A first-order system approach for diffusion equation. II: Unification of advection and diffusion
A first-order system approach for diffusion equation. II: Unification of advection and diffusion
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DOI：
10.1016/j.jcp.2009.10.040
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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
本文通过推广扩散方程的一阶系统方法，将平流和扩散统一为一个双曲系统[J]。第一版。理论物理。[j]对平流-扩散方程进行了研究。具体地说，我们将扩散项表示为一阶双曲系统，并将平流项简单地加入其中，构造了统一的双曲平流-扩散系统。自然地，我们为整个系统开发逆风方案；因此，没有必要制定两种不同的方案，即平流和扩散方案。我们表明，以这种方式构建的数值格式可以自动均匀精确，允许O(h)时间步长，并同时计算解梯度（Navier-Stokes方程的粘性应力/热通量），其精度与主变量相同，适用于所有雷诺数。我们给出了一维非均匀网格和二维不规则三角形网格上边界层型问题的数值结果，以证明该方法的各种显著优点。特别是，我们证明了解决一阶平流扩散系统的方案比传统的标量方案在CPU时间上有巨大的加速，尽管它需要携带额外的变量和求解它们的方程。我们以对未来进一步发展的讨论来结束本文。
In this paper, we unify advection and diffusion into a single hyperbolic system by extending the first-order system approach introduced for the diffusion equation [J. Comput. Phys., 227 (2007) 315–352] to the advection–diffusion equation. Specifically, we construct a unified hyperbolic advection–diffusion system by expressing the diffusion term as a first-order hyperbolic system and simply adding the advection term to it. Naturally then, we develop upwind schemes for this entire system; there is thus no need to develop two different schemes, i.e., advection and diffusion schemes. We show that numerical schemes constructed in this way can be automatically uniformly accurate, allow O(h) time step, and compute the solution gradients (viscous stresses/heat fluxes for the Navier–Stokes equations) simultaneously to the same order of accuracy as the main variable, for all Reynolds numbers. We present numerical results for boundary-layer type problems on non-uniform grids in one dimension and irregular triangular grids in two dimensions to demonstrate various remarkable advantages of the proposed approach. In particular, we show that the schemes solving the first-order advection–diffusion system give a tremendous speed-up in CPU time over traditional scalar schemes despite the additional cost of carrying extra variables and solving equations for them. We conclude the paper with discussions on further developments to come.