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Adaptation strategies for high order discontinuous Galerkin methods based on Tau-estimation

Moritz Kompenhans, Gonzalo Rubio, Esteban Ferrer, Eusebio Valero
2016 Journal of Computational Physics  
Namely, for a viscous boundary layer flow, we obtain a speedup of 6.6 and 7.6 for the quasi-a priori and quasi-a priori corrected approaches, respectively.  ...  In addition, the high order method permits the spatial decoupling for the estimated errors and enables anisotropic p-adaptation.  ...  Proof of the quasi-a priori x -estimation formula, Eq. (31) Substituting the definitions of the iteration error, u r.P estimate of the truncation error Eq. (31) and, using Taylor series, we obtain  ... 
doi:10.1016/j.jcp.2015.11.032 fatcat:rbqzflvgfjbozcrsnblamgrri4

Quasi-A Priori Truncation Error Estimation in the DGSEM

Gonzalo Rubio, François Fraysse, David A. Kopriva, Eusebio Valero
2014 Journal of Scientific Computing  
We demonstrate that this quasi-a priori approach yields a spectrally accurate estimate of the truncation error.  ...  In this paper we show how to accurately perform a quasi-a priori estimation of the truncation error of steady-state solutions computed by a discontinuous Galerkin spectral element method.  ...  Quasi-A Priori Estimates In this section we show the convergence of the quasi-a priori estimate for the truncation error and the isolated truncation error with the number of iterations of the time-iterative  ... 
doi:10.1007/s10915-014-9938-6 fatcat:q62ugstv4bcdjo4bgktct7tm5u

Truncation Error Estimation in the p-Anisotropic Discontinuous Galerkin Spectral Element Method

Andrés M. Rueda-Ramírez, Gonzalo Rubio, Esteban Ferrer, Eusebio Valero
2018 Journal of Scientific Computing  
We show that the new error estimator is cheaper to evaluate than previous implementations of the τ-estimation procedure and that it obtains more accurate extrapolations of the truncation error for representations  ...  of a higher order than the reference mesh.  ...  Acknowledgements The authors would like to thank David Kopriva for his friendly advise and cooperation.  ... 
doi:10.1007/s10915-018-0772-0 fatcat:uhbgqd3y4zdzpgzcvpwcfc3v3m

A Posteriori Error Estimator for a Non-Standard Finite Difference Scheme Applied to BVPs on Infinite Intervals [article]

Riccardo Fazio, Alessandra Jannelli
2015 arXiv   pre-print
In this paper, we present a study of an a posteriori estimator for the discretization error of a non-standard finite difference scheme applied to boundary value problems defined on an infinite interval  ...  In particular, we show how Richardson's extrapolation can be used to improve the numerical solution involving the order of accuracy and numerical solutions from two nested quasi-uniform grids.  ...  systems of ordinary or partial differential equations, see, for instance, [22] .  ... 
arXiv:1503.00037v2 fatcat:owfy5fuk7fc3zj3p5s3xqfdlz4

QUANTIFICATION OF UNCERTAINTY IN COMPUTATIONAL FLUID DYNAMICS

P. J. Roache
1997 Annual Review of Fluid Mechanics  
estimators, nonlinear dynamics, and error estimation for grid adaptation vs Quantification of Uncertainty.  ...  This review covers Verification, Validation, Confirmation and related subjects for computational fluid dynamics (CFD), including error taxonomies, error estimation and banding, convergence rates, surrogate  ...  Taxonomy for Additional Information for Error Estimates Once we have produced a CFD solution of the governing partial differential equations, it is clear that we require some additional information in  ... 
doi:10.1146/annurev.fluid.29.1.123 fatcat:2ck7wznppfdkjfxaghdj7hwlie

Extrapolation Methods for Improving the Convergence of Oligomer Calculations to the Infinite Chain Limit of Quasi-Onedimensional Stereoregular Polymers [article]

Ernst Joachim Weniger (Institute for Physical and Theoretical Chemistry, University of Regensburg, Regensburg, Germany), Bernard Kirtman (Department of Chemistry and Biochemistry, University of California, Santa Barbara, U.S.A.)
2000 arXiv   pre-print
Quasi-onedimensional stereoregular polymers as for example polyacetylene are currently of considerable interest.  ...  Several different extrapolation methods are discussed which are able to accomplish an extrapolation of energies and properties of oligomers to the infinite chain limit.  ...  Acknowledgments EJW thanks the Fonds der Chemischen Industrie for financial support.  ... 
arXiv:math/0004115v2 fatcat:6a76q46s5ba6dhl6shflknxh2m

Adaptive mesh refinement for coupled elliptic-hyperbolic systems

Frans Pretorius, Matthew W. Choptuik
2006 Journal of Computational Physics  
We present a modification to the Berger and Oliger adaptive mesh refinement algorithm designed to solve systems of coupled, non-linear, hyperbolic and elliptic partial differential equations.  ...  We also describe several other details of the code, including truncation error estimation using a self-shadow hierarchy, and the refinement-boundary interpolation operators that are used to help suppress  ...  In such a coordinate basis the field equations of general relativity consist of a set of 10, second order, quasi-linear partial differential equations (PDEs) for 10 metric coefficients that describe the  ... 
doi:10.1016/j.jcp.2006.02.011 fatcat:zbcrjeqvarf4fjnbavioamj7zi

To the boundary and back—a numerical study

S. Karni
1991 International Journal for Numerical Methods in Fluids  
We demonstrate strong focusing of asymptotic errors generated at the artificial boundary. Internal reflections due to slowly expanding grids are shown for non-linear systems.  ...  While it is widely accepted that dissipative models tend to give fewer problems than non-dissipative ones, we show a variety of cases where substantial reflections do occur even in 1D and quasi-1D set-ups  ...  Roe for his continuous interest in and advice on this work. I am also grateful to Dr. E. F. Tor0 for reading a previous version of this manuscript and making valuable remarks.  ... 
doi:10.1002/fld.1650130205 fatcat:zp4ivpa66zazhihj5icouoxrpm

Space–time adaptive finite difference method for European multi-asset options

Per Lötstedt, Jonas Persson, Lina von Sydow, Johan Tysk
2007 Computers and Mathematics with Applications  
The multi-dimensional Black-Scholes equation is solved numerically for a European call basket option using a priori-a posteriori error estimates.  ...  Bounds on the local errors and the adjoint solution are obtained by the maximum principle for parabolic equations.  ...  Acknowledgements The second author was supported by FMB, the Swedish Graduate School in Mathematics and Computing Science.  ... 
doi:10.1016/j.camwa.2006.09.014 fatcat:zcrxfny2rvb33ox34ulteubivm

Computation of derivatives of the rotation number for parametric families of circle diffeomorphisms

Alejandro Luque, Jordi Villanueva
2008 Physica D : Non-linear phenomena  
In order to justify the method, we also require the family of maps to be differentiable with respect to the parameters and the rotation number to be Diophantine.  ...  Our methodology is an extension of a recently developed approach to compute rotation numbers based on suitable averages of the iterates of the map and Richardson extrapolation.  ...  The authors have been partially supported by the Spanish MCyT/FEDER grant MTM2006-00478. Moreover, the research of A. L. has been supported by the Spanish phD grant FPU AP2005-2950.  ... 
doi:10.1016/j.physd.2008.03.047 fatcat:5utyoapk3vazpprzqqtzk6xnb4

The mathematics of finite elements and applications

1989 International Journal of Engineering Science  
This paper describes theory and methods for developing a posteriori error estimates and an adaptive strategy for hp-finite element approximations of the incompressible Navier-Stokes equations.  ...  For an error estimation, use is made of a new approach which is based on the work of Ainsworth, Wu and the author.  ...  of non-linear elliptic partial differential equations is presented.  ... 
doi:10.1016/0020-7225(89)90012-8 fatcat:z6hknn325re4tbssbqqzmcketa

Inertia–Gravity Wave Generation by Balanced Motion: Revisiting the Lorenz–Krishnamurthy Model

J. Vanneste
2004 Journal of the Atmospheric Sciences  
Boussinesq equations which are described by a (non-autonomous) linear ordinary differential equation.  ...  Numerical estimates for κ as a function of b: estimates obtained for = 0.08 (+), = 0.06 (×) and using a linear extrapolation (•) are compared with the theoretical value.  ...  Scaled inertia-gravity-wave amplitude I N /C as a function of t/(π ) 1/2 obtained numerically for = 0.1 and b = 0.5 and for N = 4, 5, 6, 7 (solid lines).  ... 
doi:10.1175/1520-0469(2004)061<0224:iwgbbm>2.0.co;2 fatcat:dnhx3zu7cnetjolor5rthju6j4

Adaptivity with moving grids

Chris J. Budd, Weizhang Huang, Robert D. Russell
2009 Acta Numerica  
We firstly examine the possible geometries of a moving mesh in both one and higher dimensions, and the discretization of partial differential equation on such meshes.  ...  In particular, we consider such issues as mesh regularity, equidistribution, variational methods, and the error in interpolating a function or truncation error on such a mesh.  ...  Acknowledgements It is a pleasure to thank Emily Walsh for Figures 5.7, 5.8, 5.9, 5.11 and 5.12 and J. F. Williams for Figures 3.7, 3.8, 3.9, 3.11, 5.2 and 5.4.  ... 
doi:10.1017/s0962492906400015 fatcat:zp36xsueyjdwfovrt6qfppl3si

Circular polarization measurement in millimeter-wavelength spectral-line VLBI observations

A. J. Kemball, L. Richter
2011 Astronomy and Astrophysics  
We present an example data reduction at λ=7 mm and derive an estimate of the predicted accuracy of the method of m_c < 0.5 or better at lambda=3 mm.  ...  We perform an analysis here of the data model containing the instrumental factors that limit the accuracy of circular polarization measurements in such observations, and present a corresponding data reduction  ...  We thank the anonymous referee for their insightful comments on this paper. This material is based upon work partially supported by the National Science Foundation under grant AST-0507473.  ... 
doi:10.1051/0004-6361/201117337 fatcat:nzvy3pbhdfd47a5jdd3ixl7is4

Fast multipole accelerated boundary knot method for inhomogeneous Helmholtz problems

Xinrong Jiang, Wen Chen, C.S. Chen
2013 Engineering analysis with boundary elements  
It is suggested that a simple point-wise error indicator can provide an a-priori indication on the number N of terms to be employed in the Gegenbauer addition formula in order to achieve a prescribed accuracy  ...  of available studies notwithstanding, arise for a wealth of different physical problems.  ...  Another class of such problems correspond to parabolic problems involving elliptic partial differential operators in the space variables and first-order time derivatives, upon using Fourier transform in  ... 
doi:10.1016/j.enganabound.2013.05.007 fatcat:2vao5saolvgmhf4scimw645mke
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