Third order maximum-principle-satisfying DG schemes for convection-diffusion problems with anisotropic diffusivity
Third order maximum-principle-satisfying DG schemes for convection-diffusion problems with anisotropic diffusivity
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DOI：
10.1016/j.jcp.2019.04.028
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发表时间：
2019-07
期刊：
影响因子：
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通讯作者：
Hui-Feng Yu;Hailiang Liu
中科院分区：
文献类型：
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作者：
Hui-Feng Yu;Hailiang Liu
对于一类具有变扩散系数的对流扩散方程，在一维和二维矩形网格上构造了三阶精度间断Galerkin（DG）格式。采用显式时间推进的DG方法可以很好地应用于非线性对流扩散方程。结果表明，在适当的时间步长限制下，在[Liu和Yu，SIAM J. Sci. Comput. 36（5）：A2296{A2325，2014]当与本DG方案结合时，保留了由初始数据指示的解边界，即，最大值原理，同时保持统一的三阶精度。这些格式可以推广到三维矩形网格。对于所有的模型方案的关键是，可以确定一个有效的测试集，以验证所需的数值解的界限。这主要是通过利用扩散通量的灵活形式和加权单元平均值的自适应分解来实现的。数值结果验证了数值方法的有效性。
For a class of convection-diffusion equations with variable diffusivity, we construct third order accurate discontinuous Galerkin (DG) schemes on both one and two dimensional rectangular meshes. The DG method with an explicit time stepping can well be applied to nonlinear convection-diffusion equations. It is shown that under suitable time step restrictions, the scaling limiter proposed in [Liu and Yu, SIAM J. Sci. Comput. 36(5): A2296{A2325, 2014] when coupled with the present DG schemes preserves the solution bounds indicated by the initial data, i.e., the maximum principle, while maintaining uniform third order accuracy. These schemes can be extended to rectangular meshes in three dimension. The crucial for all model scenarios is that an effective test set can be identified to verify the desired bounds of numerical solutions. This is achieved mainly by taking advantage of the flexible form of the diffusive flux and the adaptable decomposition of weighted cell averages. Numerical results are presented to validate the numerical methods and demonstrate their effectiveness.