ترغب بنشر مسار تعليمي؟ اضغط هنا

A discontinuous Galerkin method for nonlinear biharmonic Schr{o}dinger equations

214   0   0.0 ( 0 )
 نشر من قبل Lu Zhang
 تاريخ النشر 2021
  مجال البحث الهندسة المعلوماتية
والبحث باللغة English
 تأليف Lu Zhang




اسأل ChatGPT حول البحث

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.



قيم البحث

اقرأ أيضاً

97 - Qi Tao , Yong Liu , Yan Jiang 2021
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 99-1324.], we add the damping terms to the LDG scheme to control the spurious oscillations when solutions have a large gradient. The $L^2$-stability and optimal priori error estimates for the semi-discrete scheme are established. The numerical experiments demonstrate that the proposed method maintains the high-order accuracy and controls the spurious oscillations well.
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 esented to confirm the accuracy and efficiency of the method with several boundary conditions and several types of polygon meshes and polyhedral meshes.
243 - Yong Liu , Jianfang Lu , Qi Tao 2021
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 ich preserves exactly the hydrostatic equilibrium solutions up to machine error. Another feature is the non-oscillatory property, which is very important in the numerical simulation when there exist some shock discontinuities. To control the spurious oscillations, we construct an OFDG method with an extra damping term to the existing well-balanced DG schemes proposed in [Y. Xing and C.-W. Shu, CICP, 1(2006), 100-134.]. With a careful construction of the damping term, the proposed method achieves both the well-balanced property and non-oscillatory property simultaneously without compromising any order of accuracy. We also present a detailed procedure for the construction and a theoretical analysis for the preservation of the well-balancedness property. Extensive numerical experiments including one- and two-dimensional space demonstrate that the proposed methods possess the desired properties without sacrificing any order of accuracy.
243 - Lu Zhang 2021
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 . By introducing a velocity field, the original problem is transformed into a first-order in time system. In our formulation, the discontinuous Galerkin approximations for the original displacement field and the auxiliary velocity field are not restricted to be in the same space. In particular, a given accuracy can be achieved with the fewest degrees of freedom when the degree for the approximation space of the velocity field is two orders lower than the degree of approximation space for the displacement field. In addition, we establish the error estimates in an energy norm and demonstrate the corresponding optimal convergence in numerical experiments.
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 elativity code. As such it can accommodate (but is not limited to) elliptic problems in linear elasticity, general relativity and hydrodynamics, including problems formulated on a curved manifold. We provide practical instructions that make the scheme functional in a production code, such as instructions for imposing a range of boundary conditions, for implementing the scheme on curved and non-conforming meshes and for ensuring the scheme is compact and symmetric so it may be solved more efficiently. We report on the accuracy of the scheme for a suite of numerical test problems.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا