A conservative, discontinuous Galerkin, tracer transport scheme using compatible finite elements

Timothy C. Andrews; Thomas M. Bendall

Home/Archive/arXiv

arXiv · 2603.19075

A conservative, discontinuous Galerkin, tracer transport scheme using compatible finite elements

Timothy C. Andrews·Thomas M. Bendall

View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper outlines a conservative transport scheme for scalar tracers within a compatible finite element model for geophysical fluid equations. Instead of using the advective transport equation for a mixing ratio, a conservative transport equation is solved for the tracer density of the mixing ratio multiplied by the dry density. This ensures mass conservation in the continuous equations, which can be preserved in the discrete equations with a discontinuous Galerkin transport scheme. Our method is designed to work for two placements of the mixing ratio in a Charney-Phillips vertical staggering: either co-located with the dry density or vertically staggered from it. The new scheme is designed to conserve the tracer density and ensure consistency by maintaining a constant mixing ratio. Additionally, a mass-conserving limiter is developed to ensure non-negativity in the co-located configuration. Tests with terminator toy chemistry and a moist rising bubble show the use of the new transport scheme with physics terms and its ability to accurately model mass conservation of moisture species in a dynamical core setup.

Explore related subjects

math.NA cs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Timothy C. Andrews, Thomas M. Bendall. 2026-03-19. A conservative, discontinuous Galerkin, tracer transport scheme using compatible finite elements. https://arxiv.org/abs/2603.19075

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Discrete normalized gradient flow for two-component Bose-Einstein condensates: Energy dissipation, global convergence and sharp local convergence behavior

The gradient flow with semi-implicit discretization (GFSI) is the most widely used algorithm for computing the ground state of Gross-Pitaevskii energy functional. We apply GFSI to the two-component scenario with Josephson junction and rotating term, which is one of the most important and topical models in multi-component Bose-Einstein condensates (MBECs), and rigorously establish the following fundamental results for the first time. By introducing a Lagrange multiplier to reformulate GFSI into an equivalent form, we prove its energy dissipation property and global convergence to stationary states. More significantly, we uncover an intrinsic connection between this classical numerical PDE discretization rooted in imaginary-time evolution and Riemannian optimization, a state-of-the-art mathematical framework for manifold-constrained optimization. This connection enables us to fully characterize the local convergence behavior of GFSI within the Riemannian optimization framework. Together with the aforementioned global convergence result, this yields a complete global--local convergence theory for GFSI. Finally, numerical experiments comprehensively validate the theoretically predicted energy dissipation and convergence properties.

math.NA↗

arXiv2026

Discretization, Uniform-in-Time Estimations and Approximation of Invariant Measures for Nonlinear Stochastic Differential Equations with Non-Uniform Dissipativity

The approximation of invariant measures for nonlinear ergodic stochastic differential equations (SDEs) is a central problem in scientific computing, with important applications in stochastic sampling, physics, and ecology. We first propose an easily applicable explicit Truncated Euler-Maruyama (TEM) scheme and prove its numerical ergodicity in the $L^p$-Wasserstein distance ($p\geqslant 1$). Furthermore, by combining truncation techniques with the coupling method, we establish a uniform-in-time $1/2$-order convergence rate in moments for the TEM scheme. Additionally, leveraging the exponential ergodicity of both the numerical and exact solutions, we derive a $1/2$-order convergence rate for the invariant measures of the TEM scheme and the exact solution in the $L^1$-Wasserstein distance. Finally, two numerical experiments are conducted to validate our theoretical results.

math.NA↗

arXiv2026

Barotropic-Baroclinic Splitting for Multilayer Shallow Water Models with Exchanges

This work presents the numerical analysis of a barotropic-baroclinic splitting in a nonlinear multilayer framework with exchanges between the layers in terrain-following coordinates. The splitting is formulated as an exact operator splitting. The barotropic step handles free surface evolution and depth-averaged velocity via a well-balanced one-layer model, while the baroclinic step manages vertical exchanges between layers and adjusts velocities to their mean values. We show that the barotropic-baroclinic splitting preserves total energy conservation and meets both a discrete maximum principle and a discrete entropy inequality. Several numerical experiments are presented showing the gain in computational cost, particularly in low Froude simulations, with no loss of accuracy. The benefits of using a well-balancing strategy in the barotropic step to preserve the geostrophic equilibrium are inherited in the overall scheme.

math.NA↗