Adjoint Guided Adaptive Mesh Refinement For Hyperbolic Systems Of Equations
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Adjoint Guided Adaptive Mesh Refinement For Hyperbolic Systems Of Equations
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Author : Brisa N. Davis
language : en
Publisher:
Release Date : 2018
Adjoint Guided Adaptive Mesh Refinement For Hyperbolic Systems Of Equations written by Brisa N. Davis and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018 with categories.
One difficulty in developing numerical methods for time-dependent partial differential equations is the fact that solutions contain time-varying regions where much higher resolution is required than elsewhere in the domain. The open source Clawpack software implements block-structured adaptive mesh refinement to selectively refine around propagating waves in the AMRClaw and GeoClaw packages. In particular, GeoClaw is widely used for tsunami modeling, the application that motivated this work. For problems where the solution must be computed over a large domain but is only of interest in one small area (e.g. one coastal community when doing tsunami modeling, or the location of a pressure gauge when doing acoustics modeling), a method that allows identifying and refining the grid only in regions that influence this target area would significantly reduce the computational cost of finding a solution. The adaptive mesh refinement approach currently implemented in AMRClaw and GeoClaw often refines waves that will not impact the target area. To remedy this, we seek a method that enables the identification and refinement of only the waves that will influence the location of interest. In this work we show that solving the time-dependent adjoint equation and using a suitable inner product with either the forward solution, or the estimated one-step error in the forward solution, allows for a more precise refinement of the relevant waves. We present the adjoint methodology first in one space dimension for illustration and in a broad context since it could also be used in other adaptive software, and for other tsunami applications beyond adaptive mesh refinement. We then show how this adjoint method has been integrated into the adaptive mesh refinement strategy of the open source AMRClaw and GeoClaw software and present linear variable coefficient acoustics and tsunami modeling results showing that the accuracy of the solution is maintained and the computational time required is significantly reduced through the integration of the adjoint method into adaptive mesh refinement. The adjoint method is compared to adaptive mesh refinement methods already available in the AMRClaw software, and the advantages and disadvantages of using the adjoint method are discussed. Other capabilities of the adjoint method such as focusing on specific time ranges of interest, sensitivity analysis, and source impact analysis and design are also presented. The new algorithms are incorporated in Clawpack and code for the examples presented in this work is archived on Github.
Adaptive Mesh Refinement For Hyperbolic Partial Differential Equations
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Author : Marsha J. Berger
language : en
Publisher:
Release Date : 1982
Adaptive Mesh Refinement For Hyperbolic Partial Differential Equations written by Marsha J. Berger and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1982 with categories.
Adaptive Mesh Refinement For Hyperbolic Partial Differential Equations
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Author : Stanford University. Computer Science Department. Numerical Analysis Project
language : en
Publisher:
Release Date : 1983
Adaptive Mesh Refinement For Hyperbolic Partial Differential Equations written by Stanford University. Computer Science Department. Numerical Analysis Project and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1983 with categories.
The authors present an adaptive method based on the idea of multiple, component grids for the solution of hyperbolic partial differential equations using finite difference techniques. Based upon Richardson-type estimates of the truncation error, refined grids are created or existing ones removed to attain a given accuracy for a minimum amount of work. Their approach is recursive in that fine grids can themselves contain even finer grids. The grids with finer mesh width in space also have a smaller mesh width in time, making this a mesh refinement algorithm in time and space. This document includes algorithm, data structures and grid generation procedure, and concludes with numerical examples in one and two space dimensions. (Author).
Adaptive Mesh Refinement Theory And Applications
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Author : Tomasz Plewa
language : en
Publisher: Springer Science & Business Media
Release Date : 2005-12-20
Adaptive Mesh Refinement Theory And Applications written by Tomasz Plewa and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-12-20 with Mathematics categories.
Advanced numerical simulations that use adaptive mesh refinement (AMR) methods have now become routine in engineering and science. Originally developed for computational fluid dynamics applications these methods have propagated to fields as diverse as astrophysics, climate modeling, combustion, biophysics and many others. The underlying physical models and equations used in these disciplines are rather different, yet algorithmic and implementation issues facing practitioners are often remarkably similar. Unfortunately, there has been little effort to review the advances and outstanding issues of adaptive mesh refinement methods across such a variety of fields. This book attempts to bridge this gap. The book presents a collection of papers by experts in the field of AMR who analyze past advances in the field and evaluate the current state of adaptive mesh refinement methods in scientific computing.
Patched Based Methods For Adaptive Mesh Refinement Solutions Of Partial Differential Equations
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Author :
language : en
Publisher:
Release Date : 1997
Patched Based Methods For Adaptive Mesh Refinement Solutions Of Partial Differential Equations written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1997 with categories.
This manuscript contains the lecture notes for a course taught from July 7th through July 11th at the 1997 Numerical Analysis Summer School sponsored by C.E.A., I.N.R.I.A., and E.D.F. The subject area was chosen to support the general theme of that year's school which is ''Multiscale Methods and Wavelets in Numerical Simulation.'' The first topic covered in these notes is a description of the problem domain. This coverage is limited to classical PDEs with a heavier emphasis on hyperbolic systems and constrained hyperbolic systems. The next topic is difference schemes. These schemes are the foundation for the adaptive methods. After the background material is covered, attention is focused on a simple patched based adaptive algorithm and its associated data structures for square grids and hyperbolic conservation laws. Embellishments include curvilinear meshes, embedded boundary and overset meshes. Next, several strategies for parallel implementations are examined. The remainder of the notes contains descriptions of elliptic solutions on the mesh hierarchy, elliptically constrained flow solution methods and elliptically constrained flow solution methods with diffusion.
An Adaptive Mesh Moving And Local Refinement Method For Time Dependent Partial Differential Equations
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Author :
language : en
Publisher:
Release Date : 1990
An Adaptive Mesh Moving And Local Refinement Method For Time Dependent Partial Differential Equations written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1990 with categories.
We discuss mesh-moving, static mesh regeneration, and local mesh-refinement algorithms that can be used with a finite difference or finite element scheme to solve initial boundary value problems for vector systems of time-dependent partial differential equations in two space dimensions and time. A coarse based mesh of quadrilateral cells is moved by an algebraic mesh-movement function so as to follow and isolate spatially distinct phenomena. The local mesh-refinement method recursively divides the time step and spatial cells of the moving base mesh in regions where error indicators are high until a prescribed tolerance is satisfied. The static mesh-regeneration procedure is used to create a new base mesh when the existing one becomes to distorted. The adaptive methods have been combined with a MacCormack finite difference scheme for hyperbolic systems and an error indicator based upon estimates of the local discretization error obtained by Richardson extrapolation. Results are presented for several computational examples.
An Adaptive Mesh Algorithm For Solving Systems Of Time Dependent Partial Differential Equations
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Author : David C. Arney
language : en
Publisher:
Release Date : 1985
An Adaptive Mesh Algorithm For Solving Systems Of Time Dependent Partial Differential Equations written by David C. Arney and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1985 with categories.
This thesis discusses and adaptive mesh algorithm that can be used with a finite difference or finite element scheme to solve initial-boundary value problems for vector systems of time dependent partial differential equations in two space dimensions. This algorithm combines the adaptive technique of mesh moving, static rezoning, and local mesh refinement. The nodes of a coarse mesh of quadrilateral cells are moved by a simple algebraic node movement function. The local mesh refinement method recursively divides cells of the moving coarse mesh within clustered regions that contain nodes with large error until a user prescribed error tolerance is satisfied. Keywords: Hyperbolic equations; Expert systems; and Computations.
Parallel Object Oriented Adaptive Mesh Refinement
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Author :
language : en
Publisher:
Release Date : 1997
Parallel Object Oriented Adaptive Mesh Refinement written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1997 with categories.
In this paper we study adaptive mesh refinement (AMR) for elliptic and hyperbolic systems. We use the Asynchronous Fast Adaptive Composite Grid Method (AFACX), a parallel algorithm based upon the of Fast Adaptive Composite Grid Method (FAC) as a test case of an adaptive elliptic solver. For our hyperbolic system example we use TVD and ENO schemes for solving the Euler and MHD equations. We use the structured grid load balancer MLB as a tool for obtaining a load balanced distribution in a parallel environment. Parallel adaptive mesh refinement poses difficulties in expressing both the basic single grid solver, whether elliptic or hyperbolic, in a fashion that parallelizes seamlessly. It also requires that these basic solvers work together within the adaptive mesh refinement algorithm which uses the single grid solvers as one part of its adaptive solution process. We show that use of AMR++, an object-oriented library within the OVERTURE Framework, simplifies the development of AMR applications. Parallel support is provided and abstracted through the use of the P++ parallel array class.
An Adaptive Method With Mesh Moving And Local Mesh Refinement For Time Dependent Partial Differential Equations
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Author : David C. Arney
language : en
Publisher:
Release Date : 1988
An Adaptive Method With Mesh Moving And Local Mesh Refinement For Time Dependent Partial Differential Equations written by David C. Arney and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1988 with categories.
The authors discuss mesh moving, static mesh regeneration, and local mesh refinement algorithms that can be used with a finite difference or finite element scheme to solve initial-boundary value problems for vector systems of time-dependent partial differential equations in two-space dimensions and time. A coarse base mesh of quadrilateral cells is moved by an algebraic mesh movement function so that it may follow and isolate spatially distinct phenomena. The local mesh refinement method recursively divides the time step and spatial cells of the moving base mesh in regions were error indicators are high until a prescribed tolerance is satisfied. The static mesh regeneration procedure is used to create a new base mesh when the existing ones become too distorted. In order to test our adaptive algorithms, the authors implemented them in a system code with an initial mesh generator, a MacCormack finite difference scheme for hyperbolic systems, and an error indicator based upon estimates of the local discretization error obtained by Richardson extrapolation. Results are presented for several computational examples. (kr).
An Adaptive Finite Difference Method For Hyperbolic Systems On One Space Dimension Revision
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Author : John H. Bolstad
language : en
Publisher:
Release Date : 1982
An Adaptive Finite Difference Method For Hyperbolic Systems On One Space Dimension Revision written by John H. Bolstad and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1982 with categories.
In this paper we develop and partially analyze an adaptive finite difference mesh refinement algorithm for the initial boundary value problem for hyperbolic systems in one space dimension. The method uses clusters uniform grids which can move along with pulses or steep gradients appearing in the calculation, and which are superimposed over a uniform coarse grid. Such refinements are created, destroyed, merged, separated, recursively nested or moved based on estimates of the local truncation error. We use a four-way linked tree and sequentially allocated deques (double-ended queues) to perform these operations efficiently. The local truncation error is estimated using a three-step Richardson extrapolation procedure in the interior of the region, and differences at the boundaries. Our algorithm was implemented using a portable, extensible Fortran preprocessor, to which we added records and pointers. The method is applied to two model problems: the second order wave equation with counterstreaming Gaussian pulses, and the Riemann shock-tube problem. For both problems our algorithm is shown to be three to five times more efficient (in computing time) than the use of a uniform coarse mesh, for the same accuracy.