TY - JOUR
T1 - High-order mass- And energy-conserving sav-gauss collocation finite element methods for the nonlinear schrÖdinger equation
AU - Feng, Xiaobing
AU - Li, Buyang
AU - Ma, Shu
N1 - Funding Information:
∗Received by the editors June 11, 2020; accepted for publication (in revised form) March 10, 2021; published electronically June 9, 2021. https://doi.org/10.1137/20M1344998 Funding: The work of the first author was partially supported by the National Science Foundation grants DMS-1620168 and DMS-2012414. The work of the second author was partially supported by an internal grant of the Hong Kong Polytechnic University, project ZZKQ. The work of the third author was partially supported by the Hong Kong RGC grant 15300817. †Department of Mathematics, The University of Tennessee, Knoxville, TN 37996 USA (xfeng@ math.utk.edu). ‡Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong ([email protected]). §Corresponding author. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong ([email protected]).
Publisher Copyright:
© 2021 Society for Industrial and Applied Mathematics
PY - 2021/6/9
Y1 - 2021/6/9
N2 - A family of arbitrarily high-order fully discrete space-time finite element methods are proposed for the nonlinear Schrödinger equation based on the scalar auxiliary variable formulation, which consists of a Gauss collocation temporal discretization and the finite element spatial discretization. The proposed methods are proved to be well-posed and conserving both mass and energy at the discrete level. An error bound of the form O(hp + τk+1) in the L∞(0, T; H1)-norm is established, where h and τ denote the spatial and temporal mesh sizes, respectively, and (p, k) is the degree of the space-time finite elements. Numerical experiments are provided to validate the theoretical results on the convergence rates and conservation properties. The effectiveness of the proposed methods in preserving the shape of a soliton wave is also demonstrated by numerical results.
AB - A family of arbitrarily high-order fully discrete space-time finite element methods are proposed for the nonlinear Schrödinger equation based on the scalar auxiliary variable formulation, which consists of a Gauss collocation temporal discretization and the finite element spatial discretization. The proposed methods are proved to be well-posed and conserving both mass and energy at the discrete level. An error bound of the form O(hp + τk+1) in the L∞(0, T; H1)-norm is established, where h and τ denote the spatial and temporal mesh sizes, respectively, and (p, k) is the degree of the space-time finite elements. Numerical experiments are provided to validate the theoretical results on the convergence rates and conservation properties. The effectiveness of the proposed methods in preserving the shape of a soliton wave is also demonstrated by numerical results.
KW - Error estimates
KW - High-order conserving schemes
KW - Mass- and energy-conservation
KW - Nonlinear Schrödinger equation
KW - SAV-Gauss collocation finite element method
UR - http://www.scopus.com/inward/record.url?scp=85108635603&partnerID=8YFLogxK
U2 - 10.1137/20M1344998
DO - 10.1137/20M1344998
M3 - Journal article
AN - SCOPUS:85108635603
SN - 0036-1429
VL - 59
SP - 1566
EP - 1591
JO - SIAM Journal on Numerical Analysis
JF - SIAM Journal on Numerical Analysis
IS - 3
ER -