arXiv subjects

Marco Artiano

Publications and source records attributed to Marco Artiano.

8 recordsLinked to original sources

arXiv2026

GPU-Accelerated Nested Schur-Complement Solvers for Atmospheric Flow Simulations Using Horizontally-Explicit, Vertically-Implicit Schemes and Discontinuous Galerkin Methods

Horizontally explicit, vertically implicit (HEVI) time integration removes the acoustic time-step restriction imposed by fine vertical grids in nonhydrostatic atmospheric models, but requires fast solvers for the resulting vertical linear systems, especially on GPUs. We present GPU-accelerated nested Schur-complement solvers for these systems for a split-form discontinuous Galerkin spectral element method on Legendre-Gauss-Lobatto (LGL) nodes applied to the compressible Euler equations with gravity. In time, we use a Rosenbrock-W method, a linearly implicit Runge-Kutta scheme that retains its order and stability with approximate Jacobians. This allows a sparse vertical Jacobian that captures vertical acoustic waves and gravity on curvilinear meshes with orography and decouples into independent column systems. The solvers are nested: a global Schur complement (static condensation) exploits the LGL sparsity to condense each column onto a few element-edge unknowns, and a local Schur complement eliminates all variables but the vertical momentum within each element, exploiting that the pressure depends only on the potential-temperature density. We propose and analyze two condensation strategies, which lead to banded and block-tridiagonal reduced systems. The resulting HEVI schemes conserve total mass and potential-temperature density to round-off. The solvers are implemented for CPUs and GPUs in TrixiAtmo.jl, the atmospheric extension of Trixi.jl. We verify high-order convergence with linear inertia-gravity waves in a 2D channel and demonstrate robustness on the sphere with the baroclinic instability, a mountain-induced baroclinic instability, and the DCMIP-2025 gap-flow and vortex-shedding cases. For the baroclinic instability without orography, the scheme achieves 18.8 and 2.9 simulated years per day at 100 and 50 km resolution on one NVIDIA GH200 GPU in double precision.

math.NA↗

arXiv2026

Entropy-Stable and Well-Balanced Discontinuous Galerkin Methods for the Compressible Euler Equations in Vector-Invariant Form

We develop structure-preserving methods for the compressible Euler equations with gravity in vector-invariant form and potential temperature as a prognostic variable within the flux-differencing discontinuous Galerkin spectral element method (DGSEM) framework. By discretizing the nonconservative terms as symmetric and antisymmetric products, we derive two-point numerical fluxes that conserve both the thermodynamic entropy and the total energy. Moreover, we design an entropy-stable numerical flux that is well-balanced for both isothermal and isentropic background states. All properties are shown to carry over to the high-order DGSEM on general curvilinear meshes. Several numerical examples confirm the theoretical findings and show the robustness and accuracy of the scheme for use in modern dynamical cores for atmospheric flows.

math.NA↗

arXiv2026

Structure-Preserving High-Order Methods for the Compressible Euler Equations in Potential Temperature Formulation for Atmospheric Flows

We develop structure-preserving numerical methods for the compressible Euler equations, employing potential temperature as a prognostic variable. We construct three numerical fluxes designed to ensure the conservation of entropy and total energy within the discontinuous Galerkin framework on general curvilinear meshes. Furthermore, we introduce a generalization for the kinetic energy preservation property and total energy conservation in the presence of a gravitational potential term. To this end, we adopt a flux-differencing approach for the discretization of the source term, treated as non-conservative product. We present well-balanced schemes for different constant background states for both formulations (total energy and potential temperature) on curvilinear meshes. Finally, we validate the methods by comparing the potential temperature formulation with the traditional Euler equations formulation across a range of classical atmospheric scenarios.

math.NA↗

arXiv2026

Massively parallel numerical simulations with Julia

The Julia programming language aims to provide a modern approach to develop high-performance computing (HPC) applications. It tries to achieve this by combining a high-level, dynamic interface with just-in-time compilation to native machine code, thereby facilitating high developer productivity and native code performance at the same time. While this approach has already been shown to work well for serial applications, it is not clear if it readily translates to traditional, massively parallel HPC work loads. In this paper, we fill this gap by analyzing the parallel performance of the numerical computational fluid dynamics simulation code Trixi$.$jl, written in Julia, and compare it to the Fortran code FLUXO. We show some of the challenges of using Julia at scale and discuss possible solutions, specifically with respect to code loading and compilation at startup. Finally, we demonstrate the parallel scaling of our Julia code on up to 61440 CPU cores.

math.NA↗

arXiv2026

Asymptotic-Preserving and Well-Balanced Linearly Implicit IMEX Schemes for the Anelastic Limit of the Isentropic Euler Equations with Gravity

We consider the compressible Euler system with anelastic scaling, modeling isentropic flows under the influence of gravity. In the zero-Mach-number limit, the solution of the compressible Euler system converges to a variable density anelastic incompressible limit system. In this work, we present the design and analysis of a class of higher-order linearly implicit IMEX Runge-Kutta schemes that are asymptotic preserving, i.e., they respect the transitory nature of the governing equations in the limit. The presence of gravitational potential warrants the incorporation of the well-balancing property. The scheme is developed as a novel combination of a penalization of a linear steady state, a finite-volume balance-preserving reconstruction, and a source term discretization preserving steady states. The penalization plays a crucial role in obtaining a linearly implicit scheme, and well-balanced flux-source discretization ensures accuracy in very low Mach number regimes. Some results of numerical case studies are presented to corroborate the theoretical assertions.

math.NA↗

arXiv2026

On Affordable High-Order Entropy-Conservative/Stable and Well-Balanced Methods for Nonconservative Hyperbolic Systems

Many entropy-conservative and entropy-stable (summarized as entropy-preserving) methods for hyperbolic conservation laws rely on Tadmor's theory for two-point entropy-preserving numerical fluxes and its higher-order extension via flux differencing using summation-by-parts (SBP) operators, e.g., in discontinuous Galerkin spectral element methods (DGSEMs). The underlying two-point formulations have been extended to nonconservative systems using fluctuations by Castro et al. (2013, doi:10.1137/110845379) with follow-up generalizations to SBP methods. We propose specific forms of entropy-preserving fluctuations for nonconservative hyperbolic systems that are simple to interpret and allow an algorithmic construction of entropy-preserving methods. We analyze necessary and sufficient conditions, and obtain a full characterization of entropy-preserving three-point methods within the finite volume framework. This formulation is extended to SBP methods in multiple space dimensions on Cartesian and curvilinear meshes. Additional properties such as well-balancedness extend naturally from the underlying finite volume method to the SBP framework. We use the algorithmic construction enabled by the chosen formulation to derive several new entropy-preserving schemes for nonconservative hyperbolic systems, e.g., the compressible Euler equations of an ideal gas using the internal energy equation and a dispersive shallow-water model. Numerical experiments show the robustness and accuracy of the proposed schemes.

math.NA↗

arXiv2026

Jin-Xin relaxation as a shock-capturing method for high-order DG/FR schemes

Jin-Xin relaxation is a method for approximating non-linear hyperbolic conservation laws by a linear system of hyperbolic equations with an $\varepsilon$ dependent stiff source term. The system formally relaxes to the original conservation law as $\varepsilon \to 0$. An asymptotic analysis of the Jin-Xin relaxation system shows that it can be seen as a convection-diffusion equation with a diffusion coefficient that depends on the relaxation parameter $\varepsilon$. This work makes use of this property to use the Jin-Xin relaxation system as a shock-capturing method for high-order discontinuous Galerkin (DG) or flux reconstruction (FR) schemes. The idea is to use a smoothness indicator to choose the $\varepsilon$ value in each cell, so that we can use larger $\varepsilon$ values in non-smooth regions to add extra numerical dissipation. We show how this can be done by using a single stage method by using the compact Runge-Kutta FR method that handles the stiff source term by using IMplicit-EXplicit Runge-Kutta (IMEX-RK) schemes. Numerical results involving Burgers' equation and the compressible Euler equations are shown to demonstrate the effectiveness of the proposed method.

math.NA↗

arXiv2024

On the performances of standard and kinetic energy preserving time-integration methods for incompressible-flow simulations

The effects of kinetic-energy preservation errors due to Runge-Kutta (RK) temporal integrators have been analyzed for the case of large-eddy simulations of incompressible turbulent channel flow. Simulations have been run using the open-source solver Xcompact3D with an implicit spectral vanishing viscosity model and a variety of temporal Runge-Kutta integrators. Explicit pseudo-symplectic schemes, with improved energy preservation properties, have been compared to standard RK methods. The results show a marked decrease in the temporal error for higher-order pseudo-symplectic methods; on the other hand, an analysis of the energy spectra indicates that the dissipation introduced by the commonly used three-stage RK scheme can lead to significant distortion of the energy distribution within the inertial range. A cost-vs-accuracy analysis suggests that pseudo-symplectic schemes could be used to attain results comparable to traditional methods at a reduced computational cost.

physics.flu-dyn↗