一种基于间断Galerkin格式的新型限制器设计

An original designed limiter for discontinuous Galerkin method

使用高阶间断Galerkin格式求解守恒律方程组时,激波附近的Gibbs效应容易导致非物理解的产生。为抑制这一现象,必须构造合理的限制器对数值解进行处理。目前间断Galerkin格式中的限制器多源于有限体积法,在非结构网格上只对低阶导数项进行限制,对高阶导数项则很难给出普适判据。文章对间断Galerkin解进行广义Fourier展开,实现不同频域范围内的谱分解;在新的模板坐标系下描述各阶方向导数的变化规律;结合当前单元和相邻单元的信息,分层限制各阶方向导数,实现对非物理解的抑制。通过求解Euler方程,对二维Riemann问题、翼型周围的亚、跨声速流动问题、前台阶问题以及超燃冲压发动机内流场激波反射问题进行数值模拟,检验了新型限制器的可靠性以及向高阶格式推广的可行性。

When the high-order discretization scheme is applied for solving the hyperbolic conservation laws,the Gibbs oscillations emerging around the discontinuities may cause the fields to take unphysical val-ues,which could result in the undesired interruption of calculation process.This problem is especially severe for the discontinuous Galerkin scheme.Therefore,suitable limiting calculators are supposed to be designed in order to damp the appearance of unphysical oscillations.However,the existed limiters for the discontinu-ous Galerkin scheme originate mostly from the finite volume methods,thus for the unstructured grids they can only limit the spatial solutions of the solutions,namely the lower-order derivatives,rather than offer an general criterion for the higher order derivatives.Due to this background,the discontinuous finite element solutions are firstly processed into general Fourier expansion series for the realization of spectral decomposi-tion in different frequency domains.Afterwards,a modal coordinate system is originally proposed,descri-bing the changing patterns of the directional derivatives in different orders in a new form.Finally,after the combination of the information of the current mode and its adjacent mode,and the limitation of the direction-al derivatives in different orders for different levels,the unphysical solutions are successfully damped.Based on this method,the Euler equations are solved for the numerical simulations of the 2-D Riemann problems, the flow field of the wind-tunnel with a front step,the reflection of shockwaves in the inner field of scramjet engines as well as the subsonic and transonic flow surrounding airfoils.The obtained results demonstrate that the new limiter is relatively reliable and it can be generalized to high-order schemes.