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Book Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014

Download or read book Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014 written by Robert M. Kirby and published by Springer. This book was released on 2015-11-26 with total page 504 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book contains a selection of high quality papers, chosen among the best presentations during the International Conference on Spectral and High-Order Methods (2014), and provides an overview of the depth and breadth of the activities within this important research area. The carefully reviewed selection of papers will provide the reader with a snapshot of the state-of-the-art and help initiate new research directions through the extensive biography.

Book High order hybridized Discontinuous Galerkin Method for Geophysical Flows

Download or read book High order hybridized Discontinuous Galerkin Method for Geophysical Flows written by Shinhoo Kang and published by . This book was released on 2019 with total page 384 pages. Available in PDF, EPUB and Kindle. Book excerpt: As computational research has grown, simulation has become a standard tool in many fields of academic and industrial areas. For example, computational fluid dynamics (CFD) tools in aerospace and research facilities are widely used to evaluate the aerodynamic performance of aircraft or wings. Weather forecasts are highly dependent on numerical weather prediction (NWP) model. However, it is still difficult to simulate the complex physical phenomena of a wide range of length and time scales with modern computational resources. In this study, we develop a robust, efficient and high-order accurate numerical methods and techniques to tackle the challenges. First, we use high-order spatial discretization using (hybridized) discontinuous Galerkin (DG) methods. The DG method combines the advantages of finite volume and finite element methods. As such, it is well-suited to problems with large gradients including shocks and with complex geometries, and large-scale simulations. However, DG typically has many degrees-of-freedoms. To mitigate the expense, we use hybridized DG (HDG) method that introduces new “trace unknowns” on the mesh skeleton (mortar interfaces) to eliminate the local “volume unknowns” with static condensation procedure and reduces globally coupled system when implicit time-stepping is required. Also, since the information between the elements is exchanged through the mesh skeleton, the mortar interfaces can be used as a glue to couple multi-phase regions, e.g., solid and fluid regions, or non-matching grids, e.g., a rotating mesh and a stationary mesh. That is the HDG method provides an efficient and flexible coupling environment compared to standard DG methods. Second, we develop an HDG-DG IMEX scheme for an efficient time integrating scheme. The idea is to divide the governing equations into stiff and nonstiff parts, implicitly treat the former with HDG methods, and explicitly treat the latter with DG methods. The HDG-DG IMEX scheme facilitates high-order temporal and spatial solutions, avoiding too small a time step. Numerical results show that the HDG-DG IMEX scheme is comparable to an explicit Runge-Kutta DG scheme in terms of accuracy while allowing for much larger timestep sizes. We also numerically observe that IMEX HDG-DG scheme can be used as a tool to suppress the high-frequency modes such as acoustic waves or fast gravity waves in atmospheric or ocean models. In short, IMEX HDG-DG methods are attractive for applications in which a fast and stable solution is important while permitting inaccurate processing of the fast modes. Third, we also develop an EXPONENTIAL DG scheme for an efficient time integrators. Similar to the IMEX method, the governing equations are separated into linear and nonlinear parts, then the two parts are spatially discretized with DG methods. Next, we analytically integrate the linear term and approximate the nonlinear term with respect to time. This method accurately handles the fast wave modes in the linear operator. To efficiently evaluate a matrix exponential, we employ the cutting-edge adaptive Krylov subspace method. Finally, we develop a sliding-mesh interface by combining nonconforming treatment and the arbitrary Lagrangian-Eulerian (ALE) scheme for simulating rotating flows, which are important to estimate the characteristics of a rotating wind turbine or understanding vortical structures shown in atmospheric or astronomical phenomena. To integrate the rotating motion of the domain, we use the ALE formulation to map the governing equation to the stationary reference domain and introduce mortar interfaces between the stationary mesh and the rotating mesh. The mortar structure on the sliding interface changes dynamically as the mesh rotates

Book Discontinuous Galerkin Method

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.

Book A Discontinuous Galerkin Method for the Solution of Compressible Flows

Download or read book A Discontinuous Galerkin Method for the Solution of Compressible Flows written by Cristian Biotto and published by . This book was released on 2011 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Book A high order discontinuous Galerkin method for unsteady compressible flows with immersed boundaries

Download or read book A high order discontinuous Galerkin method for unsteady compressible flows with immersed boundaries written by Stephan Krämer-Eis and published by Cuvillier Verlag. This book was released on 2017-10-20 with total page 128 pages. Available in PDF, EPUB and Kindle. Book excerpt: Um die komplexe Physik in kompressiblen Strömungen genauer zu verstehen, kommen vermehrt Simulationen zum Einsatz. Jedoch können weit verbreitete kommerzielle Softwarepakete die Physik aufgrund ihrer niedrigen Genauigkeit oft nicht korrekt erfassen. In dieser Arbeit wird eine diskontinuierliche Galerkin Methode mit hoher Ordnung entwickelt, welche eine hohe Genauigkeit erzielt. Dabei werden insbesondere zwei Probleme, die im Kontext von Verfahren mit hoher Ordnung auftreten, behandelt. Zum einen wird die Gittergenerierung durch das Verwenden einer Immersed Boundary Methode deutlich vereinfacht. Dies bedeutet, dass die Problemgeometrie aus einem deutlich einfacheren Hintergrundgitter herausgeschnitten wird. Die Geometrie wird mit Hilfe einer Level-Set Funktion dargestellt, und die Integration auf den entstehenden geschnittenen Zellen wird mittels einer hierarchischen Moment-Fitting Quadratur durchgeführt. Das Problem der sehr kleinen oder stark gekrümmten Zellen wird durch Zellagglomeration gelöst. Zum zweiten wird die starke Zeitschrittbeschränkung durch anisotrope Gitter mit Hilfe eines lokalen Zeitschrittverfahrens behoben. Diverse numerische Experimente bestätigen die hohe Genauigkeit, Effizienz und geometrische Flexibilität der vorgestellten Methode.

Book A Multi resolution Discontinuous Galerkin Method for Unsteady Compressible Flows

Download or read book A Multi resolution Discontinuous Galerkin Method for Unsteady Compressible Flows written by Andrew Brian Shelton and published by . This book was released on 2008 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: The issue of local scale and smoothness presents a crucial and daunting challenge for numerical simulation methods in fluid dynamics. Yet in the interests of both accuracy and economy, how can one devise a general technique that efficiently resolves flow features of consequence and discriminates against others which are either ``negligible' or amenable to ``universal' modeling? This is particularly difficult because geometries of engineering interest are complex and multi-dimensional, precluding a priori knowledge of the flowfield. To address this challenge, the current work employs wavelet theory for the local scale decomposition of functions, which provides a natural mechanism for the adaptive compression of data. The resulting technique is known as the Multi-Resolution Discontinuous Galerkin (MRDG) method.

Book A Hybridized Discontinuous Petrov Galerkin Scheme for Compressible Flows

Download or read book A Hybridized Discontinuous Petrov Galerkin Scheme for Compressible Flows written by David Moro-Ludeña and published by . This book was released on 2011 with total page 117 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Hybridized Discontinuous Petrov-Galerkin scheme (HDPG) for compressible flows is presented. The HDPG method stems from a combination of the Hybridized Discontinuous Galerkin (HDG) method and the theory of the optimal test functions, suitably modified to enforce the conservativity at the element level. The new scheme maintains the same number of globally coupled degrees of freedom as the HDG method while increasing the stability in the presence of discontinuities or under-resolved features. The new scheme has been successfully tested in several problems involving shocks such as Burgers equation and the Navier-Stokes equations and delivers solutions with reduced oscillation at the shock. When combined with artificial viscosity, the oscillation can be completely eliminated using one order of magnitude less viscosity than that required by other Finite Element methods. Also, convergence studies in the sequence of meshes proposed by Peterson [49] show that, unlike other DG methods, the HDPG method is capable of breaking the suboptimal k+1/2 rate of convergence for the convective problem and thus achieve optimal k+1 convergence.

Book A Set of Parallel Implicit Methods for a Reconstructed Discontinuous Galerkin Method for Compressible Flows on 3D Hybrid Grids

Download or read book A Set of Parallel Implicit Methods for a Reconstructed Discontinuous Galerkin Method for Compressible Flows on 3D Hybrid Grids written by and published by . This book was released on 2014 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: A set of implicit methods are proposed for a third-order hierarchical WENO reconstructed discontinuous Galerkin method for compressible flows on 3D hybrid grids. An attractive feature in these methods are the application of the Jacobian matrix based on the P1 element approximation, resulting in a huge reduction of memory requirement compared with DG (P2). Also, three approaches -- analytical derivation, divided differencing, and automatic differentiation (AD) are presented to construct the Jacobian matrix respectively, where the AD approach shows the best robustness. A variety of compressible flow problems are computed to demonstrate the fast convergence property of the implemented flow solver. Furthermore, an SPMD (single program, multiple data) programming paradigm based on MPI is proposed to achieve parallelism. The numerical results on complex geometries indicate that this low-storage implicit method can provide a viable and attractive DG solution for complicated flows of practical importance.

Book Space time Discontinuous Galerkin Method for Compressible Flow

Download or read book Space time Discontinuous Galerkin Method for Compressible Flow written by Christiaan Marijn Klaij and published by . This book was released on 2006 with total page 125 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Book A Parallel Reconstructed Discontinuous Galerkin Method for the Compressible Flows on Aritrary Grids

Download or read book A Parallel Reconstructed Discontinuous Galerkin Method for the Compressible Flows on Aritrary Grids written by and published by . This book was released on 2010 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: A reconstruction-based discontinuous Galerkin method is presented for the solution of the compressible Navier-Stokes equations on arbitrary grids. In this method, an in-cell reconstruction is used to obtain a higher-order polynomial representation of the underlying discontinuous Galerkin polynomial solution and an inter-cell reconstruction is used to obtain a continuous polynomial solution on the union of two neighboring, interface-sharing cells. The in-cell reconstruction is designed to enhance the accuracy of the discontinuous Galerkin method by increasing the order of the underlying polynomial solution. The inter-cell reconstruction is devised to remove an interface discontinuity of the solution and its derivatives and thus to provide a simple, accurate, consistent, and robust approximation to the viscous and heat fluxes in the Navier-Stokes equations. A parallel strategy is also devised for the resulting reconstruction discontinuous Galerkin method, which is based on domain partitioning and Single Program Multiple Data (SPMD) parallel programming model. The RDG method is used to compute a variety of compressible flow problems on arbitrary meshes to demonstrate its accuracy, efficiency, robustness, and versatility. The numerical results demonstrate that this RDG method is third-order accurate at a cost slightly higher than its underlying second-order DG method, at the same time providing a better performance than the third order DG method, in terms of both computing costs and storage requirements.

Book A Hybrid Explicit implicit Second order TVD Method on Adaptive Unstructured Grids for Unsteady Compressible Flows

Download or read book A Hybrid Explicit implicit Second order TVD Method on Adaptive Unstructured Grids for Unsteady Compressible Flows written by Farhang Norouzi and published by . This book was released on 2015 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: " This work is devoted to the development of a hybrid, explicit-implicit, scheme for simulation of unsteady compressible flows with shock waves. The proposed scheme is of the second-order accuracy in space and time for both explicit and implicit modes, while satisfying the TVD (Total Variation Diminishing) property. The scheme is designed for simulation of the compressible flows with temporal stiffness. In this situation,the numerical time step of explicit schemes is severely limited by particular conditions in a relatively small part of the computational domain, while the rest of the domain admits much higher time steps. The hybrid scheme is designed to operate in its implicit mode in the small areas causing temporal stiffness, thus allowing to proceed with higher time steps and reduce the computational time. In this study, a new hybridization approach is suggested. On its basis, the hybrid scheme is first introduced for hyperbolic conservation laws in one dimension. In order to satisfy the TVD property and obtain monotone solutions in the presence ofdiscontinuities, TVD limiters are applied to both spatial and temporal reconstructions. The second-order accuracy in time for the implicit mode, which is the main distinction of the proposed hybrid scheme in comparison with the existing methods, is achieved through a reconstruction of the solution in time. To make the reconstruction TVD preserving, a novel time limiter is derived. The stability condition and the relation to the hybridization factor of the new scheme are obtained. Moreover, the relationship of the proposed scheme with another existing hybrid method is revealed and analyzed. A set of one-dimensional test problems is used to demonstrate the accuracy and efficiency of the new hybrid scheme as well as to show its advantagesin comparison with the existing hybrid schemes. For two-dimensional problems, the hybrid scheme is generalized for the Euler and Navier-Stokes equations on unstructured grids. Similar to one-dimensional problems, reconstructions in space and time are applied with the corresponding TVD limiters. The newly proposed time limiter of the hybrid scheme is further generalized for unstructured grids. The non-linear system of discretized equations is solved using the Lower-Upper-Symmetric-Gauss-Seidel (LU-SGS) approximate factorization method for unstructured grids. In order to eliminate the factorization and linearization errors, internal iterations are introduced at each time step. The lower and upper matrices in the LU-SGS scheme are formed via reordering of grid nodes at each time step. In addition, local transient grid adaptation is applied near solution peculiarities, such as shock waves and contact surfaces. Viscous terms of the Navier-Stokes equations are also approximated with the second order of accuracy both in space and time in both explicit and implicit modes. The new hybrid scheme is applied to a number of demonstrative problems chosen to represent typical classes of gasdynamic problems with specific source of stiffness. The numerical results demonstrate the ability of the proposed hybrid scheme to produce the same accuracy as the purely explicit scheme (the well-known MUSCLHancock scheme), while reducing the computational time." --

Book A Hybridized Discontinuous Galerkin Method for Unsteady Flows with Shock Capturing

Download or read book A Hybridized Discontinuous Galerkin Method for Unsteady Flows with Shock Capturing written by Alexander Jaust and published by . This book was released on 2014 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: