In this work we study various continuous finite element discretization for two dimensional hyperbolic partial differential equations, varying the polynomial space (Lagrangian on equispaced, Lagrangian on quadrature points (Cubature) and Bernstein), the stabilization techniques (streamline-upwind Petrov-Galerkin, continuous interior penalty, orthogonal subscale stabilization) and the time discretization (Runge-Kutta (RK), strong stability preserving RK and deferred correction). This is an extension of the one dimensional study by Michel S. et al J. Sci. Comput. (2021), whose results do not hold in multi-dimensional frameworks. The study ranks these schemes based on efficiency (most of them are mass-matrix free), stability and dispersion error, providing the best CFL and stabilization coefficients. The challenges in two-dimensions are related to the Fourier analysis. Here, we perform it on two types of periodic triangular meshes varying the angle of the advection, and we combine all the results for a general stability analysis. Furthermore, we introduce additional high order viscosity to stabilize the discontinuities, in order to show how to use these methods for tests of practical interest. All the theoretical results are thoroughly validated numerically both on linear and non-linear problems, and error-CPU time curves are provided. Our final conclusions suggest that Cubature elements combined with SSPRK and OSS stabilization is the most promising combination.

翻译：在这项工作中,我们研究了两种多元双曲部分差异方程式的各种连续限制元素分解,多种多元空间(Lagrangian on equispaced,Lagrangian on quaturation points(Cubature)和Bernstein)、稳定技术(Smillline-上风Petrov-Galerkin-Galerkin、持续的内罚、正方位次比例稳定)和时间分解(Runge-Kutta(RK)(Runge-Kutta)(RK-Kutta),坚固稳定、推迟校正)。这是Michel S. et al. J. Sci. Comput. (2021) 的一维度研究的延伸,其结果并不在多维度框架中存在。研究将这些方法归类于效率(大多数是无质量矩阵)、稳定性和分散误差,稳定和分散误差,两种方法的挑战与四流分析有关。在这里,我们用两种定期的三边线迷测模型进行,我们把所有结果合并起来进行。此外,我们引入了最高级的理论测试,以稳定和直线性检验。