The Direct Discontinuous Galerkin (DDG) Methods for Diffusion Problems
The Direct Discontinuous Galerkin (DDG) Methods for Diffusion Problems
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DOI：
10.1137/080720255
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发表时间：
2008-10
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影响因子：
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通讯作者：
Hailiang Liu;Jue Yan
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文献类型：
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作者：
Hailiang Liu;Jue Yan
提出了一种求解扩散问题的新的间断伽辽金有限元方法。与传统的局部间断Galerkin方法不同，直接间断Galerkin（DDG）方法是基于抛物方程解的直接弱形式，并仅通过数值通量$\widehat{u_x}$进行计算.我们提出了一个通用的数值通量公式的解决方案的衍生物，这是一致的和保守的，然后我们引入了一个概念的容许识别一类数值通量，使一维（1D）和多维问题的非线性稳定性得到保证。此外，当应用DDG格式与容许的数值通量的一维线性情况下，$k $阶精度的能量范数时，使用$k$次多项式证明。DDG方法具有更容易制定和实施的优势和有效的计算的解决方案。一系列数值算例证明了该方法的高精度。特别地，我们研究了不同允许数值通量下格式的数值性能。
A new discontinuous Galerkin finite element method for solving diffusion problems is introduced. Unlike the traditional local discontinuous Galerkin method, the scheme called the direct discontinuous Galerkin (DDG) method is based on the direct weak formulation for solutions of parabolic equations in each computational cell and lets cells communicate via the numerical flux $\widehat{u_x}$ only. We propose a general numerical flux formula for the solution derivative, which is consistent and conservative; and we then introduce a concept of admissibility to identify a class of numerical fluxes so that the nonlinear stability for both one-dimensional (1D) and multidimensional problems are ensured. Furthermore, when applying the DDG scheme with admissible numerical flux to the 1D linear case, $k$th order accuracy in an energy norm is proven when using $k$th degree polynomials. The DDG method has the advantage of easier formulation and implementation and efficient computation of the solution. A series of numerical examples are presented to demonstrate the high order accuracy of the method. In particular, we study the numerical performance of the scheme with different admissible numerical fluxes.