Navier-Stokes方程间断Galerkin有限元方法研究

STUDY ON DISCONTINUOUS GALERKIN METHOD FOR NAVIER-STOKES EQUATIONS

通过引入全局提升算子和局部提升算子,发展了求解Navier-Stokes方程的间断Galerkin(discontinuousGalerkin,DG)有限元方法的一般框架,并在此框架下给出了几种典型黏性离散格式的具体表达形式.对局部提升算子的求解给出了详细的计算步骤.同时还给出了一种简单有效的计算方法来对物面边界进行高阶近似.为了能够对NS方程进行精度测试,采用对原始系统添加源项的方法构造精确解.二维Euler和NS系统的精度测试表明该方法达到了DG方法的理论精度.二维圆柱无黏绕流的计算结果表明关于物面边界的高阶近似方法能够保持DG方法原有的精度.卡门涡街数值模拟则进一步验证了该方法的正确性并且显示出DG方法较高的计算精度和分辨率.

A unified framework on the discontinuous Galerkin method（DG） for Navier-Stokes equations is developed through the introduction of global lifting operator and local lifting operator.Several typical DG schemes for viscous terms are presented within the framework.Detailed steps of computing local lifting operator are given.In addition,a simple yet efficient method for the high order approximation of the body surface is proposed.In order to carry out accuracy tests for NS equation,exact solutions have been obtained by adding source terms to the original system.Accuracy tests for the two dimensional Euler and NS systems indicate that the method in this paper has achieved theoretical accuracy order.Inviscid flow computation over cylinder shows that the high order boundary approximation method proposed in this paper is able to preserve the high order accuracy of the DG method.Karmen vortex numerical simulation has further validated the method in this paper,and demonstrated the promising properties of the DG method in terms of accuracy and resolution.