Manuel Torrilhon

We present novel entropy-conservative and entropy-stable multirate Runge-Kutta methods based on Paired Explicit Runge-Kutta (P-ERK) schemes with relaxation for conservation laws and related systems of partial differential equations (PDEs). Optimized schemes up to fourth order of accuracy are derived and validated in terms of order of consistency, conservation of linear invariants, and entropy conservation/stability. We demonstrate the effectiveness of these Paired Explicit Relaxation Runge-Kutta (P-ERRK) methods when combined with a high-order, entropy-conservative/stable discontinuous Galerkin spectral element method on unstructured meshes. The P-ERRK methods are readily implemented for partitioned semidiscretizations arising from problems with equation-based scale separation, such as non-uniform meshes. We highlight that the relaxation approach acts as a time-limiting technique, which improves the nonlinear stability and thus robustness of the multirate schemes. The P-ERRK methods are applied to a range of problems, ranging from compressible Euler over compressible Navier-Stokes to the visco-resistive magnetohydrodynamics (VRMHD) equations in two and three spatial dimensions. For each test case, we compare computational load and runtime to standalone relaxed Runge-Kutta methods, which are outperformed by factors up to four. All results can be reproduced using a publicly available repository.

We introduce the concept of volume term adaptivity for high-order discontinuous Galerkin (DG) schemes solving time-dependent partial differential equations. Termed v-adaptivity, we present a novel general approach that exchanges the discretization of the volume contribution of the DG scheme at every Runge-Kutta stage based on suitable indicators. Depending on whether robustness or efficiency is the main concern, different adaptation strategies can be chosen. Precisely, the weak form volume term discretization is used instead of the entropy-conserving flux-differencing volume integral whenever the former produces more entropy than the latter, resulting in an entropy-stable scheme. Conversely, if increasing the efficiency is the main objective, the weak form volume integral may be employed as long as it does not increase entropy beyond a certain threshold or cause instabilities. Thus, depending on the choice of the indicator, the v-adaptive DG scheme improves robustness, efficiency and approximation quality compared to schemes with a uniform volume term discretization. We thoroughly verify the accuracy, linear stability, and entropy-admissibility of the v-adaptive DG scheme before applying it to various compressible flow problems in two and three dimensions.

Non-equilibrium evaporative flows play a central role in many nanoporous membrane technologies, where transport of fluids is confined by solid surfaces at the nanoscale. In this work, we propose a molecular kinetic model that consistently resolves the coupled interactions among vapour, liquid and solid surfaces in such flows. As a direct consequence of this bottom-up approach, the liquid–vapour, liquid–solid and vapour–solid interfaces form autonomously, and the effects of non-equilibrium and real fluids can be captured simultaneously, which does not need empirical models depending on ad hoc parameters such as the evaporation/condensation coefficients and contact angle. Accuracy of the model improves further by including the soft-collision effect in the pair correlation function and applying a temperature-dependent correction to the mean-field Vlasov term, as validated against the experimental data and the molecular dynamics simulations. Furthermore, when applied to unsteady, evaporation-driven liquid–vapour flows, the model reveals distinct dynamics due to surface wettability: the hydrophilic surfaces exhibit phenomena such as liquid meniscus breakage and enhanced evaporation flux, whereas the hydrophobic surfaces lead to disappearance of liquid droplets. These findings highlight the potential of the proposed molecular kinetic model as a powerful design tool for next-generation nano-technologies that leverage nano-confined phase change.

The Numerical Flow Iteration (NuFI) method has recently been proposed as a memory-slim solution method for the Vlasov--Poisson system. It stores the temporal evolution of the electric field, instead of the distribution functions, and reconstructs the solution in each time step by following the characteristics backwards in time and reconstructing the solution from the initial distribution. NuFI has been shown to be more accurate than other state-of-the-art Vlasov solvers given the same amount of degrees of freedom as well as interpolation order, essentially making NuFI a high-fidelity but low-memory cost scheme. In this paper, we build on the Hamiltonian structure of the full Vlasov--Maxwell system to extend NuFI to handle electro-magnetic kinetic plasma dynamics. We show hat the advanced structure-preserving properties of NuFI are preserved when extending to the electro-magnetic case.

The Fokker-Planck (FP) particle method accelerates rarefied-gas simulations by replacing the binary collisions of the commonly used Direct Simulation Monte Carlo (DSMC) method with a drift=diffusion process. Like all particle methods, the FP method is inherently stochastic, which leads to statistical fluctuations in macroscopic quantities and necessitates large particle numbers for accurate results. In this work, we investigate the use of quasi-random numbers, which sample distributions more evenly and thereby reduce the variance. To preserve the low-discrepancy structure across time steps, we employ the Array Randomized Quasi-Monte Carlo (Array-RQMC) technique. We combine the FP method with Array-RQMC and compare it in homogeneous and inhomogeneous problems with other commonly used variance-reduction techniques. The proposed FP-Array-RQMC approach achieves improved convergence rates compared with pseudo-random sampling and yields smaller estimator errors for sufficiently large particle numbers.

A systematic numerical investigation of the Richtmyer–Meshkov (RM) instability in shock-accelerated square and rectangular SF6 bubbles with varying aspect ratios is presented. Simulations are performed using a high-order discontinuous Galerkin method to resolve interface deformation, vorticity generation, and instability amplification. Building on our earlier study of shock-driven instabilities in square and rectangular light-gas bubbles [Singh and Torrilhon, Phys. Fluids 35, 012117 (2023)], the present work provides deeper physical insight into geometry-dependent mechanisms. The results demonstrate that bubble aspect ratio has a decisive influence on shock focusing, baroclinic vorticity deposition, and the onset of nonlinear structures. Horizontally elongated bubbles enhance confinement of transmitted shocks and circulation growth, while vertical elongation stabilizes the interface and delays nonlinear evolution. Quantitative diagnostics derived from the vorticity transport equation reveal that dilatation-induced and baroclinic contributions are amplified in horizontal configurations but suppressed in vertical ones. The evolution of enstrophy further indicates that horizontal elongation accelerates vortex roll-up, jet penetration, and transition toward turbulence-like dynamics, whereas vertical elongation promotes long-lived columnar vortices with reduced secondary instabilities. Interface-based measures of displacement, width, and height confirm these trends and establish clear links between geometry and instability growth.