Helmholtz preconditioning for the compressible Euler equations using mixed finite elements with Lorenz staggering (2403.04095v3)
Abstract: Implicit solvers for atmospheric models are often accelerated via the solution of a preconditioned system. For block preconditioners this typically involves the factorisation of the (approximate) Jacobian resulting from linearization of the coupled system into a Helmholtz equation for some function of the pressure. Here we present a preconditioner for the compressible Euler equations with a flux form representation of the potential temperature on the Lorenz grid using mixed finite elements. This formulation allows for spatial discretisations that conserve both energy and potential temperature variance. By introducing the dry thermodynamic entropy as an auxiliary variable for the solution of the algebraic system, the resulting preconditioner is shown to have a similar block structure to an existing preconditioner for the material form transport of potential temperature on the Charney-Phillips grid. This new formulation is also shown to be more efficient and stable than both the material form transport of potential temperature on the Charney-Phillips grid, and a previous Helmholtz preconditioner for the flux form transport of density weighted potential temperature on the Lorenz grid for a 1D thermal bubble configuration. The new preconditioner is further verified against standard two dimensional test cases in a vertical slice geometry.
- The CMC-MRB Global Environmental Multiscale (GEM) Model. Part III: Nonhydrostatic Formulation. Mon. Wea. Rev., 130:339–356, 2002.
- An inherently mass-conserving semi-implicit semi-Lagrangian discretization of the deep-atmosphere global non-hydrostatic equations. Q. J. R. Meteorol. Soc., 140:1505–1520, 2014.
- A mixed finite-element, finite-volume, semi-implicit discretisation for atmospheric dynamics: Cartesian geometry. Q. J. R. Meteorol. Soc., 145:1–19, 2019.
- Multigrid preconditioners for the mixed finite element dynamical core of the LFRic atmospheric model. Q. J. R. Meteorol. Soc., 146:3917–3936, 2020.
- D Lee. An energetically balanced, quasi-Newton integrator for non-hydrostatic vertical atmospheric dynamics. J. Comp. Phys., 429:109988, 2021.
- D Lee and A Palha. Exact spatial and temporal balance of energy exchanges within a horizontally explicit/vertically implicit non-hydrostatic atmosphere. J. Comp. Phys., 440:110432, 2021.
- Schur complement IMplicit-EXplicit formulations for discontinuous Galerkin non-hydrostatic atmospheric models. J. Comp. Phys., 491:112361, 2023.
- Slate: extending Firedrake’s domain-specific abstraction to hybridized solvers for geoscience and beyond. Geosci. Model. Dev., 13:735–761, 2020.
- Hybridised multigrid preconditioners for a compatible finite-element dynamical core. Q. J. R. Meteorol. Soc., 149:2454–2476, 2023.
- A Arakawa and C S Connor. Vertical differencing of the primitive equations based on the Charney-Phillips grid in hybrid σ−p𝜎𝑝\sigma-pitalic_σ - italic_p vertical coordinates. Mon. Wea. Rev., 124:511–528, 1996.
- J Thuburn and T J Woolings. Vertical discretizations for compressible Euler equation atmospheric models giving optimal representation of normal modes J. Comp. Phys., 203:386–404, 2005.
- A compatible finite-element discretisation for the moist compressible Euler equations. Q. J. R. Meteorol. Soc., 146:3187–3205, 2020.
- Choice of function spaces for thermodynamic variables in mixed finite-element methods Q. J. R. Meteorol. Soc., 144:900–916, 2018.
- An Energy Consistent Discretization of the Nonhydrostatic Equations in Primitive Variables. Journal of Advances in Modelling Earth Systems, 12:(1), 2020.
- A quasi-Hamiltonian discretization of the thermal shallow water equations. J. Comp. Phys., 379:1–31, 2019.
- Entropy and energy conservation for thermal atmospheric dynamics using mixed compatible finite elements J. Comp. Phys., 496:112605, 2024
- Compatible finite element spaces for geophysical fluid dynamics. Dyn. Stat. Climate Sys., 1:1–31, 2016.
- W Bauer and C J Cotter. Energy–enstrophy conserving compatible finite element schemes for the rotating shallow water equations with slip boundary conditions. J. Comp. Phys., 373:171–187, 2018.
- J Thuburn. Numerical entropy conservation without sacrificing Charney–Phillips grid optimal wave propagation. Q. J. R. Meteorol. Soc., 148:2755–2768, 2022.
- S. Badia and F. Verdugo. Gridap: An extensible finite element toolbox in Julia. J. Open Source Softw., 5:2520, 2020.
- GridapDistributed: a massively parallel finite element toolbox in Julia. J. Open Source Softw., 7:4157, 2022.
- A proposed baroclinic wave test case for deep‐ and shallow‐atmosphere dynamical cores. Q. J. R. Meteorol. Soc., 140:1590–1602, 2014.
- W C Skamarock and J B Klemp. Efficiency and Accuracy of the Klemp-Wilhelmson Time-Splitting Technique. Mon. Wea. Rev., 122:2623–2630, 1994.
- F X Giraldo and M Restelli. A study of spectral element and discontinuous Galerkin methods for the Navier–Stokes equations in nonhydrostatic mesoscale atmospheric modeling: Equation sets and test cases. J. Comp. Phys., 227:3849–3877, 2008.
- E Burman and A Ern. Continuous interior penalty hpℎ𝑝hpitalic_h italic_p-finite element methods for advection and advection-diffusion equations. Mathematics of Computation, 76:1119–1140, 2007.
- Numerical solutions of a non-linear density current: a benchmark solution and comparisons Int. J. Numer. Meth. Fluids, 17:, 1–22, 1993.
- D Lee. A comparison of Rosenbrock–Wanner and Crank–Nicolson time integrators for atmospheric modelling Q. J. Royal Meteorol. Soc., 758:462–483, 2024.