High-Order Schemes for Hyperbolic Conservation Laws Using Young Measures
For researchers in numerical methods for hyperbolic PDEs, this provides high-order extensions of a novel measure-valued approach, though the improvements are incremental over the existing first-order scheme.
This work extends a first-order structure-preserving scheme for random hyperbolic conservation laws to second- and fifth-order accuracy using piecewise linear reconstructions and the A-WENO framework, significantly improving discontinuity resolution as shown in numerical experiments.
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 [{\sc S. Chu, M. Herty, M. Lukáčová-Medvi{\softd}ová, and Y. Zhou}, SIAM J. Sci. Comput., 48 (2026)], 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.