NDLI: A High-Order Dirac-Delta Regularization with Optimal Scaling in the Spectral Solution of One-Dimensional Singular Hyperbolic Conservation Laws

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Don, Wai-Sun Suarez, Jean-Piero Jacobs, Gustaaf B.
Copyright Year	2014
Abstract	A regularization technique based on a class of high-order, compactly supported piecewise polynomials is developed that regularizes the time-dependent, singular Dirac-delta sources in spectral approximations of hyperbolic conservation laws. The regularization technique provides higher-order accuracy away from the singularity. A theoretical criterion that establishes a lower bound on the support length (optimal scaling) has to be satisfied to achieve optimal order of convergence. The optimal scaling parameter has been shown to be, instead of a fixed constant value, a function of the smoothness of the compactly supported piecewise polynomial and the number of vanishing moment of the polynomial when integrated with respect to mononomial of degree up to some order (moment). The effectiveness of the criterion is illustrated in the solutions of a linear and a nonlinear (Burgers) scalar hyperbolic conservation law with a singular source, as well as the nonlinear Euler equations with singular sources, a system of hyperbolic conservation laws governing compressible fluid dynamics with shocks and particles. A Chebyshev collocation method (spectral) discretizes the spatial derivatives in the scalar equation tests. A multidomain hybrid spectral-WENO method discretizes the Euler equations. The WENO scheme captures shocks and high gradients in the gas flow, whereas the spectral discretization applies in all other subdomains including the regularization of the singular source. In the advection and Burgers problems, the convergence order of the numerical solution follows the asymptotic behavior of the singular source approximation. The multidomain hybrid scheme is in excellent agreement with the computations that are based on a WENO discretization only.
Starting Page	A1831
Ending Page	A1849
Page Count	19
File Format	PDF
ISSN	10648275
DOI	10.1137/130939341
e-ISSN	10957197
Issue Number	4
Volume Number	36
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2014-08-14
Access Restriction	Subscribed
Subject Keyword	Conservation laws hyperbolic conservation laws Dirac-delta Spectral, collocation and related methods regularization WENO singular sources spectral Chebyshev collocation Shocks and singularities Finite difference methods
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computational Mathematics

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Library of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in