Mass conservation of the unified continuous and discontinuous element-based Galerkin methods on dynamically adaptive grids with application to atmospheric simulations

Abstract

We perform a comparison of mass conservation properties of the continuous (CG) and discontinuous (DG) Galerkin methods on non-conforming, dynamically adaptive meshes for two atmospheric test cases. The two methods are implemented in a unified way which allows for a direct comparison of the non-conforming edge treatment. We outline the implementation details of the non-conforming direct stiffness summation algorithm for the CG method and show that the mass conservation error is similar to the DG method. Both methods conserve to machine precision, regardless of the presence of the non-conforming edges. For lower order polynomials the CG method requires additional stabilization to run for very long simulation times. We addressed this issue by using filters and/or additional artificial viscosity. The mathematical proof of mass conservation for CG with non-conforming meshes is presented in Appendix B.Keywords: adaptive mesh refinement, continuous Galerkin method, discontinuous Galerkin method, non-conforming mesh, compressible Euler equations, atmospheric simulations IntroductionBoth element-based Galerkin methods and adaptive mesh refinement for atmospheric modeling are active fields of research. In [1] we provide an overview of the literature covering both topics, with particular attention to the discontinuous Galerkin method. In this paper we extend that work by comparing the continuous (CG) and discontinuous Galerkin (DG) methods with non-conforming adaptive mesh refinement. As a metric for the comparison we chose the mass conservation, as it is an important feature for many atmospheric applications. We believe this is a good metric because, for smooth solutions, both methods will yield similar accuracy; it is unclear how both methods will compare for non-smooth solutions but it is expected that they can achieve similarly good results, especially with the inclusion of stabilization techniques for both methods. However, this topic is beyond the scope of this paper but is something to address in the future. Both methods are implemented in a unified way, such that they use the same data structures and routines for computations, differing only in the inter-element communication, which is described in detail in Sec. 2.2. This approach allows us to be confident that the differences we see in the results are due to the different methods rather than their implementations.There is a vast collection of work in the literature on CG with geometrically non-conforming mesh refinement, starting with the early work of [2,3] and later [4,5]. Those approaches were based on the mortar element method, which employs integral projection to ensure a weak continuity across the non-conforming edges. In this paper we follow the approach by Fischer et al. [6], that uses an interpolation based method for reconciling the continuity condition on non-conforming edges. The two methods were compared in [7].

Multi-layer non-hydrostatic free surface modelling using the discontinuous Galerkin method

“…In addition, in the atmospheric sciences DG-FE methods have also been applied to non-hydrostatic modelling. For example, the studies of Restelli and Giraldo (2009), Kelly and Giraldo (2012), Kopera and Giraldo (2015), Abdi and Giraldo (2016), and Abdi et al (2017) consider the compressible Euler equations where a non-hydrostatic pressure is obtained from the equation of state. In contrast to the majority of this prior work, the multi-layer nonhydrostatic modelling approach we employ here does not solve the 3D equations, but rather adopts an edge-based approach (following zStelling and Zijlema (2003)) to handle the inter-layer connections associated with the vertical velocities at the layer interfaces.…”

Discontinuous Galerkin scheme for the spherical shallow water equations with applications to tsunami modeling and prediction

“…We briefly outline our method of handling such non-conforming meshes and restrict ourselves to the analysis of balanced non-conforming grids, in which only one "hanging node" is permitted at each interface. We refer the reader to [25][26][27] for the details of the presented mesh refinement procedures.…”

A Review of Element-Based Galerkin Methods for Numerical Weather Prediction: Finite Elements, Spectral Elements, and Discontinuous Galerkin

“…An application of the pointwise-matching scheme to geophysical simulations can be found in [258]. In [185], Kopera and Giraldo presented a unified framwework including both CG and DG methods, as well as integral projection (for DG) and pointwise-matching (for CG) schemes for non-conforming interfaces and found that similar mass conservation properties can be obtained for both configurations.…”

“…28 presents schematics for both methods, the actual implementation in the code may di er. Details of CG interpolation method implementation is presented in [185], along with a unified implementation of the CG/DG method.…”