基于谱元法的三维欧拉-拉格朗日浅水模型

3D Eulerian-Lagrangian shallow water model using spectral element method

基于欧拉-拉格朗日法(Eulerian-Lagrangian method,ELM)计算对流项的三维水流数学模型,大时间步长计算仍保持稳定,但空间线性插值计算会产生附加数值阻力,限制了该类模型在河流工程中的应用。谱元法(spectral element method,SEM)是一种适合复杂区域计算的高阶数值方法。本文引入谱元法,将垂向线性插值改为多项式插值,构建了新的欧拉-拉格朗日法三维浅水流动数值模型。通过顺直明渠的算例分析表明,新三维模型结合了欧拉-拉格朗日法与谱元法的优点,时间步长能突破柯朗数的限制,同时消除了经典欧拉-拉格朗日法的数值阻力,计算精度有数量级的提高。

A 3 D hydrodynamic model that calculates advection terms with an Eulerian-Lagrangian method(ELM)can maintain numerical stability at large time steps,while its linear interpolation generates a significant amount of additional numerical resistance,restricting its application in river engineering.Spectral element method(SEM)is a high-order method suitable for irregular domain calculations.This paper uses SEM to replace linear interpolation with polynomial interpolation in vertical direction,and develops a new 3 D shallow water model by combining with ELM.Numerical experiment on open-channel flow shows that this 3 D model,taking advantage of the spectral element method and ELM,is capable of breaking through the Courant number constraint on its time steps,and eliminates most of the numerical resistance of a classical ELM,thus improving its accuracy by an order of magnitude.