On the implementation of WENO schemes for a class of polydisperse sedimentation models
On the implementation of WENO schemes for a class of polydisperse sedimentation models
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DOI：
10.1016/j.jcp.2010.12.019
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发表时间：
2011-03
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影响因子：
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通讯作者：
R. Bürger;R. Donat;P. Mulet;Carlos A. Vega
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文献类型：
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作者：
R. Bürger;R. Donat;P. Mulet;Carlos A. Vega
相同密度但属于有限种类（尺寸类别）的小刚性球体的多分散悬浮液的沉降可以用一阶非线性强耦合守恒定律的空间一维系统来描述。未知数是每个物种的体积分数（浓度）作为深度和时间的函数。典型的解决方案，例如，在一个柱中批量沉淀，包括不连续（运动冲击）分离不同成分的区域。这些解的精确数值逼近是一个挑战，因为通量雅可比矩阵的闭型特征值和特征向量通常不可用，而且特征场既不是真正的非线性也不是线性退化的。然而，与Masliyah， Lockett和Bassoon （MLB模型）以及Höfler和Schwarzer （HS模型）等广泛使用的模型相关的通量向量产生的雅可比矩阵是对角矩阵的低秩扰动。这个性质允许应用一种方便的双曲性判据，这种判据被称为“长期方程”[J]。关于对角矩阵摄动特征值的长期方程，林。Alg。应用学报，246(1996)49-70]。这一标准最近得到了应用[R]。[8]李建军，李建军，李建军，李建军，李建军。基于时间方程的多分散沉积模型的双曲线分析。Math. 70(2010) 2186-2213]证明MLB和HS模型在易于验证的条件下是严格双曲的，它们的特征值与形成通量向量的物种的速度相交（因此速度是根查找器的良好起始值），并且可以用可接受的努力计算相应的特征向量。在目前的工作中，新获得的特征信息被用于实现MLB和HS模型的特征（谱）加权基本非振荡（WENO）方案。数值算例表明，使用该光谱信息的WENO方案在分辨率上优于基于分量的WENO方案，甚至在相同的总分辨率下效率也优于基于分量的WENO方案。
The sedimentation of a polydisperse suspension of small rigid spheres of the same density, but which belong to a finite number of species (size classes), can be described by a spatially one-dimensional system of first-order, nonlinear, strongly coupled conservation laws. The unknowns are the volume fractions (concentrations) of each species as functions of depth and time. Typical solutions, e.g. for batch settling in a column, include discontinuities (kinematic shocks) separating areas of different composition. The accurate numerical approximation of these solutions is a challenge since closed-form eigenvalues and eigenvectors of the flux Jacobian are usually not available, and the characteristic fields are neither genuinely nonlinear nor linearly degenerate. However, the flux vectors associated with the widely used models by Masliyah, Lockett and Bassoon (MLB model) and Höfler and Schwarzer (HS model) give rise to Jacobians that are low-rank perturbations of a diagonal matrix. This property allows to apply a convenient hyperbolicity criterion that has become known as the “secular equation” [J. Anderson, A secular equation for the eigenvalues of a diagonal matrix perturbation, Lin. Alg. Appl. 246 (1996) 49–70]. This criterion was recently applied [R. Bürger, R. Donat, P. Mulet, C.A. Vega, Hyperbolicity analysis of polydisperse sedimentation models via a secular equation for the flux Jacobian, SIAM J. Appl. Math. 70 (2010) 2186–2213] to prove that the MLB and HS models are strictly hyperbolic under easily verifiable conditions, that their eigenvalues interlace with the velocities of the species that form the flux vector (so the velocities are good starting values for a root finder), and that the corresponding eigenvectors can be calculated with acceptable effort. In the present work, the newly available characteristic information is exploited for the implementation of characteristic-wise (spectral) weighted essentially non-oscillatory (WENO) schemes for the MLB and HS models. Numerical examples illustrate that WENO schemes which use this spectral information are superior in resolution, and even in efficiency for the same overall resolution, to component-wise WENO schemes.