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arXiv 2209.13656math.NAcs.NA
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非线性扩散方程的直接间断Galerkin方法的推广框架

A generalized framework for direct discontinuous Galerkin methods for nonlinear diffusion equations

Mustafa Engin Danis, Jue Yan

摘要:

在这项研究中,我们提出了一个统一的,一般的框架直接间断伽辽金方法。在新的框架中,不需要非线性扩散矩阵的反导数。这使得一个简单的定义的数值通量,它可以用于一般的扩散方程,没有进一步的修改。我们还提出了新的直接间断Galerkin方法的非线性稳定性分析,并进行了几个数值实验,以评估其性能。数值试验表明,该方法的对称修正和界面修正形式均能达到最优收敛,且优于非对称修正形式,后者仅对对角扩散矩阵问题具有最优收敛性,而对非对角扩散矩阵的偶数次多项式则失序.奇异或爆破的解决方案也很好地捕捉新的直接间断伽辽金方法。

英文摘要:

In this study, we propose a unified, general framework for the direct discontinuous Galerkin methods. In the new framework, the antiderivative of the nonlinear diffusion matrix is not needed. This allows a simple definition of the numerical flux, which can be used for general diffusion equations with no further modification. We also present the nonlinear stability analyses of the new direct discontinuous Galerkin methods and perform several numerical experiments to evaluate their performance. The numerical tests show that the symmetric and the interface correction versions of the method achieve optimal convergence and are superior to the nonsymmetric version, which demonstrates optimal convergence only for problems with diagonal diffusion matrices but loses order for even degree polynomials with a non-diagonal diffusion matrix. Singular or blow-up solutions are also well captured with the new direct discontinuous Galerkin methods.

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