arXiv subjects

Guido Lodato

Publications and source records attributed to Guido Lodato.

4 recordsLinked to original sources

arXiv2026

A novel compact scheme for second-order fluxes applied to the Spectral Difference method

The discretization of second-order (viscous) terms in Discontinuous Spectral Element Methods (DSEMs) typically relies on an auxiliary gradient variable, whose treatment at element interfaces affects the accuracy and stability of the scheme. The Bassi-Rebay (BR1) formulation is attractive for its simplicity and parameter-free character, but suffers from sub-optimal convergence at even polynomial orders and requires an extended five-element stencil. Inspired by Huynh's Flux Reconstruction formulation, we develop a compact, fully-centered scheme for second-order fluxes within the Spectral Difference (SD) method. The proposed approach modifies the reconstruction of the auxiliary gradient using interface-dependent, one-sided continuous fluxes, reducing the stencil from five to three elements while preserving the centered and parameter-free nature of BR1. The formulation is developed in one dimension and extended to multiple dimensions. Temporal eigenanalysis is used to characterize its dissipation and dispersion properties, including the effects of interior penalty terms. Numerical tests consider the linear diffusion equation, an under-resolved localized Dirac's delta, the nonlinear porous medium equation, and implicit large-eddy simulations of the three-dimensional Taylor-Green vortex at $\mathrm{Re}=1600$ and $5000$. The compact scheme restores the expected convergence order for all polynomial degrees, including even orders, and reduces spurious oscillations in under-resolved and nonlinear regimes. It also remains stable in turbulent cases where the standard formulation fails, owing to improved damping of high-wavenumber numerical modes. The proposed approach provides an attractive compact alternative to BR1 for second-order fluxes in the SD method.

math.NA↗

arXiv2024

Generalisation of the Spectral Difference scheme for the diffused-interface five equation model

The present work focuses on the generalisation of the Spectral Difference (SD) scheme to the reduced Baer-Nunziato system known as five-equation model for the simulation of two immiscible compressible fluids. This five equation model is considered with the additional Allen-Cahn regularisation to avoid both over-diffusion and over-thinning of the phase field representing the interface. Finally, in order to preserve contact discontinuities, in the reconstruction step of the spectral difference scheme, a change of variables from conservative to primitive is used. This approach is shown to be beneficial in avoiding pressure oscillations at material interfaces. An extensive series of numerical tests are proposed to assess accuracy and robustness of the present method. Both kinematic (Rider-Kothe vortex) and two-phase flow problems (Rayleigh-Taylor instability, shock-droplet interaction, Taylor-Green vortex) are considered.

physics.flu-dyn↗

arXiv2021

Fully-discrete spatial eigenanalysis of discontinuous spectral element methods: insights into well-resolved and under-resolved vortical flows

This study presents a comprehensive spatial eigenanalysis of fully-discrete discontinuous spectral element methods, now generalizing previous spatial eigenanalysis that did not include time integration errors. The influence of discrete time integration is discussed in detail for different explicit Runge-Kutta (1st to 4th order accurate) schemes combined with either Discontinuous Galerkin (DG) or Spectral Difference (SD) methods, both here recovered from the Flux Reconstruction (FR) scheme. Selected numerical experiments using the improved SD method by Liang and Jameson [1] are performed to quantify the influence of time integration errors on actual simulations. These involve test cases of varied complexity, from one-dimensional linear advection equation studies to well-resolved and under-resolved inviscid vortical flows. It is shown that, while both well-resolved and under-resolved simulations of linear problems correlate well with the eigenanalysis prediction of time integration errors, the correlation can be much worse for under-resolved nonlinear problems. The effect of mesh regularity is also considered, where time integration errors are found to be, in the case of irregular grids, less pronounced than those of the spatial discretisation. In fact, for the under-resolved vortical flows considered, the predominance of spatial errors made it practically impossible for time integration errors to be distinctly identified. Nevertheless, for well-resolved nonlinear simulations, the effect of time integration errors could still be recognized. This highlights that the interaction between space and time discretisation errors is more complex than otherwise anticipated, contributing to the current understanding about when eigenanalysis can effectively predict the behaviour of numerical errors in practical under-resolved nonlinear problems, including under-resolved turbulence computations.

physics.flu-dyn↗

arXiv2017

Spectral energy cascade in thermoacoustic shock waves

We have performed a theoretical and numerical investigation of thermoacoustically amplified quasi-planar nonlinear waves driven to the limit of shock-wave formation in a variable-area 2.58 m long looped resonator designed via eigenvalue analysis to maximize the growth rate of the second harmonic (265 Hz). High-order unstructured fully compressible Navier-Stokes simulations reveal the presence of three regimes: (i) Monochromatic harmonic growth, governed by linear thermoacoustics; (ii) Hierarchical spectral broadening, characterized by nonlinear energy cascade; (iii) Shock-wave dominated limit cycle, where energy production is balanced by dissipation occurring at the captured shock-thickness scale. We have modeled these regimes of thermoacoustic wave amplification based on first-principle governing equations and elucidated the physics of energy density production and saturation to a limit cycle.

physics.flu-dyn↗