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An explicit Euler method for Sobolev vector fields with applications to the continuity equation on non cartesian grids

Tommaso Cortopassi

TL;DR

The paper develops an explicit Euler-type Lagrangian scheme to approximate the regular Lagrangian flow of Sobolev vector fields and uses this to obtain quantitative, non-Cartesian-grid approximations of the continuity equation's density. A novel L^p stability estimate between the true regular Lagrangian flow and the Euler approximation enables two density-approximation strategies: a probabilistic scheme on unstructured meshes, yielding Wasserstein and logarithmic Wasserstein convergence rates in expectation, and a deterministic, diffuse scheme based on a mean-flow construction that achieves order-1 convergence in the logarithmic Wasserstein metric for p>d. The results advance numerical analysis for transport equations with low-regularity velocity fields, removing CFL constraints and enabling parallelizable, mesh-agnostic schemes with provable convergence in Wasserstein distances. Open questions include extending results to 1<p≤d in deterministic settings and exploring stronger convergence notions or couplings with other PDEs.

Abstract

We prove a novel stability estimate in $L^\infty _t (L^p _x)$ between the regular Lagrangian flow of a Sobolev vector field and a piecewise affine approximation of such flow. This approximation of the flow is obtained by a (sort of) explicit Euler method, and it is the crucial tool to prove approximation results for the solution of the continuity equation by using the representation of the solution as the push-forward via the regular Lagrangian flow of the initial datum. We approximate the solution in two ways, one probabilistic and one deterministic, using different approximations for both the flow and the initial datum. Such estimates for the solution of the continuity equation are derived on non Cartesian grids and without the need to assume a CFL condition.

An explicit Euler method for Sobolev vector fields with applications to the continuity equation on non cartesian grids

TL;DR

The paper develops an explicit Euler-type Lagrangian scheme to approximate the regular Lagrangian flow of Sobolev vector fields and uses this to obtain quantitative, non-Cartesian-grid approximations of the continuity equation's density. A novel L^p stability estimate between the true regular Lagrangian flow and the Euler approximation enables two density-approximation strategies: a probabilistic scheme on unstructured meshes, yielding Wasserstein and logarithmic Wasserstein convergence rates in expectation, and a deterministic, diffuse scheme based on a mean-flow construction that achieves order-1 convergence in the logarithmic Wasserstein metric for p>d. The results advance numerical analysis for transport equations with low-regularity velocity fields, removing CFL constraints and enabling parallelizable, mesh-agnostic schemes with provable convergence in Wasserstein distances. Open questions include extending results to 1<p≤d in deterministic settings and exploring stronger convergence notions or couplings with other PDEs.

Abstract

We prove a novel stability estimate in between the regular Lagrangian flow of a Sobolev vector field and a piecewise affine approximation of such flow. This approximation of the flow is obtained by a (sort of) explicit Euler method, and it is the crucial tool to prove approximation results for the solution of the continuity equation by using the representation of the solution as the push-forward via the regular Lagrangian flow of the initial datum. We approximate the solution in two ways, one probabilistic and one deterministic, using different approximations for both the flow and the initial datum. Such estimates for the solution of the continuity equation are derived on non Cartesian grids and without the need to assume a CFL condition.
Paper Structure (7 sections, 18 theorems, 143 equations)

This paper contains 7 sections, 18 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Stability estimates
  3. Singular probabilistic approximation of $\rho$
  4. Diffuse deterministic approximation of $\rho$
  5. Final discussion, questions and future perspectives
  6. Appendix
  7. Acknowledgments

Key Result

Theorem 1.1

(Cauchy-Lipschitz) There exist many different versions of this theorem. The boundedness of $u$ is not usually required, but we assume it since in the following the velocity $u$ that we consider will always be bounded. Let $u : [0,T] \times \mathbb{R}^d \to \mathbb{R}^d$ be bounded, continuous in $t exists for all $t \in [0, T]$ and it is unique.

Theorems & Definitions (48)

  • Theorem 1.1
  • Definition 1.1
  • Remark 1
  • Theorem 2.1
  • Definition 2.1
  • Proposition 2.1
  • proof
  • Remark 2
  • Remark 3: The choice of $\delta$
  • Corollary 2.1
  • ...and 38 more