We propose a new high order accurate nodal discontinuous Galerkin (DG) method for the solution of nonlinear hyperbolic systems of partial differential equations (PDE) on unstructured polygonal Voronoi meshes. Rather than using classical polynomials of degree N inside each element, in our new approach the discrete solution is represented by piecewise continuous polynomials of degree N within each Voronoi element, using a continuous finite element basis defined on a subgrid inside each polygon. We call the resulting subgrid basis an agglomerated finite element (AFE) basis for the DG method on general polygons, since it is obtained by the agglomeration of the finite element basis functions associated with the subgrid triangles. The basis functions on each sub-triangle are defined, as usual, on a universal reference element, hence allowing to compute universal mass, flux and stiffness matrices for the subgrid triangles once and for all in a pre-processing stage for the reference element only. Consequently, the construction of an efficient quadrature-free algorithm is possible, despite the unstructured nature of the computational grid. High order of accuracy in time is achieved thanks to the ADER approach, making use of an element-local space-time Galerkin finite element predictor. The novel schemes are carefully validated against a set of typical benchmark problems for the compressible Euler and Navier-Stokes equations. The numerical results have been checked with reference solutions available in literature and also systematically compared, in terms of computational efficiency and accuracy, with those obtained by the corresponding modal DG version of the scheme.
翻译:我们提出一个新的高顺序、准确的交点不连续 Galerkin (DG) 方法,用于解决非结构化多边多边形Voronoi meshes的非线性双曲偏差方程(PDE) 。 与在每种元素中使用典型的N级多语种, 不同的解决方案在我们的新方法中, 在每个Voronoi元素中使用小盘连续的N级多语种, 使用每个多边形内的子网格定义的连续有限元素基础。 我们将由此产生的子网基在一般多边形中为DG法的超离线性偏差方方程(PDE) 基数限制要素(AFE) 基数, 因为它是通过与子网格三角关系相关的定基数函数的加亮度函数获得的。 每个子网格的基础函数通常以通用的参考元素来定义, 从而可以一次对亚格三角形三角形三角形三角形进行全局的连续的量、通和坚硬度矩阵基基基。 因此, 在普通多边多边方程式中构建一个高效的平方形的平方程式中, 比较的平方程式的平级平级平级平流的平流的平流计算中, 使这些平面平流的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局的平局法是可能的平局的平局的平局 。