Convergence order of upwind type schemes for transport equations with discontinuous coefficients
Provides a rigorous convergence rate for numerical schemes solving transport equations with discontinuous coefficients, a problem relevant to computational fluid dynamics and hyperbolic conservation laws.
The paper proves that the upwind scheme for transport equations with discontinuous coefficients converges with order 1/2 in Wasserstein distance, and shows this rate is optimal. The result also extends to other diffusive first-order schemes and a forward semi-Lagrangian scheme.
An analysis of the error of the upwind scheme for transport equation with discontinuous coefficients is provided. We consider here a velocity field that is bounded and one-sided Lipschitz continuous. In this framework, solutions are defined in the sense of measures along the lines of Poupaud and Rascle's work. We study the convergence order of the upwind scheme in the Wasserstein distances. More precisely, we prove that in this setting the convergence order is 1/2. We also show the optimality of this result. In the appendix, we show that this result also applies to other "diffusive" "first order" schemes and to a forward semi-Lagrangian scheme.