Download or read book Efficient Reduced basis Approximation of Scalar Nonlinear Time dependent Convection diffusion Problems and Extension to Compressible Flow Problems written by Han Abby Men and published by . This book was released on 2006 with total page 65 pages. Available in PDF, EPUB and Kindle. Book excerpt: (cont.) The operation count for the online stage depends only on the dimension of our reduced-basis approximation space and the dimension of our coefficient-function approximation space. The extension of the reduced-order model to a system of equations is also explored.

Download or read book Finite Difference Methods of Solution of Nonlinear Flow Processes with Application to the Ben ard Problem written by Jacob E. Fromm and published by . This book was released on 1966 with total page 138 pages. Available in PDF, EPUB and Kindle. Book excerpt: A general method of numerical calculation of compressible flows is outlined in which such flows are divided into irrotational and solenoidal parts. The general equations are reduced to the Boussinesq approximation for consideration of the B©Øenard problem. The B©Øenard problem, both in method of solution and result, is used to analyse a number of crucial aspects of finite difference calculation. In particular, the nonlinear formulations in current use are developed and related in a systematic way; and, in addition, some higher order methods are derived. Examples of the time-dependent behavior of the thermal convection problem are examined for physical interpretation in terms of gross property measurements and character of instantaneous solutions with the hope that the experience so gained will be valuable to extensions of the numerical method to more general problems.

Download or read book Discontinuous Galerkin Method written by Vít Dolejší and published by Springer. This book was released on 2015-07-17 with total page 575 pages. Available in PDF, EPUB and Kindle. Book excerpt: The subject of the book is the mathematical theory of the discontinuous Galerkin method (DGM), which is a relatively new technique for the numerical solution of partial differential equations. The book is concerned with the DGM developed for elliptic and parabolic equations and its applications to the numerical simulation of compressible flow. It deals with the theoretical as well as practical aspects of the DGM and treats the basic concepts and ideas of the DGM, as well as the latest significant findings and achievements in this area. The main benefit for readers and the book’s uniqueness lie in the fact that it is sufficiently detailed, extensive and mathematically precise, while at the same time providing a comprehensible guide through a wide spectrum of discontinuous Galerkin techniques and a survey of the latest efficient, accurate and robust discontinuous Galerkin schemes for the solution of compressible flow.

Download or read book Certified Reduced Basis Methods for Parametrized Partial Differential Equations written by Jan S Hesthaven and published by Springer. This book was released on 2015-08-20 with total page 139 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a thorough introduction to the mathematical and algorithmic aspects of certified reduced basis methods for parametrized partial differential equations. Central aspects ranging from model construction, error estimation and computational efficiency to empirical interpolation methods are discussed in detail for coercive problems. More advanced aspects associated with time-dependent problems, non-compliant and non-coercive problems and applications with geometric variation are also discussed as examples.

Download or read book Discontinuous Galerkin Methods written by Bernardo Cockburn and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 468 pages. Available in PDF, EPUB and Kindle. Book excerpt: A class of finite element methods, the Discontinuous Galerkin Methods (DGM), has been under rapid development recently and has found its use very quickly in such diverse applications as aeroacoustics, semi-conductor device simula tion, turbomachinery, turbulent flows, materials processing, MHD and plasma simulations, and image processing. While there has been a lot of interest from mathematicians, physicists and engineers in DGM, only scattered information is available and there has been no prior effort in organizing and publishing the existing volume of knowledge on this subject. In May 24-26, 1999 we organized in Newport (Rhode Island, USA), the first international symposium on DGM with equal emphasis on the theory, numerical implementation, and applications. Eighteen invited speakers, lead ers in the field, and thirty-two contributors presented various aspects and addressed open issues on DGM. In this volume we include forty-nine papers presented in the Symposium as well as a survey paper written by the organiz ers. All papers were peer-reviewed. A summary of these papers is included in the survey paper, which also provides a historical perspective of the evolution of DGM and its relation to other numerical methods. We hope this volume will become a major reference in this topic. It is intended for students and researchers who work in theory and application of numerical solution of convection dominated partial differential equations. The papers were written with the assumption that the reader has some knowledge of classical finite elements and finite volume methods.

Download or read book Reduced Order Methods for Modeling and Computational Reduction written by Alfio Quarteroni and published by Springer. This book was released on 2014-06-05 with total page 338 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph addresses the state of the art of reduced order methods for modeling and computational reduction of complex parametrized systems, governed by ordinary and/or partial differential equations, with a special emphasis on real time computing techniques and applications in computational mechanics, bioengineering and computer graphics. Several topics are covered, including: design, optimization, and control theory in real-time with applications in engineering; data assimilation, geometry registration, and parameter estimation with special attention to real-time computing in biomedical engineering and computational physics; real-time visualization of physics-based simulations in computer science; the treatment of high-dimensional problems in state space, physical space, or parameter space; the interactions between different model reduction and dimensionality reduction approaches; the development of general error estimation frameworks which take into account both model and discretization effects. This book is primarily addressed to computational scientists interested in computational reduction techniques for large scale differential problems.

Download or read book Fundamentals of Computational Fluid Dynamics written by H. Lomax and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 256 pages. Available in PDF, EPUB and Kindle. Book excerpt: The chosen semi-discrete approach of a reduction procedure of partial differential equations to ordinary differential equations and finally to difference equations gives the book its distinctiveness and provides a sound basis for a deep understanding of the fundamental concepts in computational fluid dynamics.

Download or read book Applied mechanics reviews written by and published by . This book was released on 1948 with total page 400 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Physics Briefs written by and published by . This book was released on 1990 with total page 1354 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Efficient Parallel Solvers for Two Dimensional Potential Flow and Convection Diffusion Problems on Nonmatching Grids written by Jurij Iliaš and published by . This book was released on 1996 with total page 66 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book On the Stabilization and Enhancement of the Reduced order Models for Compressible Flows written by Elnaz Rezaianzadeh and published by . This book was released on 2020 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Projection-based model reduction offers a physically informed, and mathematically rigorous framework to bypass the prohibitive amount of computational resources required by the direct numerical simulations in fluid dynamics, and enable the recurrent computations that dominate many-queries applications. Projection of the governing equations onto a low-dimensional space, however, does not guarantee to naturally inherit the stability properties of the high-fidelity model. Symmetrization of the Reduced-Order Model (ROM) through a least squares Petrov-Galerkin projection, or by Galerkin projection using the symmetry inner product, provides theoretical error bounds, and generates more stable ROMs. This study shows that besides being more stable, the symmetrized ROMs are more controllable and robust. The stability guarantees by symmetrization or energy-based inner products, assume that the subspace constructed for projection, accurately captures the coherent structures that are the main ingredients in the dynamics of the flow. However, when the high-fidelity simulations contain nonlinear phenomena (e.g. unsteady shock waves, and turbulence), truncation of the high-frequency modes through dimensionality reduction with a linear approach like Proper Orthogonal Decomposition (POD), that is biased towards the most energetic modes, may result in losing structures with critical contributions in the dynamical evolution of the system. As a result, especially when the governing equations lack any intrinsic dissipative mechanisms to contain the generated errors (e.g. the Euler equations), symmetrization alone is not sufficient to preserve stability. Therefore, a complete framework is proposed in this study for the enhancement of ROMs for compressible flows, through ROM symmetrization, and post-ROM stabilization. Two optimization-based non-intrusive stabilization methods are developed here: a Hybrid method for the stabilization of ROMs as Linear Time-Invariant (LTI) systems, and an eigenvalue reassignment method for stabilization of nonlinear ROMs (ERN algorithm). The Hybrid method is a two-step approach: in step one (efficiency-oriented), the left reduced order basis of the ROM is minimally modified in a convex optimization problem; in step two (accuracy-oriented), an eigenvalue reassignment method is used to stabilize the most energetic eigen-modes. The ERN algorithm, on the other hand, confines the nonlinear ROM to maintain a negative total power for stability; and the distance between the nonlinear ROM and Full-Order Model (FOM) attractors is directly minimized as the eigenvalues of the linear dynamics matrix (control parameters) are reassigned in the complex plane. A computational bottleneck occurs in strongly nonlinear systems (e.g. advection-dominated flows), where the slow decay of the projection error requires more base functions to accurately span the high-fidelity solutions with a linear subspace. Hence to sufficiently describe a strongly nonlinear system, ROMs have higher dimensions intrinsically. Nevertheless, the truncation of such ROMs may still bring in instability, and their relatively higher dimension (i.e. large coefficient matrices) leads to a large number of control parameters which may potentially prevent the stabilization algorithm being feasible in computation. As a remedy, this study introduces a multi-stage layout for robust stabilization of nonlinear ROMs with the ERN algorithm, in strongly nonlinear systems, where a linear ROM typically fails to capture the true dynamics. The proposed methods are applied on POD-Galerkin ROMs based on the snapshots of two supersonic flow applications. The high-fidelity simulations are performed with a Weighted Essentially Non-Oscillatory (WENO) shock capturing scheme, integrated with the immersed boundary method. The two applications involve strong shock-wake interactions in the downstream, where the unsteady shock oscillations as a result of the interaction of shock waves with vortices, exhibit strong nonlinearities that are not completely resolved in the leading POD modes. Thus, the missing high-frequency contributions of this phenomenon trigger strong instabilities in the linear and nonlinear ROMs, and enable a thorough investigation of the ideas that are developed in this research for the stabilization and enhancement of ROMs.

Download or read book Numerical Solution of Time Dependent Advection Diffusion Reaction Equations written by Willem Hundsdorfer and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 479 pages. Available in PDF, EPUB and Kindle. Book excerpt: Unique book on Reaction-Advection-Diffusion problems

Download or read book New Multi Layer Compact High Order Finite Difference Methods with Spectral Like Resolution for Compressible Flow Simulations written by Zeyu Bai and published by . This book was released on 2019 with total page 288 pages. Available in PDF, EPUB and Kindle. Book excerpt: Numerical simulations of multi-scale flow problems such as hypersonic boundary layer transition, turbulent flows, computational aeroacoustics and other flow problems with complex physics require high-order methods with high spectral resolutions. For instance, the receptivity mechanisms in the hypersonic boundary layer are the resonant interactions between forcing waves and boundary-layer waves, and the complex wave interactions are difficult to be accurately predicted by conventional low-order numerical methods. High-order methods, which are robust and accurate in resolving a wide range of time and length scales, are required. The objective of this dissertation is to develop and analyze new very high-order numerical methods with spectral-like resolution for flow simulations on structured grids, with focus on smooth flow problems involving multiple scales. These numerical methods include: the multi-layer compact (MLC) scheme, the directional multi-layer compact (DMLC) scheme, and the least square multi-layer compact (LSMLC) scheme. In the first place, a new upwind multi-layer compact (MLC) scheme up to seventh order is derived in a finite difference framework. By using the 'multi-layer' idea, which introduces first derivatives into the MLC schemes and approximates the second derivatives, the resolution of the MLC schemes can be significantly improved within a compact grid stencil. The auxiliary equations are introduced, and they are the only nontrivial equations. The original equation requires no approximation which contributes to good computational efficiency. In addition, the upwind MLC schemes are derived on centered stencils with adjustable parameters to control the dissipation. Fourier analysis is performed to show that the new MLC schemes have very small dissipation and dispersion in a wide range of wavenumbers in both one and two-dimensional cases, and the anisotropic error is much smaller than conventional finite difference methods in the two-dimensional case. Comparison with discontinuous-Galerkin methods is performed with Fourier analysis as well. Furthermore, stability analysis with matrix method shows that high-order boundary closure schemes are stable because of compactness of the stencils. The accuracies and rates of convergence of the new schemes are validated by numerical experiments of the linear advection equation, the nonlinear Euler equations, and the Navier-Stokes equations in both one and two-dimensional settings. The numerical results show that good computational efficiency, very high-order accuracies, and high spectral resolutions especially on coarse meshes can be attained with the MLC scheme. On the other hand, even though the MLC scheme is promising in most test cases, it shows weak numerical instabilities for a small range of wavenumbers when it is applied to multi-dimensional flows, which are mainly triggered by the inconsistency between its one and two-dimensional formulations. The instability could lead to divergence in long-time multi-dimensional simulations. Moreover, the cross-derivative approximation in the MLC scheme requires an ad-hoc selection of supporting grid points, and the cross-derivative approximation is relatively inefficient for very high-order cases. To address the remaining challenges of the MLC scheme and achieve better performance for multi-dimensional flow simulations, another two new schemes are developed - the directional multi-layer compact (DMLC) scheme, and the least square multi-layer compact (LSMLC) scheme. In the second place, a new upwind directional multi-layer compact (DMLC) scheme is developed for multi-dimensional simulations. The main idea of the DMLC scheme is to introduce auxiliary equation for cross derivative in multi-dimensional cases. Consequently, the spatial discretization can be fulfilled along each dimension independently. With this directional discretization technique, the one-dimensional formulation of the MLC scheme can be applied to all spatial derivatives in a multi-dimensional governing equation. Therefore, the DMLC scheme overcomes the inconsistency between one and two-dimensional formulations of the MLC scheme, and it also avoids the ad-hoc cross-derivative approximations. Two-dimensional Fourier analysis demonstrates that all modes of the DMLC scheme are stable in the full range of wavenumbers, and it has better spectral resolution and smaller anisotropic error than the MLC scheme. Stability analysis with matrix method indicates that stable boundary closure schemes are much easier to be obtained in the DMLC scheme. Numerical tests in the linear advection equation and the nonlinear Euler equations validate that the DMLC scheme are more accurate and require less CPU time than the MLC scheme on the same mesh. In particular, the long-time simulation results reveal that the DMLC scheme is always stable for both periodic and non-periodic boundary conditions in two-dimensional cases. In the third place, a new upwind least square multi-layer compact (LSMLC) scheme is developed for multi-dimensional simulations. The main idea of the LSMLC scheme is using the weighted least square approximation to redesign the two-dimensional formulation for cross derivatives. It avoids the ad-hoc selection of grid points in the MLC scheme. Meanwhile, the two-dimensional upwind scheme can be derived by introducing upwind correction into the weight function. The upwind factor can adjust the dissipation and stability of the LSMLC scheme. Lagrange multiplier is used to ensure that the LSMLC scheme satisfies both the consistency constraint at the base point and the one-dimensional constraint from the MLC scheme. The LSMLC scheme does not increase computational cost on structured meshes, and can be implemented in the same way as the MLC scheme. A parametric study based on two-dimensional Fourier analysis shows that the truncated Gaussian distribution (TGD) weight function leads to better LSMLC scheme among other weight functions because it removes the numerical instability and maintain small dissipations. The LSMLC scheme has larger dissipation than the MLC scheme, and shows similar spectral resolution. Stability analysis with matrix method indicates that a combination of an interior LSMLC scheme and MLC boundary closure schemes can improve the boundary stability while maintaining small dissipation. Numerical tests in the linear advection equation and the nonlinear Euler equations validate that the LSMLC scheme produces slightly larger errors compared with MLC scheme. The long-time simulation results reveal that the LSMLC scheme is always stable for both periodic and non-periodic boundary conditions in two-dimensional cases. Overall, the new very high-order multi-layer compact finite difference methods have the properties of simple formulations, high-order accuracies, spectral-like resolutions, and compact stencils, and they are suitable for accurate simulation of smooth multi-scale flows with complex physics. Among the three schemes developed in this dissertation, the DMLC scheme is always the best choice for multi-dimensional simulations because it shows comprehensive improvements from the MLC scheme with consistent stability, higher accuracy and spectral resolution, and better computational efficiency. The LSMLC scheme is also appropriate considering it has consistent stability and it is easy to be implemented.

Download or read book Numerical Methods for Conservation Laws written by LEVEQUE and published by Birkhäuser. This book was released on 2013-11-11 with total page 221 pages. Available in PDF, EPUB and Kindle. Book excerpt: These notes developed from a course on the numerical solution of conservation laws first taught at the University of Washington in the fall of 1988 and then at ETH during the following spring. The overall emphasis is on studying the mathematical tools that are essential in de veloping, analyzing, and successfully using numerical methods for nonlinear systems of conservation laws, particularly for problems involving shock waves. A reasonable un derstanding of the mathematical structure of these equations and their solutions is first required, and Part I of these notes deals with this theory. Part II deals more directly with numerical methods, again with the emphasis on general tools that are of broad use. I have stressed the underlying ideas used in various classes of methods rather than present ing the most sophisticated methods in great detail. My aim was to provide a sufficient background that students could then approach the current research literature with the necessary tools and understanding. vVithout the wonders of TeX and LaTeX, these notes would never have been put together. The professional-looking results perhaps obscure the fact that these are indeed lecture notes. Some sections have been reworked several times by now, but others are still preliminary. I can only hope that the errors are not too blatant. Moreover, the breadth and depth of coverage was limited by the length of these courses, and some parts are rather sketchy.

Download or read book Numerical Methods for Fluid Dynamics written by Dale R. Durran and published by Springer Science & Business Media. This book was released on 2010-09-14 with total page 527 pages. Available in PDF, EPUB and Kindle. Book excerpt: This scholarly text provides an introduction to the numerical methods used to model partial differential equations, with focus on atmospheric and oceanic flows. The book covers both the essentials of building a numerical model and the more sophisticated techniques that are now available. Finite difference methods, spectral methods, finite element method, flux-corrected methods and TVC schemes are all discussed. Throughout, the author keeps to a middle ground between the theorem-proof formalism of a mathematical text and the highly empirical approach found in some engineering publications. The book establishes a concrete link between theory and practice using an extensive range of test problems to illustrate the theoretically derived properties of various methods. From the reviews: "...the books unquestionable advantage is the clarity and simplicity in presenting virtually all basic ideas and methods of numerical analysis currently actively used in geophysical fluid dynamics." Physics of Atmosphere and Ocean

Download or read book Chebyshev and Fourier Spectral Methods written by John P. Boyd and published by Courier Corporation. This book was released on 2001-12-03 with total page 690 pages. Available in PDF, EPUB and Kindle. Book excerpt: Completely revised text focuses on use of spectral methods to solve boundary value, eigenvalue, and time-dependent problems, but also covers Hermite, Laguerre, rational Chebyshev, sinc, and spherical harmonic functions, as well as cardinal functions, linear eigenvalue problems, matrix-solving methods, coordinate transformations, methods for unbounded intervals, spherical and cylindrical geometry, and much more. 7 Appendices. Glossary. Bibliography. Index. Over 160 text figures.

Download or read book Nonlinear problems of compressible flow tr written by Aleksandr Georgievich Bagdoev and published by . This book was released on with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: