| 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. Lukakova-Medvidova, 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. |
