#### Title

Efficient Time-Integration Schemes for Discontinuous Galerkin Non-Hydrostatic Atmospheric Models

#### Date of Award

Spring 1-1-2016

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Applied Mathematics

#### First Advisor

Henry M. Tufo

#### Second Advisor

Thomas A. Manteuffel

#### Third Advisor

Keith Julien

#### Fourth Advisor

Ram D. Nair

#### Fifth Advisor

Congming Li

#### Abstract

This thesis presents the ongoing work on the numerical aspects of designing a numerical frame- work on which to build a high-resolution atmospheric model using the discontinuous Galerkin (DG) methods. As the horizontal resolution exceeds the hydrostatic limit (1/10◦ or 10 km), which is usu- ally referred as the non-hydrostatic (NH) scale, the compressible Euler system must be employed to characterize the motion of the air flow.

To simulate this system numerically, we consider the DG method for the spatial discretization and cubed-sphere grid system. The High-Order Method Modeling Environment (HOMME) is a highly scalable hydrostatic dynamical core based on spectral element and/or DG methods. It utilizes cube-sphere geometry and shows great scalability. Our goal is to extend HOMME-DG model to the non-hydrostatic scale.

We use the global shallow water equations to study the influence of the full conservative equation sets in conserving integral invariants is rigorously compared against the vector-invariant form. Several important components, such as the horizontal discretization and numerical diffusion are also discussed briefly.

The terrain-following height-based coordinate transform is adopted to handle the orography. For the time discretization, we consider a Horizontally explicit and Vertically implicit operator splitting based on Strang-splitting approach. HEVI treats the vertical component implicitly and the horizontal component explicitly. As a consequence, the maximum allowed time-step size is only constrained by the horizontal grid spacing, which is usually several orders of magnitude higher than the vertical. We compare HEVI operator splitting with Implicit-explict (IMEX) linear-nonlinear splitting ideas. We also perform the linear stability study of various IMEX Runge-Kutta schemes. HEVI-Strang splitting shows large stability region in the well-resolved scale and only requires one implicit solve compared with other IMEX-RK schemes. This study is the first time testing the DG scheme with the dimensional splitting approach. The HEVI-Strang scheme shows qualitatively comparable results at a more lower computational cost. The efficiency of the linear solver resulting from the Newton’s method is also investigated. A right preconditioner is suggested to improve the convergence of the GMRES iterative solver. Numerical results show that the preconditioned GMRES and the direct solvers are both viable options to solve the vertical implicit component.

The global 3D DG-NH model is constructed by vertical stacking of the horizontal cubes- sphere layers. The 3D global advection problem is tested using two DCMIP test cases. We also present some preliminary results for the non-hydrostatic inertia gravity wave test utilizing the HEVI-Strang time integration scheme. The results of 3D DG-NH model based on HEVI-Strang time integration scheme are qualitatively in line with other non-hydrostatic models. The time-step size of the HEVI-Strang scheme is not affected as the vertical grid spacing varies and the 3D DG-NH model maintains the scalability of the HOMME framework.

#### Recommended Citation

Bao, Lei, "Efficient Time-Integration Schemes for Discontinuous Galerkin Non-Hydrostatic Atmospheric Models" (2016). *Applied Mathematics Graduate Theses & Dissertations*. 73.

https://scholar.colorado.edu/appm_gradetds/73