Search for: All records

Abstract Extending the work of Yang–Zumbrun for the hydrodynamically stable case of Froude number$$F<2$$ $F<2$, we categorize completely the existence and convective stability of hydraulic shock profiles of the Saint Venant equations of inclined thin film flow. Moreover, we confirm by numerical experiment that asymptotic dynamics for general Riemann data is given in the hydrodynamic instability regime by either stable hydraulic shock waves, or a pattern consisting of an invading roll wave front separated by a finite terminating Lax shock from a constant state at plus infinity. Notably, profiles, and existence and stability diagrams, are all rigorously obtained by mathematical analysis and explicit calculation. 

We study for the Richard-Gavrilyuk model of inclined shallow water flow, an extension of the classical Saint Venant equations incorporating vorticity, the new feature of convective-wave solutions analogous to contact discontinuitis in inviscid conservation laws. These are traveling waves for which fluid velocity is constant and equal to the speed of propagation of the wave, but fluid height and/or enstrophy (thus vorticity) varies. Together with hydraulic shocks, they play an important role in the structure of Riemann solutions. 

Neural Radiance Field (NeRF) based rendering has attracted growing attention thanks to its state-of-the-art (SOTA) rendering quality and wide applications in Augmented and Virtual Reality (AR/VR). However, immersive real-time (> 30 FPS) NeRF based rendering enabled interactions are still limited due to the low achievable throughput on AR/VR devices. To this end, we first profile SOTA efficient NeRF al- gorithms on commercial devices and identify two primary causes of the aforementioned inefficiency: (1) the uniform point sampling and (2) the dense accesses and computations of the required embeddings in NeRF. Furthermore, we propose RT-NeRF, which to the best of our knowledge is the first algorithm-hardware co-design acceleration of NeRF. Specifically, on the algorithm level, RT-NeRF integrates an efficient rendering pipeline for largely alleviating the inefficiency due to the commonly adopted uniform point sampling method in NeRF by directly computing the geometry of pre-existing points. Additionally, RT-NeRF leverages a coarse-grained view-dependent computing ordering scheme for eliminating the (unnecessary) pro- cessing of invisible points. On the hardware level, our proposed RT-NeRF accelerator (1) adopts a hybrid encoding scheme to adap- tively switch between a bitmap- or coordinate-based sparsity encoding format for NeRF’s sparse embeddings, aiming to maximize the storage savings and thus reduce the required DRAM accesses while supporting efficient NeRF decoding; and (2) integrates both a high-density sparse search unit and a dual-purpose bi-direction adder & search tree to coordinate the two aforementioned encod- ing formats. Extensive experiments on eight datasets consistently validate the effectiveness of RT-NeRF, achieving a large throughput improvement (e.g., 9.7×∼3,201×) while maintaining the rendering quality as compared with SOTA efficient NeRF solutions. 

For strong detonation waves of the inviscid Majda model, spectral stability was established by Jung and Yao for waves with step-type ignition functions, by a proof based largely on explicit knowledge of wave profiles. In the present work, we extend their stability results to strong detonation waves with more general ignition functions where explicit profiles are unknown. Our proof is based on reduction to a generalized Sturm-Liouville problem, similar to that used by Sukhtayev, Yang, and Zumbrun to study spectral stability of hydraulic shock profiles of the Saint-Venant equations.