AI 中文总结
研究针对波传播问题提出辛哈密顿直接间断伽辽金方法，通过证明数值通量双线性形式对称性与离散哈密顿结构的关系，构建全离散辛格式，推导误差估计，经数值实验验证该方法在能量守恒和精度上表现优异。
AI 中文摘要
本文提出一种用于逼近波传播问题（包括线性和半线性波动方程）的辛哈密顿直接间断伽辽金（DDG）方法。在无辅助变量的DG框架内，证明数值通量双线性形式的对称性等同于离散哈密顿结构的存在。由此得出对称内部罚方法和对称DDG（SDDG）方法等具有离散哈密顿结构，而鲍曼 - 奥登、DDG和BR2方法等不具备。利用此结构，通过将SDDG空间离散与辛时间积分器结合构建全离散辛格式。还推导了应用于半线性波动方程的SDDG方法的误差估计，展示了位移的最优收敛速率和速度的次优收敛速率。数值实验验证了理论收敛速率，并表明辛哈密顿DDG方法实现了卓越的长时间能量守恒和精度。
英文摘要
This paper presents a symplectic Hamiltonian direct discontinuous Galerkin (DDG) method for approximating wave propagation problems, including the linear and semilinear wave equations. Within an auxiliary-variable-free DG framework, we prove that the symmetry of the numerical flux bilinear form is equivalent to the existence of a discrete Hamiltonian structure. It follows that methods such as the symmetric interior penalty method and the symmetric DDG (SDDG) method admit a discrete Hamiltonian structure, whereas schemes including the Baumann--Oden, DDG, and BR2 methods do not possess this property. Exploiting this structure, we construct fully discrete symplectic schemes by combining the SDDG spatial discretization with symplectic time integrators. We further derive error estimates for the SDDG method applied to semilinear wave equations, showing the optimal convergence rate for the displacement and the suboptimal convergence rate for the velocity. Numerical experiments validate the theoretical convergence rates and demonstrate that the symplectic Hamiltonian DDG method achieves superior long-time energy conservation and accuracy.
Comments41 pages, 11 figures
