High-Order Schemes for Hyperbolic Conservation Laws Using Young Measures
Abstract
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. Lukáčová-Medviová, 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.
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