Non-hydrostatic mesoscale atmospheric modeling by the anisotropic mesh adaptive discontinuous Galerkin method

TL;DR

This paper develops a fully implicit space-time discontinuous Galerkin method for non-hydrostatic mesoscale atmospheric flows, integrated with anisotropic hp-mesh adaptation to minimize degrees of freedom under a numerical error tolerance. By combining a Newton-like nonlinear solver with residual-based adaptive time stepping and hp-AMA-driven mesh refinement, the approach accommodates different meshes at successive time levels while maintaining stability and accuracy. Numerical experiments on inertia-gravity waves, rising thermal bubbles, density currents, and mountain flows demonstrate high-order accuracy, robust boundary treatments, and substantial computational savings relative to fixed-mesh baselines. The work provides a practical, high-order framework for efficient non-hydrostatic atmospheric modeling with rigorous error control and conservation considerations.

Abstract

We deal with non-hydrostatic mesoscale atmospheric modeling using the fully implicit space-time discontinuous Galerkin method in combination with the anisotropic $hp$-mesh adaptation technique. The time discontinuous approximation allows the treatment of different meshes at different time levels in a natural way which can significantly reduce the number of degrees of freedom. The presented approach generates a sequence of triangular meshes consisting of possible anisotropic elements and varying polynomial approximation degrees such that the interpolation error is below the given tolerance and the number of degrees of freedom at each time step is minimal. We describe the discretization of the problem together with several implementation issues related to the treatment of boundary conditions, algebraic solver and adaptive choice of the size of the time steps.The computational performance of the proposed method is demonstrated on several benchmark problems.

Figures (10)

• Figure 1: Illustration of meshes ${{\mathscr T}_{h,m-1}}$ (red) and ${{\mathscr T}_{h,m}}$ (blue), total view (left) and a detail (right).
• Figure 2: Comparison of the semi-implicit (blue) and fully implicit (red) schemes: the dependence of the size of the time step $\tau_m$ (left) and the accumulated number of GMRES iterations (right) with respect to the physical time $t\in(0,700)\,\mathrm{s}$. Each dot corresponds to one time step $m=1,\dots, r$.
• Figure 3: Performance of the adaptive algebraic stopping criteria and the adaptive choice of the time step: estimators $\eta^{\mathrm{A}}_{m}$, $\eta^{\mathrm{S}}_{m}$, $\eta^{\mathrm{T}}_{m}$ (cf. \ref{['etas']}) and $\tau_m$ for $m=1,\dots, 15$ (left) and $m=141,\dots, 156$ (right), each dot corresponds to one iterative step $k$ in \ref{['SS1']}.
• Figure 4: Smooth rising thermal bubble: meshes (first line), isolines of the potential temperature perturbation with the contour interval of $0.05$ (second line) and the isolines of the density with the contour interval of $0.05$ (third line); the computations using the uniform grid (left column) and anisotropic meshes generated with the tolerances $\omega= 0.02$, $0.015$ and $0.01$ (from second to fourth columns) at $T=500\,\mathrm{s}$. Note: The numbers of $\#{\mathscr T}_h$ and $\mathrm{DoF}$ are the same for $\omega=0.02$ and $\omega=0.015$ which is just a coincidence.
• Figure 5: Interior gravity wave: isolines of the potential temperature perturbation (the contour interval of $5\cdot10^{-4}\,\mathrm{K}$) with the corresponding mesh (left) and the potential temperature perturbation along $x_2=5\, 000\,\mathrm{m}$ at $t=0\,\mathrm{s}$ (top), $t=\,\numprint{2000}\,\mathrm{s}$ (center) and $t=\,\numprint{3000}\,\mathrm{s}$ (bottom).

Theorems & Definitions (2)

• Definition 3.1
• Remark 5.1