论文标题
部分可观测时空混沌系统的无模型预测
A generalized framework for direct discontinuous Galerkin methods for nonlinear diffusion equations
论文作者
论文摘要
在这项研究中，我们提出了一个直接不连续的Galerkin方法的统一的一般框架。在新框架中，不需要非线性扩散矩阵的抗体。这允许对数值通量进行简单的定义，该定义可用于不进一步修改的一般扩散方程。我们还介绍了新的直接不连续的盖尔金方法的非线性稳定性分析，并执行几个数值实验以评估其性能。发性测试表明，该方法的对称和界面校正版本可实现最佳收敛性，并且优于非对称版本，该版本仅对对角度扩散矩阵的问题证明了最佳收敛性，但偶数偶数订单的均匀度均具有非diagongonal扩散矩阵。新的直接不连续的盖尔金方法也可以很好地捕获奇异或爆炸解决方案。
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.
