Using a unified framework, the formulation of a super-convergent discontinuous Galerkin (SDG) method and a hybridized discontinuous Galerkin (HDG) version, both applied to a general nonlinear Schrödinger equation is presented. Conservation of the mass and the energy is studied, theoretically for the semi-discrete formulation; and, for the fully discrete method using the Modified Crank–Nicolson time scheme. Conservation of both quantities is numerically validated on two dimensional problems and high order approximations. A numerical study of convergence illustrates the advantages of the new formulations over the traditional Local Discontinuous Galerkin (LDG) method. Numerical experiments show that the approximation of the initial discrete energy converges with order 2k+1, which is better than that obtained by the standard (continuous) finite element, which is only of order 2k when polynomials of degree k are used.

Castillo, P., Gomez, S. (2020). Conservative super-convergent and hybrid discontinuous Galerkin methods applied to nonlinear Schrodinger equations. APPLIED MATHEMATICS AND COMPUTATION, 371(15 April 2020) [10.1016/j.amc.2019.124950].

Conservative super-convergent and hybrid discontinuous Galerkin methods applied to nonlinear Schrodinger equations

Sergio Gomez
2020

Abstract

Using a unified framework, the formulation of a super-convergent discontinuous Galerkin (SDG) method and a hybridized discontinuous Galerkin (HDG) version, both applied to a general nonlinear Schrödinger equation is presented. Conservation of the mass and the energy is studied, theoretically for the semi-discrete formulation; and, for the fully discrete method using the Modified Crank–Nicolson time scheme. Conservation of both quantities is numerically validated on two dimensional problems and high order approximations. A numerical study of convergence illustrates the advantages of the new formulations over the traditional Local Discontinuous Galerkin (LDG) method. Numerical experiments show that the approximation of the initial discrete energy converges with order 2k+1, which is better than that obtained by the standard (continuous) finite element, which is only of order 2k when polynomials of degree k are used.
Articolo in rivista - Articolo scientifico
Hybridized discontinuous Galerkin; Mass and energy conservation; Nonlinear Schrödinger equation; Super-convergent local discontinuous Galerkin;
English
20-dic-2019
2020
371
15 April 2020
124950
none
Castillo, P., Gomez, S. (2020). Conservative super-convergent and hybrid discontinuous Galerkin methods applied to nonlinear Schrodinger equations. APPLIED MATHEMATICS AND COMPUTATION, 371(15 April 2020) [10.1016/j.amc.2019.124950].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/442747
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