Inviscid Burgers as a degenerate elliptic problem

TL;DR

The paper develops a duality-based variational framework for the inviscid Burgers equation in both conservation and Hamilton–Jacobi forms by recasting the PDE as constraints and optimizing a convex auxiliary potential. A dual-to-primal (DtP) mapping links dual fields to the primal solution, yielding a degenerate-elliptic space–time formulation solved via Galerkin FEM on time-concatenated stages. Extensive numerical experiments on classic Riemann problems show that the dual scheme reproduces entropy solutions for Burgers while exposing a richer set of weak solutions in the Burgers–HJ setting, with viscosity-regularized base states providing a mechanism to enforce entropy-like behavior. The approach highlights how base states and dual formulations can stabilize and guide the computation of nonlinear hyperbolic equations, with potential connections to optimal transport via the dual structure. Overall, the work introduces a novel, flexible computational pathway for hyperbolic PDEs through a convex dual framework and base-state-driven iterations.

Abstract

We demonstrate the feasibility of a scheme to obtain approximate weak solutions to the (inviscid) Burgers equation in conservation and Hamilton-Jacobi form, treated as degenerate elliptic problems. We show different variants recover non-unique weak solutions as appropriate, and also specific constructive approaches to recover the corresponding entropy solutions.

Figures (20)

• Figure 1: Domain of interest
• Figure 2: (a) A sample $2\times2$ mesh with element numbers at center. Red dashed lines, passing through the nodal points (red dots), represent nodal timelines. $g_{1}$ and $g_{2}$ represent the first and the second Gauss timelines passing through the Gauss points of the elements $1$ and $2$. (b) represents the concatenated domains after truncation. Additionally, $t_f^{(s)} = t_i^{(s+1)}$.
• Figure 3: DtP mapping generated primal field $u$ for the expansion fan. Fig. (c) shows minor overshoots as the fan opens up which may be attributed to the $C^0$ approximation of the dual fields.
• Figure 4: DtP mapping generated primal field $u$ for the shock problem . In Figure (a), the black asterisks represent the shock trajectory based on the exact entropy solution, which is being superimposed on the original plot.
• Figure 5: DtP mapping generated primal field $u$ for a double shock. In Figure (a), the black asterisks represent the shock trajectories for the two shocks based on the exact entropy solution, which is being superimposed on the original plot. The two shocks merge at $(x,t)=(0.625,0.5)$