Experimental Convergence Rate Study for Three Shock-Capturing Schemes and Development of Highly Accurate Combined Schemes

We study experimental convergence rates of three shock-capturing schemes for hyperbolic systems of conservation laws: the second-order central-upwind (CU) scheme, the third-order Rusanov-Burstein-Mirin (RBM), and the fifth-order alternative weighted essentially non-oscillatory (A-WENO) scheme. We use three imbedded grids to define the experimental pointwise, integral, and $W^{-1,1}$ convergence rates. We apply the studied schemes to the shallow water equations and conduct their comprehensive numerical convergence study. We verify that while the studied schemes achieve their formal orders of accuracy on smooth solutions, after the shock formation, a part of the computed solutions is affected by shock propagation and both the pointwise and integral convergence rates reduce there. Moreover, while the $W^{-1,1}$ convergence rates for the CU and A-WENO schemes, which rely on nonlinear stabilization mechanisms, reduce to the first order, the RBM scheme, which utilizes a linear stabilization, is clearly second-order accurate. Finally, relying on the conducted experimental convergence rate study, we develop two new combined schemes based on the RBM and either the CU or A-WENO scheme. The obtained combined schemes can achieve the same high-order of accuracy as the RBM scheme in the smooth areas while being non-oscillatory near the shocks.Comment: 33 page

Investigation of High-Q Lithium Niobate-Based Double Ring Resonator Used in RF Signal Modulation

In recent years, millimeter-wave communication has played a crucial role in satellite communication, 5G, and even 6G applications. The millimeter-wave electro-optic modulator is capable of receiving and processing millimeter-wave signals effectively. However, the large attenuation of millimeter waves in the air remains a primary limiting factor for their future applications. Therefore, finding a waveguide structure with a high quality factor (Q-factor) is critical for millimeter-wave electro-optic modulators. This manuscript presents the demonstration of a double ring modulator made of lithium niobate with the specific goal of modulating an RF signal at approximately 35 GHz. By optimizing the microring structure, the double ring resonator with high Q-factor is studied to obtain high sensitivity modulation of the RF signal. This manuscript employs the transfer matrix method to investigate the operational principles of the double ring structure and conducts simulations to explore the influence of structural parameters on its performance. Through a comparison with the traditional single ring structure, it is observed that the Q-factor of the double ring modulator can reach 7.05 × 108, which is two orders of magnitude greater than that of the single ring structure. Meanwhile, the electro-optical tunability of the double ring modulator is 6 pm/V with a bandwidth of 2.4 pm, which only needs 0.4 V driving voltage. The high Q double ring structure proposed in this study has potential applications not only in the field of communication but also as a promising candidate for a variety of chemical and biomedical sensing applications