TY - JOUR
T1 - Superconvergence of the local discontinuous Galerkin method for the sine-Gordon equation in one space dimension
AU - Baccouch, Mahboub
N1 - Funding Information:
The author would also like to thank the anonymous referees for their constructive comments and remarks which helped improve the quality and readability of the paper. This research was partially supported by the NASA Nebraska Space Grant Program and UCRCA at the University of Nebraska at Omaha (UCRCA Proposal 2016-01-F ).
Funding Information:
The author would also like to thank the anonymous referees for their constructive comments and remarks which helped improve the quality and readability of the paper. This research was partially supported by the NASA Nebraska Space Grant Program and UCRCA at the University of Nebraska at Omaha (UCRCA Proposal2016-01-F).
Publisher Copyright:
© 2017 Elsevier B.V.
PY - 2018/5/1
Y1 - 2018/5/1
N2 - In this paper, we present superconvergence results for the local discontinuous Galerkin (LDG) method for the sine-Gordon nonlinear hyperbolic equation in one space dimension. We identify a special numerical flux and a suitable projection of the initial conditions for the LDG scheme for which the L2-norm of the LDG solution and its spatial derivative are of order p+1, when piecewise polynomials of degree at most p are used. Our numerical experiments demonstrate optimal order of convergence. We further prove superconvergence toward particular projections of the exact solutions. More precisely, we prove that the LDG solution and its spatial derivative are O(hp+3∕2) super close to particular projections of the exact solutions, while computational results show higher O(hp+2) convergence rate. Our analysis is valid for arbitrary regular meshes and for Pp polynomials with arbitrary p≥1. Numerical experiments validating these theoretical results are presented.
AB - In this paper, we present superconvergence results for the local discontinuous Galerkin (LDG) method for the sine-Gordon nonlinear hyperbolic equation in one space dimension. We identify a special numerical flux and a suitable projection of the initial conditions for the LDG scheme for which the L2-norm of the LDG solution and its spatial derivative are of order p+1, when piecewise polynomials of degree at most p are used. Our numerical experiments demonstrate optimal order of convergence. We further prove superconvergence toward particular projections of the exact solutions. More precisely, we prove that the LDG solution and its spatial derivative are O(hp+3∕2) super close to particular projections of the exact solutions, while computational results show higher O(hp+2) convergence rate. Our analysis is valid for arbitrary regular meshes and for Pp polynomials with arbitrary p≥1. Numerical experiments validating these theoretical results are presented.
KW - Error estimates
KW - Local discontinuous Galerkin method
KW - Projections
KW - Sine-Gordon equation
KW - Superconvergence
UR - http://www.scopus.com/inward/record.url?scp=85035016049&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85035016049&partnerID=8YFLogxK
U2 - 10.1016/j.cam.2017.11.007
DO - 10.1016/j.cam.2017.11.007
M3 - Article
AN - SCOPUS:85035016049
SN - 0377-0427
VL - 333
SP - 292
EP - 313
JO - Journal of Computational and Applied Mathematics
JF - Journal of Computational and Applied Mathematics
ER -