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Author: Hendrik Ranocha Publisher: Cuvillier Verlag ISBN: 3736987358 Category : Mathematics Languages : en Pages : 304
Book Description
This thesis is dedicated to the investigation and development of numerical methods for hyperbolic partial differential equations arising in continuum physics and contains several new theoretical and practical insights which have resulted in novel numerical algorithms that are provably stable and robust, presented here for the first time as a whole. After extending the theory of conservative discretisations using summation-by-parts operators and symmetric numerical fluxes, the application of these methods to nonlinear balance laws such as the shallow water equations and the Euler equations is studied. While it is not clear whether entropy stable schemes can be formulated in this way for the Euler equations and general summation-by-parts operators, it is possible to construct such schemes using classical summation-by-parts operators. Following again the idea to mimic properties of the continuous level discretely, several numerical methods are investigated and new ones are developed. Moreover, stability of fully discrete schemes using explicit Runge-Kutta methods is investigate. Finally, an underlying concept of the previous investigations is studied in detail. Since the entropy plays a crucial role in the theory of hyperbolic balance laws, it has been used as a design principle of numerical methods as described before. Extending these studies, variational principles for the entropy are investigated with respect to their applicability in numerical schemes.
Author: Hendrik Ranocha Publisher: Cuvillier Verlag ISBN: 3736987358 Category : Mathematics Languages : en Pages : 304
Book Description
This thesis is dedicated to the investigation and development of numerical methods for hyperbolic partial differential equations arising in continuum physics and contains several new theoretical and practical insights which have resulted in novel numerical algorithms that are provably stable and robust, presented here for the first time as a whole. After extending the theory of conservative discretisations using summation-by-parts operators and symmetric numerical fluxes, the application of these methods to nonlinear balance laws such as the shallow water equations and the Euler equations is studied. While it is not clear whether entropy stable schemes can be formulated in this way for the Euler equations and general summation-by-parts operators, it is possible to construct such schemes using classical summation-by-parts operators. Following again the idea to mimic properties of the continuous level discretely, several numerical methods are investigated and new ones are developed. Moreover, stability of fully discrete schemes using explicit Runge-Kutta methods is investigate. Finally, an underlying concept of the previous investigations is studied in detail. Since the entropy plays a crucial role in the theory of hyperbolic balance laws, it has been used as a design principle of numerical methods as described before. Extending these studies, variational principles for the entropy are investigated with respect to their applicability in numerical schemes.
Author: Jens M. Melenk Publisher: Springer Nature ISBN: 3031204328 Category : Mathematics Languages : en Pages : 571
Book Description
The volume features high-quality papers based on the presentations at the ICOSAHOM 2020+1 on spectral and high order methods. The carefully reviewed articles cover state of the art topics in high order discretizations of partial differential equations. The volume presents a wide range of topics including the design and analysis of high order methods, the development of fast solvers on modern computer architecture, and the application of these methods in fluid and structural mechanics computations.
Author: Emmanuel Franck Publisher: Springer Nature ISBN: 3031408608 Category : Mathematics Languages : en Pages : 296
Book Description
This volume comprises the second part of the proceedings of the 10th International Conference on Finite Volumes for Complex Applications, FVCA, held in Strasbourg, France, during October 30 to November 3, 2023. The Finite Volume method, and several of its variants, is a spatial discretization technique for partial differential equations based on the fundamental physical principle of conservation. Recent decades have brought significant success in the theoretical understanding of the method. Many finite volume methods are also built to preserve some properties of the continuous equations, including maximum principles, dissipativity, monotone decay of the free energy, asymptotic stability, or stationary solutions. Due to these properties, finite volume methods belong to the wider class of compatible discretization methods, which preserve qualitative properties of continuous problems at the discrete level. This structural approach to the discretization of partial differential equations becomes particularly important for multiphysics and multiscale applications. In recent years, the efficient implementation of these methods in numerical software packages, more specifically to be used in supercomputers, has drawn some attention. The first volume contains all invited papers, as well as the contributed papers focusing on finite volume schemes for elliptic and parabolic problems. They include structure-preserving schemes, convergence proofs, and error estimates for problems governed by elliptic and parabolic partial differential equations. This volume is focused on finite volume methods for hyperbolic and related problems, such as methods compatible with the low Mach number limit or able to exactly preserve steady solutions, the development and analysis of high order methods, or the discretization of kinetic equations.
Author: Philipp Öffner Publisher: Springer Nature ISBN: 3658426209 Category : Mathematics Languages : en Pages : 486
Book Description
The book focuses on stability and approximation results concerning recent numerical methods for the numerical solution of hyperbolic conservation laws. The work begins with a detailed and thorough introduction of hyperbolic conservation/balance laws and their numerical treatment. In the main part, recent results in such context are presented focusing on the investigation of approximation properties of discontinuous Galerkin and flux reconstruction methods, the construction of (entropy) stable numerical methods and the extension of existing (entropy) stability results for both semidiscrete and fully discrete schemes, and development of new high-order methods.
Author: Hannes Frenander Publisher: Linköping University Electronic Press ISBN: 9176855953 Category : Languages : en Pages : 54
Book Description
In this thesis, we use finite difference operators with the Summation-By-Partsproperty (SBP) and a weak boundary treatment, known as SimultaneousApproximation Terms (SAT), to construct high-order accurate numerical schemes.The SBP property and the SAT’s makes the schemes provably stable. The numerical procedure is general, and can be applied to most problems, but we focus on hyperbolic problems such as the shallow water, Euler and wave equations. For a well-posed problem and a stable numerical scheme, data must be available at the boundaries of the domain. However, there are many scenarios where additional information is available inside the computational domain. In termsof well-posedness and stability, the additional information is redundant, but it can still be used to improve the performance of the numerical scheme. As a first contribution, we introduce a procedure for implementing additional data using SAT’s; we call the procedure the Multiple Penalty Technique (MPT). A stable and accurate scheme augmented with the MPT remains stable and accurate. Moreover, the MPT introduces free parameters that can be used to increase the accuracy, construct absorbing boundary layers, increase the rate of convergence and control the error growth in time. To model infinite physical domains, one need transparent artificial boundary conditions, often referred to as Non-Reflecting Boundary Conditions (NRBC). In general, constructing and implementing such boundary conditions is a difficult task that often requires various approximations of the frequency and range of incident angles of the incoming waves. In the second contribution of this thesis,we show how to construct NRBC’s by using SBP operators in time. In the final contribution of this thesis, we investigate long time error bounds for the wave equation on second order form. Upper bounds for the spatial and temporal derivatives of the error can be obtained, but not for the actual error. The theoretical results indicate that the error grows linearly in time. However, the numerical experiments show that the error is in fact bounded, and consequently that the derived error bounds are probably suboptimal.
Author: Remi Abgrall Publisher: Elsevier ISBN: 0444637958 Category : Mathematics Languages : en Pages : 668
Book Description
Handbook of Numerical Methods for Hyperbolic Problems explores the changes that have taken place in the past few decades regarding literature in the design, analysis and application of various numerical algorithms for solving hyperbolic equations. This volume provides concise summaries from experts in different types of algorithms, so that readers can find a variety of algorithms under different situations and readily understand their relative advantages and limitations. Provides detailed, cutting-edge background explanations of existing algorithms and their analysis Ideal for readers working on the theoretical aspects of algorithm development and its numerical analysis Presents a method of different algorithms for specific applications and the relative advantages and limitations of different algorithms for engineers or readers involved in applications Written by leading subject experts in each field who provide breadth and depth of content coverage
Author: Giacomo Albi Publisher: Springer Nature ISBN: 3031298756 Category : Mathematics Languages : en Pages : 241
Book Description
A broad range of phenomena in science and technology can be described by non-linear partial differential equations characterized by systems of conservation laws with source terms. Well known examples are hyperbolic systems with source terms, kinetic equations, and convection-reaction-diffusion equations. This book collects research advances in numerical methods for hyperbolic balance laws and kinetic equations together with related modelling aspects. All the contributions are based on the talks of the speakers of the Young Researchers’ Conference “Numerical Aspects of Hyperbolic Balance Laws and Related Problems”, hosted at the University of Verona, Italy, in December 2021.
Author: María Luz Muñoz-Ruiz Publisher: Springer Nature ISBN: 3030728501 Category : Mathematics Languages : en Pages : 269
Book Description
The present volume contains selected papers issued from the sixth edition of the International Conference "Numerical methods for hyperbolic problems" that took place in 2019 in Málaga (Spain). NumHyp conferences, which began in 2009, focus on recent developments and new directions in the field of numerical methods for hyperbolic partial differential equations (PDEs) and their applications. The 11 chapters of the book cover several state-of-the-art numerical techniques and applications, including the design of numerical methods with good properties (well-balanced, asymptotic-preserving, high-order accurate, domain invariant preserving, uncertainty quantification, etc.), applications to models issued from different fields (Euler equations of gas dynamics, Navier-Stokes equations, multilayer shallow-water systems, ideal magnetohydrodynamics or fluid models to simulate multiphase flow, sediment transport, turbulent deflagrations, etc.), and the development of new nonlinear dispersive shallow-water models. The volume is addressed to PhD students and researchers in Applied Mathematics, Fluid Mechanics, or Engineering whose investigation focuses on or uses numerical methods for hyperbolic systems. It may also be a useful tool for practitioners who look for state-of-the-art methods for flow simulation.
Author: Remi Abgrall Publisher: Elsevier ISBN: 044463911X Category : Mathematics Languages : en Pages : 612
Book Description
Handbook on Numerical Methods for Hyperbolic Problems: Applied and Modern Issues details the large amount of literature in the design, analysis, and application of various numerical algorithms for solving hyperbolic equations that has been produced in the last several decades. This volume provides concise summaries from experts in different types of algorithms, so that readers can find a variety of algorithms under different situations and become familiar with their relative advantages and limitations. Provides detailed, cutting-edge background explanations of existing algorithms and their analysis Presents a method of different algorithms for specific applications and the relative advantages and limitations of different algorithms for engineers or those involved in applications Written by leading subject experts in each field, the volumes provide breadth and depth of content coverage
Author: Hendrik Ranocha
Author: Hendrik Ranocha
Author: Jens M. Melenk
Author: Emmanuel Franck
Author: Philipp Öffner
Author: Lucas Friedrich
Author: Hannes Frenander
Author: Remi Abgrall
Author: Giacomo Albi
Author: María Luz Muñoz-Ruiz
Author: Remi Abgrall