双曲系统的高阶波传播算法
High-order Wave Propagation Algorithms for Hyperbolic Systems
AI总结：
提出一种基于黎曼问题求解和波动（而非通量）计算的有限体积法，通过WENO重构和SSP-RK时间积分实现高阶精度，并利用f波黎曼求解器和波斜率WENO重构实现平衡律的近稳态平衡。
摘要：
我们提出了一种适用于双曲型偏微分方程的有限体积方法，包括空间变化和半线性非保守系统。空间离散化与著名的Clawpack软件类似，基于求解黎曼问题和计算波动（而非通量）。实现中采用了空间上的加权本质无振荡重构和时间上的强稳定性保持龙格-库塔积分。该方法可以扩展到任意高阶精度，并允许实现平衡律在稳态附近的平衡捕捉。这种平衡通过$f$-波黎曼求解器和一种新颖的波斜率WENO重构过程实现。通过数值示例展示了该方法的广泛适用性和优越性能，包括非保守形式的问题、空间变化通量的问题以及涉及平衡律近平衡解的问题。
英文摘要：
We present a finite volume method that is applicable to hyperbolic PDEs including spatially varying and semilinear nonconservative systems. The spatial discretization, like that of the well-known Clawpack software, is based on solving Riemann problems and calculating fluctuations (not fluxes). The implementation employs weighted essentially non-oscillatory reconstruction in space and strong stability preserving Runge-Kutta integration in time. The method can be extended to arbitrarily high order of accuracy and allows a well-balanced implementation for capturing solutions of balance laws near steady state. This well-balancing is achieved through the $f$-wave Riemann solver and a novel wave-slope WENO reconstruction procedure. The wide applicability and advantageous properties of the method are demonstrated through numerical examples, including problems in nonconservative form, problems with spatially varying fluxes, and problems involving near-equilibrium solutions of balance laws.
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