Xu Zhiliang, Chen Xu-Yan, Liu Yingjie
Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN 46556.
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332.
J Comput Phys. 2014 Dec 1;278:348-377. doi: 10.1016/j.jcp.2014.08.042.
We present a new formulation of the Runge-Kutta discontinuous Galerkin (RKDG) method [9, 8, 7, 6] for solving conservation Laws with increased CFL numbers. The new formulation requires the computed RKDG solution in a cell to satisfy additional conservation constraint in adjacent cells and does not increase the complexity or change the compactness of the RKDG method. Numerical computations for solving one-dimensional and two-dimensional scalar and systems of nonlinear hyperbolic conservation laws are performed with approximate solutions represented by piecewise quadratic and cubic polynomials, respectively. The hierarchical reconstruction [17, 33] is applied as a limiter to eliminate spurious oscillations in discontinuous solutions. From both numerical experiments and the analytic estimate of the CFL number of the newly formulated method, we find that: 1) this new formulation improves the CFL number over the original RKDG formulation by at least three times or more and thus reduces the overall computational cost; and 2) the new formulation essentially does not compromise the resolution of the numerical solutions of shock wave problems compared with ones computed by the RKDG method.
我们提出了一种新的龙格 - 库塔间断伽辽金（RKDG）方法[9, 8, 7, 6]的公式，用于求解具有更高库朗数（CFL数）的守恒律。新公式要求在一个单元中计算得到的RKDG解满足相邻单元中的附加守恒约束，并且不会增加RKDG方法的复杂度或改变其紧凑性。分别使用分段二次多项式和三次多项式表示的近似解，对求解一维和二维标量以及非线性双曲守恒律方程组进行了数值计算。应用分层重构[17, 33]作为限制器，以消除间断解中的虚假振荡。从数值实验以及新公式化方法的CFL数的解析估计中，我们发现：1）这种新公式比原始的RKDG公式将CFL数提高了至少三倍或更多，从而降低了总体计算成本；2）与通过RKDG方法计算的数值解相比，新公式在本质上不会损害激波问题数值解的分辨率。
