A Fourth-Order Accurate Finite-Volume Method with Structured Adaptive Mesh Refinement for Solving the Advection-Diffusion Equation
A Fourth-Order Accurate Finite-Volume Method with Structured Adaptive Mesh Refinement for Solving the Advection-Diffusion Equation
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DOI：
10.1137/110820105
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发表时间：
2012-04
期刊：
影响因子：
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通讯作者：
Qinghai Zhang;H. Johansen;P. Colella
中科院分区：
文献类型：
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作者：
Qinghai Zhang;H. Johansen;P. Colella
我们提出了一种四阶精确算法，用于在块结构、自适应细化网格的层次结构上求解泊松方程、热方程和平流扩散方程。对于空间离散化，在结构化自适应网格细化和各种边界条件的背景下，为散度算子和拉普拉斯算子导出了有限体积模板；使用多重网格算法求解所得线性系统。对于时间积分，我们将椭圆求解器与由 Kennedy 和 Carpenter 引入的四阶精确 Runge-Kutta 方法结合起来 [Appl.数字。 Math., 44 (2003), pp. 139-181]，这使我们能够显式地处理非刚性平流项并隐式地处理刚性扩散项。我们通过将结果与解析解进行比较来证明空间和时间的准确性。由于该方法的通用表述，该算法很容易扩展到更复杂的物理系统。
We present a fourth-order accurate algorithm for solving Poisson's equation, the heat equation, and the advection-diffusion equation on a hierarchy of block-structured, adaptively refined grids. For spatial discretization, finite-volume stencils are derived for the divergence operator and Laplacian operator in the context of structured adaptive mesh refinement and a variety of boundary conditions; the resulting linear system is solved with a multigrid algorithm. For time integration, we couple the elliptic solver to a fourth-order accurate Runge-Kutta method, introduced by Kennedy and Carpenter [Appl. Numer. Math., 44 (2003), pp. 139-181], which enables us to treat the nonstiff advection term explicitly and the stiff diffusion term implicitly. We demonstrate the spatial and temporal accuracy by comparing results with analytical solutions. Because of the general formulation of the approach, the algorithm is easily extensible to more complex physical systems.