Superconvergence of Discontinuous Galerkin Method for Scalar Nonlinear Hyperbolic Equations
Superconvergence of Discontinuous Galerkin Method for Scalar Nonlinear Hyperbolic Equations
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DOI：
10.1137/17m1128605
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发表时间：
2018-03
期刊：
影响因子：
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通讯作者：
Waixiang Cao;Chi-Wang Shu;Yang Yang-Yang;Zhimin Zhang
中科院分区：
文献类型：
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作者：
Waixiang Cao;Chi-Wang Shu;Yang Yang-Yang;Zhimin Zhang
研究了一维非线性双曲型方程半离散间断Galerkin（DG）方法的超收敛性。建立了固定风向和交变风向问题的超收敛结果。一方面，我们证明，如果风向是固定的（即，通量函数的导数有界远离零），当使用逆风通量和k次分段多项式时，单元平均误差和单元界面处的数值通量误差都以2k +1的速率收敛。此外，我们还证明了DG解的函数值逼近在右内Radau点处是超收敛的，导数值逼近在左内Radau点处是超收敛的，其阶分别为k+2和k+1.作为一个副产品，我们显示了（k+2）阶超收敛的DG解决方案的高斯-Radau投影的精确解。另一方面，超收敛结果...
In this paper, we study the superconvergence behavior of the semi-discrete discontinuous Galerkin (DG) method for scalar nonlinear hyperbolic equations in one spatial dimension. Superconvergence results for problems with fixed and alternating wind directions are established. On the one hand, we prove that, if the wind direction is fixed (i.e., the derivative of the flux function is bounded away from zero), both the cell average error and numerical flux error at cell interfaces converge at a rate of $2k+1$ when upwind fluxes and piecewise polynomials of degree $k$ are used. Moreover, we also prove that the function value approximation of the DG solution is superconvergent at interior right Radau points, and the derivative value approximation is superconvergent at interior left Radau points, with an order of k+2 and k+1, respectively. As a byproduct, we show a (k+2)th order superconvergence of the DG solution towards the Gauss--Radau projection of the exact solution. On the other hand, superconvergence resu...