一类Hermite型平流方程离散化的稳定性壁垒
The stability barrier of a class of Hermite-type discretizations of advection equations
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- Lehigh University（利哈伊大学）
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中文总结 AI 辅助
本文对线性平流方程的混合变量离散化方法进行了完整分类，证明了其稳定性壁垒与经典有限差分格式相似，并验证了中心格式的中性稳定性。
中文摘要 AI 辅助
本文根据稳定性性质对线性平流方程的所有混合变量（HV）离散化进行了完整分类，并确立了这些方法的稳定性壁垒。在HV离散化框架中，我们同时寻找单元平均解和节点解的数值近似，并随时间同步演化它们。我们证明了HV方法具有与经典有限差分格式相似的稳定性壁垒，即半离散HV格式是稳定的当且仅当其使用的模板满足：迎风方向未知数数量恰好等于背风方向未知数数量加一或加二，唯一例外是当两个数量分别为三和零时。此外，还证明了所有中心HV格式都是中性稳定的，即解的L2范数在所有时刻保持一致有界。理论预测通过大量数值测试得到验证。
英文摘要
In this paper we fully categorize all hybrid-variable (HV) discretizations of linear advection equations according to their stability property and establish the stability barrier of these methods. In the HV discretization framework, we find numerical approximations to both cell-averaged solutions and nodal solutions and evolve them in time simultaneously. We prove that the HV methods have a similar stability barrier as the classical finite difference schemes, that the semi-discretized HV scheme is stable if and only if it uses a stencil such that the number of unknowns in the upwind direction is precisely that in the downwind direction plus one or two, with only one exception when the two numbers are three and zero, respectively. It is also proved that all central HV schemes are neutrally stable, in the sense that the L2-norm of the solution remain uniformly bounded at all times. The theoretical predictions are verify by extensive numerical tests.
