HDG方法中关于波数的稳定化
Stabilization in relation to wavenumber in HDG methods
AI总结：
本文研究了HDG方法在复杂介质中波传播的稳定性，分析了稳定参数对实复波数解的影响，提出了一种保证唯一解的条件，并通过色散分析展示了稳定参数的选择对误差的影响。
摘要：
波通过复杂介质的模拟依赖于对数值方法在实复波数下性质的正确理解。本文考虑了混合不连续伽辽金（HDG）类型数值方法用于模拟满足亥姆霍兹和麦克斯韦方程的波。研究表明，当HDG方法使用不当，会为复波数产生奇异系统。对于复波数，得到了保证HDG系统唯一可解性的HDG稳定参数充分条件。对于实波数，给出了色散分析的结果。通过网格元素数量增加时色散关系的渐近展开，发现某些稳定参数选择优于其他。总结发现，某些HDG稳定参数值会导致HDG方法在复波数下失效，但若稳定参数的实部与波数的虚部符号相反，则可修复此失败。当波数为实数时，发现稳定参数在虚轴上渐近最小化HDG波数误差。最后，混合混合拉维亚-托马斯方法的色散分析表明，其波数误差比HDG方法小一个数量级。
英文摘要：
Simulation of wave propagation through complex media relies on proper understanding of the properties of numerical methods when the wavenumber is real and complex. Numerical methods of the Hybrid Discontinuous Galerkin (HDG) type are considered for simulating waves that satisfy the Helmholtz and Maxwell equations. It is shown that these methods, when wrongly used, give rise to singular systems for complex wavenumbers. A sufficient condition on the HDG stabilization parameter for guaranteeing unique solvability of the numerical HDG system, both for Helmholtz and Maxwell systems, is obtained for complex wavenumbers. For real wavenumbers, results from a dispersion analysis are presented. An asymptotic expansion of the dispersion relation, as the number of mesh elements per wave increase, reveal that some choices of the stabilization parameter are better than others. To summarize the findings, there are values of the HDG stabilization parameter that will cause the HDG method to fail for complex wavenumbers. However, this failure is remedied if the real part of the stabilization parameter has the opposite sign of the imaginary part of the wavenumber. When the wavenumber is real, values of the stabilization parameter that asymptotically minimize the HDG wavenumber errors are found on the imaginary axis. Finally, a dispersion analysis of the mixed hybrid Raviart-Thomas method showed that its wavenumber errors are an order smaller than those of the HDG method.
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