Finite volume WENO schemes for nonlinear parabolic problems with degenerate diffusion on non-uniform meshes
Finite volume WENO schemes for nonlinear parabolic problems with degenerate diffusion on non-uniform meshes
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DOI：
10.1016/j.jcp.2019.108921
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发表时间：
2019-12
期刊：
影响因子：
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通讯作者：
T. Arbogast;Chieh-Sen Huang;X. Zhao
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文献类型：
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作者：
T. Arbogast;Chieh-Sen Huang;X. Zhao
我们考虑简并平流扩散方程的数值近似，该方程在形式上是抛物线的，但可能表现出双曲行为。我们在非均匀计算网格上的多个空间维度上开发了显式和隐式有限体积加权本质上非振荡（WENO）方案。通过使用基尔霍夫变换重新表述扩散简并性。使用具有自适应顺序的 WENO 重建（WENO-AO）对空间进行离散化，它具有多个优点，包括避免负线性权重以及处理不规则计算网格的能力。开发了一种特殊的两阶段 WENO 重建程序来处理简并扩散。首先重建解的元素平均值以给出解的点值，并且这些点值依次用于重建扩散通量的基尔霍夫变换变量。使用直线法和龙格-库塔时间积分器对时间进行离散化。我们对显式方案使用强稳定性保持（SSP）龙格-库塔方法，该方法具有严格的抛物线缩放时间步长限制以保持稳定性。我们还开发了隐式龙格-库塔方法。 SSP 方法仅在条件下稳定，因此我们讨论 L 稳定 Runge-Kutta 方法的使用。我们提出了使用间隔或四边形的非均匀网格在一维和二维空间中的空间和时间三阶详细方案。描述了逻辑上矩形的计算网格的有效实现。通过冯诺依曼（或傅立叶模式）稳定性分析，我们表明，当使用隐式 Radau IIA Runge-Kutta 方法时，线性问题的平滑解在均匀计算网格上是无条件 L 稳定的。计算结果表明该方案能够准确地近似具有挑战性的测试问题。
We consider numerical approximation of the degenerate advection-diffusion equation, which is formally parabolic but may exhibit hyperbolic behavior. We develop both explicit and implicit finite volume weighted essentially non-oscillatory (WENO) schemes in multiple space dimensions on non-uniform computational meshes. The diffusion degeneracy is reformulated through the use of the Kirchhoff transformation. Space is discretized using WENO reconstructions with adaptive order (WENO-AO), which have several advantages, including the avoidance of negative linear weights and the ability to handle irregular computational meshes. A special two-stage WENO reconstruction procedure is developed to handle degenerate diffusion. Element averages of the solution are first reconstructed to give point values of the solution, and these point values are in turn used to reconstruct the Kirchhoff transform variable of the diffusive flux. Time is discretized using the method of lines and a Runge-Kutta time integrator. We use Strong Stability Preserving (SSP) Runge-Kutta methods for the explicit schemes, which have a severe parabolically scaled time step restriction to maintain stability. We also develop implicit Runge-Kutta methods. SSP methods are only conditionally stable, so we discuss the use of L-stable Runge-Kutta methods. We present in detail schemes that are third order in both space and time in one and two space dimensions using non-uniform meshes of intervals or quadrilaterals. Efficient implementation is described for computational meshes that are logically rectangular. Through a von Neumann (or Fourier mode) stability analysis, we show that smooth solutions to the linear problem are unconditionally L-stable on uniform computational meshes when using an implicit Radau IIA Runge-Kutta method. Computational results show the ability of the schemes to accurately approximate challenging test problems.