Sensitivity analysis of Burgers' equation with shocks

TL;DR

This paper demonstrates that, despite singularities in the solution profile of Burgers' equation, key physical quantities remain smooth functions of uncertainties, allowing for accurate polynomial chaos-based uncertainty quantification through proper shifting techniques.

Contribution

The paper introduces a method to accurately quantify uncertainties in hyperbolic PDEs by focusing on physical quantities, overcoming the regularity limitations of gPC in such problems.

Findings

Physical quantities are smooth functions of uncertainties.

Proper shifting enables accurate polynomial approximation.

Error decays with increasing polynomial order.

Abstract

Generalized polynomial chaos (gPC) method has been extensively used in uncertainty quantification problems where equations contain random variables. For gPC to achieve high accuracy, PDE solutions need to have high regularity in the random space, but this is what hyperbolic type problems cannot provide. We provide a counter-argument in this paper, and show that even though the solution profile develops singularities in the random space, which destroys the spectral accuracy of gPC, the physical quantities (such as the shock emergence time, the shock location, and the shock strength) are all smooth functions of the uncertainties coming from both initial data and the wave speed: with proper shifting, the solution's polynomial interpolation approximates the real solution accurately, and the error decays as the order of the polynomial increases. Therefore this work provides a new perspective to "quantify uncertainties" and significantly improves the accuracy of the gPC method with a slight reformulation. We use the Burgers' equation as an example for the thorough analysis, and the analysis could be extended to general conservation laws with convex fluxes.