The obstacle problem for linear scalar conservation laws with constant velocity

TL;DR

This paper introduces a new method for solving obstacle problems in linear scalar conservation laws by dynamically scaling the velocity to respect inequality constraints, ensuring physically reasonable solutions.

Contribution

The paper proposes a novel velocity scaling approach for obstacle problems in conservation laws, including a discontinuous scaling variant, with numerical demonstrations.

Findings

Velocity scaling effectively enforces obstacle constraints.

Discontinuous scaling yields valid solutions to discontinuous conservation laws.

Numerical results confirm the approach's practicality.

Abstract

In this contribution, we present a novel approach for solving the obstacle problem for (linear) conservation laws. Usually, given a conservation law with an initial datum, the solution is uniquely determined. How to incorporate obstacles, i.e., inequality constraints on the solution so that the resulting solution is still "physically reasonable" and obeys the obstacle, is unclear. The proposed approach involves scaling down the velocity of the conservation law when the solution approaches the obstacle. We demonstrate that this leads to a reasonable solution and show that, when scaling down is performed in a discontinuous fashion, we still obtain a suitable velocity - and the solution satisfying a discontinuous conservation law. We illustrate the developed solution concept using numerical approximations.