The orthogonal gradients method: A radial basis functions method for solving partial differential equations on arbitrary surfaces
KAUST Grant NumberKUK-C1-013-04
Permanent link to this recordhttp://hdl.handle.net/10754/599943
MetadataShow full item record
AbstractMuch work has been done on reconstructing arbitrary surfaces using the radial basis function (RBF) method, but one can hardly find any work done on the use of RBFs to solve partial differential equations (PDEs) on arbitrary surfaces. In this paper, we investigate methods to solve PDEs on arbitrary stationary surfaces embedded in . R3 using the RBF method. We present three RBF-based methods that easily discretize surface differential operators. We take advantage of the meshfree character of RBFs, which give us a high accuracy and the flexibility to represent the most complex geometries in any dimension. Two out of the three methods, which we call the orthogonal gradients (OGr) methods are the result of our work and are hereby presented for the first time. © 2012 Elsevier Inc.
CitationPiret C (2012) The orthogonal gradients method: A radial basis functions method for solving partial differential equations on arbitrary surfaces. Journal of Computational Physics 231: 4662–4675. Available: http://dx.doi.org/10.1016/j.jcp.2012.03.007.
SponsorsThe work of this author was supported by a FSR post-doctoral grant from the catholic University of Louvain. Part of the present work was conducted when the author was a Visiting Post-Doctoral Research Assistant at OCCAM (Oxford Centre for Collaborative Applied Mathematics) under support provided by Award No. KUK-C1-013-04 to the University of Oxford, UK, by King Abdullah University of Science and Technology (KAUST).
JournalJournal of Computational Physics
CollectionsPublications Acknowledging KAUST Support
Showing items related by title, author, creator and subject.
Materials For Gas Capture, Methods Of Making Materials For Gas Capture, And Methods Of Capturing GasPolshettiwar, Vivek; Patil, Umesh (2013-06-20) [Patent]In accordance with the purpose(s) of the present disclosure, as embodied and broadly described herein, embodiments of the present disclosure, in one aspect, relate to materials that can be used for gas (e.g., CO.sub.2) capture, methods of making materials, methods of capturing gas (e.g., CO.sub.2), and the like, and the like.
Propagation of internal errors in explicit Runge–Kutta methods and internal stability of SSP and extrapolation methodsKetcheson, David I.; Loczi, Lajos; Parsani, Matteo (2014-04-11) [Technical Report]In practical computation with Runge--Kutta methods, the stage equations are not satisfied exactly, due to roundoff errors, algebraic solver errors, and so forth. We show by example that propagation of such errors within a single step can have catastrophic effects for otherwise practical and well-known methods. We perform a general analysis of internal error propagation, emphasizing that it depends significantly on how the method is implemented. We show that for a fixed method, essentially any set of internal stability polynomials can be obtained by modifying the implementation details. We provide bounds on the internal error amplification constants for some classes of methods with many stages, including strong stability preserving methods and extrapolation methods. These results are used to prove error bounds in the presence of roundoff or other internal errors.
The Method of Manufactured Universes for validating uncertainty quantification methodsStripling, H.F.; Adams, M.L.; McClarren, R.G.; Mallick, B.K. (Reliability Engineering & System Safety, Elsevier BV, 2011-09) [Article]The Method of Manufactured Universes is presented as a validation framework for uncertainty quantification (UQ) methodologies and as a tool for exploring the effects of statistical and modeling assumptions embedded in these methods. The framework calls for a manufactured reality from which experimental data are created (possibly with experimental error), an imperfect model (with uncertain inputs) from which simulation results are created (possibly with numerical error), the application of a system for quantifying uncertainties in model predictions, and an assessment of how accurately those uncertainties are quantified. The application presented in this paper manufactures a particle-transport universe, models it using diffusion theory with uncertain material parameters, and applies both Gaussian process and Bayesian MARS algorithms to make quantitative predictions about new experiments within the manufactured reality. The results of this preliminary study indicate that, even in a simple problem, the improper application of a specific UQ method or unrealized effects of a modeling assumption may produce inaccurate predictions. We conclude that the validation framework presented in this paper is a powerful and flexible tool for the investigation and understanding of UQ methodologies. © 2011 Elsevier Ltd. All rights reserved.