Beyond Interface Gradient: A General Principle for Constructing Diffusion Schemes
Beyond Interface Gradient: A General Principle for Constructing Diffusion Schemes
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DOI：
10.2514/6.2010-5093
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发表时间：
2010-06
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通讯作者：
H. Nishikawa
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作者：
H. Nishikawa
本文介绍了构造时间精确扩散格式的一般原理，它适用于各种离散方法，包括有限体积法、残差分布法、间断Galerkin法和谱体积法。该原则是基于双曲型弛豫系统模型的扩散。这是离散的双曲型系统的平流计划，然后推导出扩散计划，使其平衡极限在一个相当简单的方式。所提出的原则的一个显着特点是，它会自动引入一个阻尼项到派生的扩散计划，这是必不可少的有效的高频误差阻尼，在某些情况下，一致性也。另一个有用的功能是，派生的扩散方案具有相同的实现结构作为相应的平流方案，这使得它非常简单，将其与对流计划的对流扩散问题。我们通过在一维均匀网格和二维非结构网格上构造扩散格式，为节点/单元格中心nite体积法、剩余分布法、间断Galerkin法和谱体积法证明了一般原理。数值结果验证了扩散格式的准确性，并说明了阻尼项的重要性。它还表明，派生的扩散计划产生更高或更准确的解决方案比广泛使用的各向同性/各向异性不规则三角形网格上的时间依赖性扩散问题的计划。
In this paper, we introduce a general principle for constructing time-accurate diffusion schemes, which is applicable to various discretization methods, includingnite-volume, residual- distribution, discontinuous-Galerkin, and spectral-volume methods. The principle is based on a hyperbolic relaxation-system model for diffusion. It is to discretize the hyperbolic system by an advection scheme, and then derive a diffusion scheme by bringing it to the equilibrium limit in a rather simple manner. A distinguished feature of the proposed principle is that it automatically introduces a damping term into the derived diffusion scheme, which is essential for effective high-frequency error damping and, in some cases, for consistency also. Another useful feature is that the derived diffusion scheme has the same implementation structure as a corresponding advection scheme, which makes it remarkably simple to integrate it with the advection scheme for advection-diffusion problems. We demonstrate the general principle by constructing diffusion schemes on uniform grids in one dimension and unstructured grids in two dimensions, for node/cell-centerednite-volume, residual-distribution, discontinuous- Galerkin, and spectral-volume methods. Numerical results are presented to verify the accuracy of the diffusion schemes and to illustrate the importance of the damping term. It is also shown that derived diffusion schemes yield comparably or more accurate solutions than widely-used schemes for time-dependent diffusion problems on isotropic/anisotropic irregular triangular grids.