High Order Space Time Methods For Conservation Laws
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High Order Space Time Methods For Conservation Laws
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Author : National Aeronautics and Space Administration (NASA)
language : en
Publisher: Createspace Independent Publishing Platform
Release Date : 2018-05-22
High Order Space Time Methods For Conservation Laws written by National Aeronautics and Space Administration (NASA) and has been published by Createspace Independent Publishing Platform this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-05-22 with categories.
Current high-order methods such as discontinuous Galerkin and/or flux reconstruction can provide effective discretization for the spatial derivatives. Together with a time discretization, such methods result in either too small a time step size in the case of an explicit scheme or a very large system in the case of an implicit one. To tackle these problems, two new high-order space-time schemes for conservation laws are introduced: the first is explicit and the second, implicit. The explicit method here, also called the moment scheme, achieves a Courant-Friedrichs-Lewy (CFL) condition of 1 for the case of one-spatial dimension regardless of the degree of the polynomial approximation. (For standard explicit methods, if the spatial approximation is of degree p, then the time step sizes are typically proportional to 1/p(exp 2)). Fourier analyses for the one and two-dimensional cases are carried out. The property of super accuracy (or super convergence) is discussed. The implicit method is a simplified but optimal version of the discontinuous Galerkin scheme applied to time. It reduces to a collocation implicit Runge-Kutta (RK) method for ordinary differential equations (ODE) called Radau IIA. The explicit and implicit schemes are closely related since they employ the same intermediate time levels, and the former can serve as a key building block in an iterative procedure for the latter. A limiting technique for the piecewise linear scheme is also discussed. The technique can suppress oscillations near a discontinuity while preserving accuracy near extrema. Preliminary numerical results are shown Huynh, H. T. Glenn Research Center CONSERVATION LAWS; GALERKIN METHOD; COMPUTATIONAL FLUID DYNAMICS; RUNGE-KUTTA METHOD; DIFFERENTIAL EQUATIONS; SPACE-TIME FUNCTIONS; DISCONTINUITY; COMPUTATIONAL GRIDS; TIME MARCHING
Compact Higher Order Numerical Methods For Hyperbolic Conservation Laws
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Author : Robert Byron Lowrie
language : en
Publisher:
Release Date : 1996
Compact Higher Order Numerical Methods For Hyperbolic Conservation Laws written by Robert Byron Lowrie and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1996 with categories.
Space Time Conservation Element And Solution Element Method
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Author : Chih-Yung Wen
language : en
Publisher: Springer Nature
Release Date : 2023-05-10
Space Time Conservation Element And Solution Element Method written by Chih-Yung Wen and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-05-10 with Computers categories.
This open access book introduces the fundamentals of the space–time conservation element and solution element (CESE) method, which is a novel numerical approach for solving equations of physical conservation laws. It highlights the recent progress to establish various improved CESE schemes and its engineering applications. With attractive accuracy, efficiency, and robustness, the CESE method is particularly suitable for solving time-dependent nonlinear hyperbolic systems involving dynamical evolutions of waves and discontinuities. Therefore, it has been applied to a wide spectrum of problems, e.g., aerodynamics, aeroacoustics, magnetohydrodynamics, multi-material flows, and detonations. This book contains algorithm analysis, numerical examples, as well as demonstration codes. This book is intended for graduate students and researchers who are interested in the fields such as computational fluid dynamics (CFD), mechanical engineering, and numerical computation.
Numerical Methods For Conservation Laws
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Author : Jan S. Hesthaven
language : en
Publisher: SIAM
Release Date : 2018-01-30
Numerical Methods For Conservation Laws written by Jan S. Hesthaven and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-01-30 with Science categories.
Conservation laws are the mathematical expression of the principles of conservation and provide effective and accurate predictive models of our physical world. Although intense research activity during the last decades has led to substantial advances in the development of powerful computational methods for conservation laws, their solution remains a challenge and many questions are left open; thus it is an active and fruitful area of research. Numerical Methods for Conservation Laws: From Analysis to Algorithms offers the first comprehensive introduction to modern computational methods and their analysis for hyperbolic conservation laws, building on intense research activities for more than four decades of development; discusses classic results on monotone and finite difference/finite volume schemes, but emphasizes the successful development of high-order accurate methods for hyperbolic conservation laws; addresses modern concepts of TVD and entropy stability, strongly stable Runge-Kutta schemes, and limiter-based methods before discussing essentially nonoscillatory schemes, discontinuous Galerkin methods, and spectral methods; explores algorithmic aspects of these methods, emphasizing one- and two-dimensional problems and the development and analysis of an extensive range of methods; includes MATLAB software with which all main methods and computational results in the book can be reproduced; and demonstrates the performance of many methods on a set of benchmark problems to allow direct comparisons. Code and other supplemental material will be available online at publication.
The Space Time Conservation Element And Solution Element Method
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Author : National Aeronautics and Space Adm Nasa
language : en
Publisher:
Release Date : 2018-11-14
The Space Time Conservation Element And Solution Element Method written by National Aeronautics and Space Adm Nasa and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-11-14 with categories.
A new high resolution and genuinely multidimensional numerical method for solving conservation laws is being, developed. It was designed to avoid the limitations of the traditional methods. and was built from round zero with extensive physics considerations. Nevertheless, its foundation is mathmatically simple enough that one can build from it a coherent, robust. efficient and accurate numerical framework. Two basic beliefs that set the new method apart from the established methods are at the core of its development. The first belief is that, in order to capture physics more efficiently and realistically, the modeling, focus should be placed on the original integral form of the physical conservation laws, rather than the differential form. The latter form follows from the integral form under the additional assumption that the physical solution is smooth, an assumption that is difficult to realize numerically in a region of rapid chance. such as a boundary layer or a shock. The second belief is that, with proper modeling of the integral and differential forms themselves, the resulting, numerical solution should automatically be consistent with the properties derived front the integral and differential forms, e.g., the jump conditions across a shock and the properties of characteristics. Therefore a much simpler and more robust method can be developed by not using the above derived properties explicitly. Chang, Sin-Chung and Wang, Xiao-Yen and Chow, Chuen-Yen Glenn Research Center SPACE-TIME FUNCTIONS; CONSERVATION; SHOCK WAVES; RELATIVITY; HIGH RESOLUTION; CONSERVATION LAWS; BOUNDARY LAYERS; FLUX DENSITY...
Adaptive High Order Methods In Computational Fluid Dynamics
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Author : Zhi Jian Wang
language : en
Publisher: World Scientific
Release Date : 2011-03-24
Adaptive High Order Methods In Computational Fluid Dynamics written by Zhi Jian Wang and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-03-24 with Science categories.
This book consists of important contributions by world-renowned experts on adaptive high-order methods in computational fluid dynamics (CFD). It covers several widely used, and still intensively researched methods, including the discontinuous Galerkin, residual distribution, finite volume, differential quadrature, spectral volume, spectral difference, PNPM, and correction procedure via reconstruction methods. The main focus is applications in aerospace engineering, but the book should also be useful in many other engineering disciplines including mechanical, chemical and electrical engineering. Since many of these methods are still evolving, the book will be an excellent reference for researchers and graduate students to gain an understanding of the state of the art and remaining challenges in high-order CFD methods.
Property Preserving Numerical Schemes For Conservation Laws
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Author : Dmitri Kuzmin
language : en
Publisher: World Scientific
Release Date : 2023-08-28
Property Preserving Numerical Schemes For Conservation Laws written by Dmitri Kuzmin and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-08-28 with Mathematics categories.
High-order numerical methods for hyperbolic conservation laws do not guarantee the validity of constraints that physically meaningful approximations are supposed to satisfy. The finite volume and finite element schemes summarized in this book use limiting techniques to enforce discrete maximum principles and entropy inequalities. Spurious oscillations are prevented using artificial viscosity operators and/or essentially nonoscillatory reconstructions.An introduction to classical nonlinear stabilization approaches is given in the simple context of one-dimensional finite volume discretizations. Subsequent chapters of Part I are focused on recent extensions to continuous and discontinuous Galerkin methods. Many of the algorithms presented in these chapters were developed by the authors and their collaborators. Part II gives a deeper insight into the mathematical theory of property-preserving numerical schemes. It begins with a review of the convergence theory for finite volume methods and ends with analysis of algebraic flux correction schemes for finite elements. In addition to providing ready-to-use algorithms, this text explains the design principles behind such algorithms and shows how to put theory into practice. Although the book is based on lecture notes written for an advanced graduate-level course, it is also aimed at senior researchers who develop and analyze numerical methods for hyperbolic problems.
Frontiers In Physics Rising Stars
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Author : Alex Hansen
language : en
Publisher: Frontiers Media SA
Release Date : 2021-10-04
Frontiers In Physics Rising Stars written by Alex Hansen and has been published by Frontiers Media SA this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-10-04 with Science categories.
The A 4 Scheme A High Order Neutrally Stable Cese Solver
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Author : National Aeronautics and Space Administration (NASA)
language : en
Publisher: Createspace Independent Publishing Platform
Release Date : 2018-05-20
The A 4 Scheme A High Order Neutrally Stable Cese Solver written by National Aeronautics and Space Administration (NASA) and has been published by Createspace Independent Publishing Platform this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-05-20 with categories.
The CESE development is driven by a belief that a solver should (i) enforce conservation laws in both space and time, and (ii) be built from a nondissipative (i.e., neutrally stable) core scheme so that the numerical dissipation can be controlled effectively. To provide a solid foundation for a systematic CESE development of high order schemes, in this paper we describe a new high order (4-5th order) and neutrally stable CESE solver of a 1D advection equation with a constant advection speed a. The space-time stencil of this two-level explicit scheme is formed by one point at the upper time level and two points at the lower time level. Because it is associated with four independent mesh variables (the numerical analogues of the dependent variable and its first, second, and third-order spatial derivatives) and four equations per mesh point, the new scheme is referred to as the a(4) scheme. As in the case of other similar CESE neutrally stable solvers, the a(4) scheme enforces conservation laws in space-time locally and globally, and it has the basic, forward marching, and backward marching forms. Except for a singular case, these forms are equivalent and satisfy a space-time inversion (STI) invariant property which is shared by the advection equation. Based on the concept of STI invariance, a set of algebraic relations is developed and used to prove the a(4) scheme must be neutrally stable when it is stable. Numerically, it has been established that the scheme is stable if the value of the Courant number is less than 1/3 Chang, Sin-Chung Glenn Research Center COMPUTATIONAL GRIDS; SPACE-TIME CE/SE METHOD; INVERSIONS; DEPENDENT VARIABLES; INDEPENDENT VARIABLES; CONSERVATION LAWS; ANALOGS
Magnetohydrodynamic Modeling Of The Solar Corona And Heliosphere
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Author : Xueshang Feng
language : en
Publisher: Springer
Release Date : 2019-08-01
Magnetohydrodynamic Modeling Of The Solar Corona And Heliosphere written by Xueshang Feng and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-08-01 with Science categories.
The book covers intimately all the topics necessary for the development of a robust magnetohydrodynamic (MHD) code within the framework of the cell-centered finite volume method (FVM) and its applications in space weather study. First, it presents a brief review of existing MHD models in studying solar corona and the heliosphere. Then it introduces the cell-centered FVM in three-dimensional computational domain. Finally, the book presents some applications of FVM to the MHD codes on spherical coordinates in various research fields of space weather, focusing on the development of the 3D Solar-InterPlanetary space-time Conservation Element and Solution Element (SIP-CESE) MHD model and its applications to space weather studies in various aspects. The book is written for senior undergraduates, graduate students, lecturers, engineers and researchers in solar-terrestrial physics, space weather theory, modeling, and prediction, computational fluid dynamics, and MHD simulations. It helps readers to fully understand and implement a robust and versatile MHD code based on the cell-centered FVM.