Symplectic Hamiltonian HDG methods for wave propagation phenomena
Symplectic Hamiltonian HDG methods for wave propagation phenomena
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DOI：
10.1016/j.jcp.2017.09.010
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发表时间：
2017-12
期刊：
影响因子：
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通讯作者：
M. Sánchez;C. Ciucă;N. Nguyen;J. Peraire;Bernardo Cockburn
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文献类型：
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作者：
M. Sánchez;C. Ciucă;N. Nguyen;J. Peraire;Bernardo Cockburn
我们设计了第一个用于声波方程的辛哈密顿可杂化间断伽辽金（HDG）方法。我们在空间上使用哈密顿 HDG 方案（即保留波动方程哈密顿结构的 HDG 方法）进行离散，在时间上使用辛、对角隐式和显式分区龙格-库塔方法进行离散。由此产生的方案的基本特征是保证了离散能量的守恒，而离散能量只不过是原始哈密顿量的离散版本。我们提出的数值实验表明，当使用 k≥ 0 阶多项式和 k+ 1 阶龙格-库塔时间推进方法时，该方法在 L 2-范数中实现了 k+ 1 阶最优近似。此外，通过后处理技术以及将龙格-库塔方法的阶数增加到k+2，我们获得了位移和速度的L 2-范数中k+2阶的超收敛近似。我们还提供了数值示例，证实这些方法可以节省能量，并且在长时间模拟中，它们与具有相似精度特性的耗散 HDG 方案相比具有优势。
We devise the first symplectic Hamiltonian hybridizable discontinuous Galerkin (HDG) methods for the acoustic wave equation. We discretize in space by using a Hamiltonian HDG scheme, that is, an HDG method which preserves the Hamiltonian structure of the wave equation, and in time by using symplectic, diagonally implicit and explicit partitioned Runge–Kutta methods. The fundamental feature of the resulting scheme is that the conservation of a discrete energy, which is nothing but a discrete version of the original Hamiltonian, is guaranteed. We present numerical experiments which indicate that the method achieves optimal approximations of order k+ 1 in the L 2-norm when polynomials of degree k≥ 0 and Runge–Kutta time-marching methods of order k+ 1 are used. In addition, by means of post-processing techniques and by increasing the order of the Runge–Kutta method to k+ 2, we obtain superconvergent approximations of order k+ 2 in the L 2-norm for the displacement and the velocity. We also present numerical examples that corroborate that the methods conserve energy and that they compare favorably with dissipative HDG schemes, of similar accuracy properties, for long-time simulations.