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首页> 外文期刊>Journal of Computational Physics >A spectral Galerkin method for the coupled Orr-Sommerfeld and induction equations for free-surface MHD
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A spectral Galerkin method for the coupled Orr-Sommerfeld and induction equations for free-surface MHD

机译：用于自由表面MHD的Orr-Sommerfeld耦合方程和归纳方程的谱Galerkin方法

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We develop and test spectral Galerkin schemes to solve the coupled Orr-Sommerfeld and induction equations for parallel. incompressible MHD in free-surface and fixed-boundary, geometries. The schemes' discrete bases consist of Legendre internal shape functions, supplemented with nodal shape functions for the weak imposition of the stress and insulating boundary conditions. The orthogonality properties of the basis polynomials solve the matrix-coefficient growth problem, and eigenvalue-eigenfunction pairs can be computed stably at spectral orders at least as large as p = 3000 with p-independent roundoff error. Accuracy is limited instead by roundoff sensitivity due to non-normality of the stability operators at large hydrodynamic and/or magnetic Reynolds numbers (Re. Rm greater than or similar to 4 x 10(4)). In problems with Hartmann velocity and magnetic-field profiles we employ suitable Gauss quadrature rules to evaluate the associated exponentially weighted sesquilinear forms without error. An alternative approach. which involves approximating the forms by means of Legendre-Gauss-Lobatto quadrature at the 2p - 1 precision level, is found to yield equal eigenvalues within roundoff error. As a consistency check, we compare modal growth rates to energy growth rates in nonlinear simulations and record relative discrepancy smaller that, 10(5) for the least stable mode in free-surface flow at Re = 3 x 10(4). Moreover, we confirm that the computed normal modes satisfy an energy conservation law for free-surface MHD with error smaller than 10(6). The critical Reynolds number in free-surface MHD is found to be sensitive to the magnetic Prandtl number Pin, even at the Pm = O(10(5)) regime of liquid metals. (C) 2008 Elsevier Inc. All rights reserved.
机译：我们开发和测试频谱Galerkin方案，以解决并行的Orr-Sommerfeld耦合方程和感应方程。在自由表面和固定边界几何形状中不可压缩的MHD。该方案的离散基础包括Legendre内部形状函数，并补充了节点形状函数，以弱加应力和绝缘边界条件。基本多项式的正交性解决了矩阵系数增长的问题，并且特征值-特征函数对可以稳定地以至少与p无关的舍入误差在p = 3000的频谱阶上稳定地计算。由于在大的水动力和/或磁雷诺数（Re。Rm大于或类似于4 x 10（4））下稳定算子不正常，因此精度受舍入灵敏度的限制而取而代之。在有关Hartmann速度和磁场分布的问题中，我们采用合适的高斯正交规则来评估相关的指数加权倍半线性形式，而没有错误。另一种方法。通过在2p-1精度级别上通过Legendre-Gauss-Lobatto正交近似形式，发现在舍入误差内产生相等的特征值。作为一致性检查，我们在非线性模拟中将模态增长率与能量增长率进行了比较，并记录了相对误差，对于Re = 3 x 10（4）的自由表面流中最小稳定模式，其相对误差小于10（5）。此外，我们确认所计算的法线满足自由表面MHD的能量守恒定律，误差小于10（6）。发现即使在液态金属的Pm = O（10（5））态下，自由表面MHD中的临界雷诺数也对磁Prandtl数Pin敏感。 （C）2008 Elsevier Inc.保留所有权利。

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