We propose a generalized Eulerian-Lagrangian (GEL) discontinuous Galerkin (DG) method. The method is a generalization of the Eulerian-Lagrangian (EL) DG method for transport problems proposed in [arXiv preprint arXiv: 2002.02930 (2020)], which tracks solution along approximations to characteristics in the DG framework, allowing extra large time stepping size with stability. The newly proposed GEL DG method in this paper is motivated for solving linear hyperbolic systems with variable coefficients, where the velocity field for adjoint problems of the test functions is frozen to constant. In this paper, in a simplified scalar setting, we propose the GEL DG methodology by freezing the velocity field of adjoint problems, and by formulating the semi-discrete scheme over the space-time region partitioned by linear lines approximating characteristics. The fully-discrete schemes are obtained by method-of-lines Runge-Kutta methods. We further design flux limiters for the schemes to satisfy the discrete geometric conservation law (DGCL) and maximum principle preserving (MPP) properties. Numerical results on 1D and 2D linear transport problems are presented to demonstrate great properties of the GEL DG method. These include the high order spatial and temporal accuracy, stability with extra large time stepping size, and satisfaction of DGCL and MPP properties.

翻译：我们建议采用通用的Eulerian-Lagrangian(GEL)不连续加列尔金(GG)方法。该方法是对Eulerian-Lagrangian(EL)DG(EL)在[arXiv print arXiv: 2002/02930(202020)]中提议的运输问题方法的概括化方法,该方法跟踪解决方案与DG框架中的特征近似,允许有稳定性的超大时间跨步尺寸。本文件中新提出的GEL DG方法用于用可变系数解决线性双曲线系统,将测试功能的配套问题速度域冻结在固定状态上。在本文件中,在简化的标度设置中,我们提议GEL DGD(GL) 方法,通过冻结连接连接问题的速度域域域,并针对时空区域配有线线性平线的半分流系统,通过Rung-Kutta方法获得完全分解的系统。我们进一步设计了用于各种机制的通量限制,以便满足离式的地理-D节时段的准确性保护特性和最大平坦度的MD(D) 原则包括这些高分辨率的MD 和最大递制。