TY - JOUR

T1 - Superconvergence of the semi-discrete local discontinuous galerkin method for nonlinear KDV-type problems

AU - Baccouch, Mahboub

N1 - Funding Information:
Acknowledgments. The author would like to thank the three anonymous reviewers for the valuable comments and suggestions which improved the quality of the paper. This research was supported by the University Committee on Research and Creative Activity (UCRCA Proposal 2017-01-F) at the University of Nebraska at Omaha.

PY - 2019/1

Y1 - 2019/1

N2 - In this paper, we present and analyze a superconvergent local discontinuous Galerkin (LDG) scheme for the numerical solution of nonlinear KdV-type partial differential equations. Optimal a priori error estimates for the LDG solution and for the two auxiliary variables that approximate the firstand second-order derivative are derived in the L2-norm for the semi-discrete formulation. The order of convergence is proved to be p + 1, when piecewise polynomials of degree at most p are used. We further prove that the derivative of the LDG solution is superconvergent with order p + 1 towards the derivative of a special projection of the exact solution. We use this results to prove that the LDG solution is superconvergent with order p + 3/2 toward a special Gauss-Radau projection of the exact solution. Finally, several numerical examples are given to validate the theoretical results. Our proofs are valid for arbitrary regular meshes using Pp polynomials with p ≥ 1 and under the condition that |f'(u)| possesses a uniform positive lower bound, where f(u) is the nonlinear flux function. Our experiments demonstrate that our results hold true for KdV equations with general flux functions.

AB - In this paper, we present and analyze a superconvergent local discontinuous Galerkin (LDG) scheme for the numerical solution of nonlinear KdV-type partial differential equations. Optimal a priori error estimates for the LDG solution and for the two auxiliary variables that approximate the firstand second-order derivative are derived in the L2-norm for the semi-discrete formulation. The order of convergence is proved to be p + 1, when piecewise polynomials of degree at most p are used. We further prove that the derivative of the LDG solution is superconvergent with order p + 1 towards the derivative of a special projection of the exact solution. We use this results to prove that the LDG solution is superconvergent with order p + 3/2 toward a special Gauss-Radau projection of the exact solution. Finally, several numerical examples are given to validate the theoretical results. Our proofs are valid for arbitrary regular meshes using Pp polynomials with p ≥ 1 and under the condition that |f'(u)| possesses a uniform positive lower bound, where f(u) is the nonlinear flux function. Our experiments demonstrate that our results hold true for KdV equations with general flux functions.

KW - Local Discontinuous Galerkin Method Kdv Equations

KW - Stability A Priori Error Estimates Superconvergence Gauss-Radau Projections

UR - http://www.scopus.com/inward/record.url?scp=85055203165&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85055203165&partnerID=8YFLogxK

U2 - 10.3934/dcdsb.2018104

DO - 10.3934/dcdsb.2018104

M3 - Article

AN - SCOPUS:85055203165

VL - 24

SP - 19

EP - 54

JO - Discrete and Continuous Dynamical Systems - Series B

JF - Discrete and Continuous Dynamical Systems - Series B

SN - 1531-3492

IS - 1

ER -