论文标题
具有地形的浅水方程的界限和熵稳定的代数通量校正方案
Bound-preserving and entropy-stable algebraic flux correction schemes for the shallow water equations with topography
论文作者
论文摘要
浅水方程（SWE）的精心设计的数值方法应确保均衡性，水高度的非负性和熵稳定性。对于没有源术语的非线性双曲系统的连续有限元离散化，可以使用代数通量校正（AFC）实施阳性保存和熵稳定性。在这项工作中，我们为SWE系统开发了均衡的AFC方案，包括地形源术语。我们的方法将湖泊保持静止平衡至机器精度。低阶版本代表了有限元设置的现有有限体积方法的概括。高阶扩展名配备了属性保护的通量限制器。在标准的CFL条件下，保证水高度的非负性。此外，经过通量校正的空间离散化满足了半混凝土熵的不等式。提出了新的算法，以对润湿和干燥过程进行现实模拟。提出了众所周知的基准测试的数值示例，以评估该方案的性能。
A well-designed numerical method for the shallow water equations (SWE) should ensure well-balancedness, nonnegativity of water heights, and entropy stability. For a continuous finite element discretization of a nonlinear hyperbolic system without source terms, positivity preservation and entropy stability can be enforced using the framework of algebraic flux correction (AFC). In this work, we develop a well-balanced AFC scheme for the SWE system including a topography source term. Our method preserves the lake at rest equilibrium up to machine precision. The low-order version represents a generalization of existing finite volume approaches to the finite element setting. The high-order extension is equipped with a property-preserving flux limiter. Nonnegativity of water heights is guaranteed under a standard CFL condition. Moreover, the flux-corrected space discretization satisfies a semi-discrete entropy inequality. New algorithms are proposed for realistic simulation of wetting and drying processes. Numerical examples for well-known benchmarks are presented to evaluate the performance of the scheme.
