ﻻ يوجد ملخص باللغة العربية
This paper proposes and analyzes an ultra-weak local discontinuous Galerkin scheme for one-dimensional nonlinear biharmonic Schr{o}dinger equations. We develop the paradigm of the local discontinuous Galerkin method by introducing the second-order spatial derivative as an auxiliary variable instead of the conventional first-order derivative. The proposed semi-discrete scheme preserves a few physically relevant properties such as the conservation of mass and the conservation of Hamiltonian accompanied by its stability for the targeted nonlinear biharmonic Schr{o}dinger equations. We also derive optimal $L^2$-error estimates of the scheme that measure both the solution and the auxiliary variable. Several numerical studies demonstrate and support our theoretical findings.
In this paper, we develop an oscillation free local discontinuous Galerkin (OFLDG) method for solving nonlinear degenerate parabolic equations. Following the idea of our recent work [J. Lu, Y. Liu, and C.-W. Shu, SIAM J. Numer. Anal. 59(2021), pp. 12
We propose a new discontinuous Galerkin method based on the least-squares patch reconstruction for the biharmonic problem. We prove the optimal error estimate of the proposed method. The two-dimensional and three-dimensional numerical examples are pr
In this paper, we develop a well-balanced oscillation-free discontinuous Galerkin (OFDG) method for solving the shallow water equations with a non-flat bottom topography. One notable feature of the constructed scheme is the well-balanced property, wh
In this paper, an energy-based discontinuous Galerkin method for dynamic Euler-Bernoulli beam equations is developed. The resulting method is energy-dissipating or energy-conserving depending on the simple, mesh-independent choice of numerical fluxes
We present a discontinuous Galerkin internal-penalty scheme that is applicable to a large class of linear and non-linear elliptic partial differential equations. The scheme constitutes the foundation of the elliptic solver for the SpECTRE numerical r