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A first-order system approach for diffusion equation. II: Unification of advection and diffusion

机译：扩散方程的一阶系统方法。二：对流和扩散的统一

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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.
机译：在本文中，我们通过扩展为扩散方程引入的一阶系统方法，将对流和扩散统一为一个双曲系统[J.计算对流扩散方程[Phys。，227（2007）315-352]。具体来说，我们通过将扩散项表示为一阶双曲系统并简单地将对流项添加到其中来构造一个统一的双曲对流扩散系统。然后，我们自然会为整个系统开发迎风方案;因此，不需要开发两种不同的方案，即对流和扩散方案。我们表明，以这种方式构造的数值方案可以自动统一准确，允许O（h）时间步长，并且可以同时以与计算精度相同的顺序来计算溶液梯度（Navier-Stokes方程的粘应力/热通量）。主变量，用于所有雷诺数。我们提供一维非均匀网格和二维不规则三角形网格上边界层类型问题的数值结果，以证明该方法的各种显着优势。尤其是，我们证明，解决一阶对流扩散系统的方案比传统的标量方案能显着提高CPU时间，尽管需要额外的成本来携带额外的变量并为其求解方程。最后，本文讨论了未来的发展。

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