346 pages. Available in PDF, EPUB and Kindle. Book excerpt: Substantial effort has been drawn for years onto the development of (possibly high-order) numerical techniques for the scalar homogeneous conservation law, an equation which is strongly dissipative in L1 thanks to shock wave formation. Such a dissipation property is generally lost when considering hyperbolic systems of conservation laws, or simply inhomogeneous scalar balance laws involving accretive or space-dependent source terms, because of complex wave interactions. An overall weaker dissipation can reveal intrinsic numerical weaknesses through specific nonlinear mechanisms: Hugoniot curves being deformed by local averaging steps in Godunov-type schemes, low-order errors propagating along expanding characteristics after having hit a discontinuity, exponential amplification of truncation errors in the presence of accretive source terms... This book aims at presenting rigorous derivations of different, sometimes called well-balanced, numerical schemes which succeed in reconciling high accuracy with a stronger robustness even in the aforementioned accretive contexts. It is divided into two parts: one dealing with hyperbolic systems of balance laws, such as arising from quasi-one dimensional nozzle flow computations, multiphase WKB approximation of linear Schrödinger equations, or gravitational Navier-Stokes systems. Stability results for viscosity solutions of onedimensional balance laws are sketched. The other being entirely devoted to the treatment of weakly nonlinear kinetic equations in the discrete ordinate approximation, such as the ones of radiative transfer, chemotaxis dynamics, semiconductor conduction, spray dynamics or linearized Boltzmann models. “Caseology” is one of the main techniques used in these derivations. Lagrangian techniques for filtration equations are evoked too. Two-dimensional methods are studied in the context of non-degenerate semiconductor models.
Book Synopsis Multicomponent and Multiscale Systems by : Juergen Geiser
Download or read book Multicomponent and Multiscale Systems written by Juergen Geiser and published by Springer. This book was released on 2015-08-21 with total page 343 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book examines the latest research results from combined multi-component and multi-scale explorations. It provides theory, considers underlying numerical methods and presents brilliant computational experimentation. Engineering computations featured in this monograph further offer particular interest to many researchers, engineers and computational scientists working in frontier modeling and applications of multicomponent and multiscale problems. Professor Geiser gives specific attention to the aspects of decomposing and splitting delicate structures and controlling decomposition and the rationale behind many important applications of multi-component and multi-scale analysis. Multicomponent and Multiscale Systems: Theory, Methods and Applications in Engineering also considers the question of why iterative methods can be powerful and more appropriate for well-balanced multiscale and multicomponent coupled nonlinear problems. The book is ideal for engineers and scientists working in theoretical and applied areas.
