In the context of preserving stationary states, e.g. lake at rest and moving equilibria, a new formulation of the shallow water system, called Flux Globalization has been introduced by Cheng et al. (2019). This approach consists in including the integral of the source term in the global flux and reconstructing the new global flux rather than the conservative variables. The resulting scheme is able to preserve a large family of smooth and discontinuous steady state moving equilibria. In this work, we focus on an arbitrary high order WENO Finite Volume (FV) generalization of the global flux approach. The most delicate aspect of the algorithm is the appropriate definition of the source flux (integral of the source term) and the quadrature strategy used to match it with the WENO reconstruction of the hyperbolic flux. When this construction is correctly done, one can show that the resulting WENO FV scheme admits exact discrete steady states characterized by constant global fluxes. We also show that, by an appropriate quadrature strategy for the source, we can embed exactly some particular steady states, e.g. the lake at rest for the shallow water equations. It can be shown that an exact approximation of global fluxes leads to a scheme with better convergence properties and improved solutions. The novel method has been tested and validated on classical cases: subcritical, supercritical and transcritical flows.
翻译:在保护静止状态的背景下,例如,在休息时的湖泊和移动平衡的背景下,成正等人(2019年)提出了浅水系统的新提法,称为“通量全球化”(2019年),这一方法包括将源词的内在组成部分纳入全球通量,并重建新的全球通量,而不是保守变量。由此形成的计划能够保持一个由平滑和不连续稳定状态组成的庞大大家庭,移动平衡。在这项工作中,我们侧重于全球通量方法的任意高顺序WENO Finite 卷(FV)的概括。算法的最微妙的方面是源通量的适当定义(源词的整体性)和用于使其与WENO对超曲流的重建相匹配的二次曲线战略。当这一构建正确时,人们可以表明,由此形成的WENO FV计划承认了以持续全球通量为特征的离散稳定状态。我们还表明,通过对源的适当的二次曲线战略,我们可以插入某些特定的稳定状态,例如,对于浅水流(源术语的整体)的休息状态,以及用来使其与WENO的二次通度相适应,可以显示一个更精确的临界的转变的路径。它可以显示,一个精确的精确的精确的路径,一个对准的精确的走向,一个精确的走向。