Michael Herty

This work provides a comparative assessment of several low‐dissipation numerical schemes for hyperbolic conservation laws, highlighting their performance relative to the classical Harten‐Lax‐van Leer (HLL) schemes. The schemes under consideration include the classical Harten‐Lax‐van Leer‐Contact (HLLC), the recently proposed TV flux splitting, the low‐dissipation Central‐Upwind (LDCU), and the local characteristic decomposition‐based Central‐Upwind (LCDCU) schemes. These methods are extended to higher orders of accuracy, up to the fifth order, within both finite‐volume and finite‐difference frameworks. A series of numerical experiments for the one‐ and two‐dimensional Euler equations of gas dynamics are performed to evaluate the accuracy, robustness, and computational efficiency of the studied schemes. The comparison highlights the trade‐offs between resolution of contact and shear waves, robustness in the presence of shocks, and computational cost. The investigated low‐dissipation schemes show comparable levels of numerical dissipation, with only subtle differences appearing in selected benchmark problems. The results provide practical guidance for selecting efficient low‐dissipation solvers for the simulation of complex compressible flows.

A novel structure-preserving numerical method to solve random hyperbolic systems of conservation laws is presented. The method uses a concept of generalized, measure-valued solutions to random conservation laws. This yields a linear partial differential equation with respect to the Young measure and allows for the computation of the approximation based on linear programming problems. We analyze structure-preserving properties of the derived numerical method and discuss its advantages and disadvantages. We numerically demonstrate the approach on the one-dimensional Burgers and isentropic Euler equations and compare with stochastic collocation. In addition, we introduce a discontinuous-flux test in which different entropies used in the linear-program objective select different weak entropy solutions, and we report the corresponding changes in the moments and supports of the Young measure.

A method for the uncertainty quantification of nonlinear hyperbolic conservation laws with many uncertain parameters is presented. The method combines stochastic finite volume methods and ...

We develop high-order numerical schemes to solve random hyperbolic conservation laws using linear programming. The proposed schemes are high-order extensions of the existing first-order scheme introduced in [S. Chu, M. Herty, M. Lukacova-Medvicova, and Y. Zhou, solving random hyperbolic conservation laws using linear programming], where a novel structure-preserving numerical method using a concept of generalized, measure-valued solutions to solve random hyperbolic systems of conservation laws is proposed, yielding a linear partial differential equation concerning the Young measure and allowing the computation of approximations based on linear programming problems. The second-order extension is obtained using piecewise linear reconstructions of the one-sided point values of the unknowns. The fifth-order scheme is developed using the finite-difference alternative weighted essentially non-oscillatory (A-WENO) framework. These extensions significantly improve the resolution of discontinuities, as demonstrated by a series of numerical experiments on both random (Burgers equation, isentropic Euler equations) and deterministic (discontinuous flux, pressureless gas dynamics, Burgers equation with non-atomic support) hyperbolic conservation laws.

This work provides a comparative assessment of several low-dissipation numerical schemes for hyperbolic conservation laws, highlighting their performance relative to the classical Harten-Lax-van Leer (HLL) schemes. The schemes under consideration include the classical Harten-Lax-van Leer-Contact (HLLC), the recently proposed TV flux splitting, the low-dissipation Central-Upwind (LDCU), and the local characteristic decomposition-based Central-Upwind (LCDCU) schemes. These methods are extended to higher orders of accuracy, up to the fifth order, within both finite-volume and finite-difference frameworks. A series of numerical experiments for the one- and two-dimensional Euler equations of gas dynamics are performed to evaluate the accuracy, robustness, and computational efficiency of the studied schemes. The comparison highlights the trade-offs between resolution of contact and shear waves, robustness in the presence of shocks, and computational cost. The investigated low-dissipation schemes show comparable levels of numerical dissipation, with only subtle differences appearing in selected benchmark problems. The results provide practical guidance for selecting efficient low-dissipation solvers for the simulation of complex compressible flows.

We propose a Bayesian framework for feedback boundary control for hyperbolic balance laws. The method propagates a probability distribution over feedback parameters by using Lyapunov decay estimates as a likelihood. In the linear setting, the framework recovers the available analytical results, while simultaneously extending them to nonlinear regimes where such results are not readily accessible. We first validate the method using the first-order local Lax-Friedrichs (LLF) discretizations on linear models -- the decoupled wave system and the linearized Saint-Venant equations -- recovering the known stability intervals and mixed boundary couplings reported in the control literature. We then consider nonlinear and stochastic settings, including the nonlinear Saint-Venant system and Burgers equation with random initial data, as well as a nonconservative perturbation with source terms, and demonstrate that the computed stability domains remain accurate and robust with respect to the choice of indicator and the initial prior. We further show that the methodology carries over to a second-order semi-discrete LLF scheme and to a model with two interacting control parameters for the temperature field development in laser powder bed fusion with feedback power regulation. These numerical experiments confirm consistency with available theory on benchmark cases and highlight the practicality of the proposed, discretization-agnostic feedback selection procedure.

We propose a variant of consensus-based optimization (CBO) algorithms, controlled-CBO, which introduces a feedback control term to improve convergence towards global minimizers of nonconvex functions in multiple dimensions. The feedback law is a gradient of a numerical approximation to the Hamilton–Jacobi–Bellman (HJB) equation, which serves as a proxy of the original objective function. Thus, the associated control signal furnishes gradient-like information to facilitate the identification of the global minimum without requiring derivative computation from the objective function itself. The proposed method exhibits significantly improved performance over standard CBO methods in numerical experiments, particularly in scenarios involving a limited number of particles, or where the initial particle ensemble is not well positioned with respect to the global minimum. At the same time, the modification keeps the algorithm amenable to theoretical analysis in the mean-field sense. The superior convergence rates are assessed experimentally. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/AmberYuyangHuang/ControlledCBO and in the supplementary material ( M170644_Supplementary_Materials.pdf [259KB], ControlledCBO-main.zip [20.0KB]). [Formula: see text]

We present a modified simulated annealing method with a dynamical choice of the cooling temperature. The latter is determined via a closed-loop control and is proven to yield exponential decay of the entropy of the particle system. The analysis is carried out through kinetic equations for interacting particle systems describing the simulated annealing method in an extended phase space. Decay estimates are derived under the quasi-invariant scaling of the resulting system of Boltzmann-type equations to assess the consistency with their mean-field limit. Numerical results are provided to illustrate and support the theoretical findings.