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Approximation of viscous transport and conservative equations with one sided Lipschitz velocity fields

Fabio Camilli, Adriano Festa, Luciano Marzufero

TL;DR

This work develops semi-Lagrangian schemes on unstructured grids for viscous transport ($-$∂t u$-term with $a$) and viscous conservative ($\,partial_t f$) equations with measurable coefficients under a one-sided Lipschitz condition ($OSLC$). By leveraging probabilistic representations and the duality between time-measurable viscosity solutions and duality measure-valued solutions, the authors prove convergence of the schemes to the correct continuous limits; the transport scheme uses a semi-discrete-in-time approach and Barles–Souganidis stability, while the conservative scheme relies on discrete duality arguments and Wasserstein contractions. Numerical experiments in 1D and 2D validate convergence and robustness on unstructured meshes, including discontinuous velocity fields and variable diffusion, highlighting applicability to high-dimensional and geometrically complex domains and to mean-field game contexts. The framework extends classical semi-Lagrangian methods beyond the DiPerna–Lions setting and suggests future work on fully nonlinear second-order equations and coupled systems.

Abstract

The aim of this work is to investigate semi-Lagrangian approximation schemes on unstructured grids for viscous transport and conservative equations with measurable coefficients that satisfy a one-sided Lipschitz condition. To establish the convergence of the schemes, we exploit the characterization of the solution for these equations expressed in terms of measurable time-dependent viscosity solution and, respectively, duality solution. We supplement our theoretical analysis with various numerical examples to illustrate the features of the schemes.

Approximation of viscous transport and conservative equations with one sided Lipschitz velocity fields

TL;DR

This work develops semi-Lagrangian schemes on unstructured grids for viscous transport (∂t ua\,partial_t fOSLC$). By leveraging probabilistic representations and the duality between time-measurable viscosity solutions and duality measure-valued solutions, the authors prove convergence of the schemes to the correct continuous limits; the transport scheme uses a semi-discrete-in-time approach and Barles–Souganidis stability, while the conservative scheme relies on discrete duality arguments and Wasserstein contractions. Numerical experiments in 1D and 2D validate convergence and robustness on unstructured meshes, including discontinuous velocity fields and variable diffusion, highlighting applicability to high-dimensional and geometrically complex domains and to mean-field game contexts. The framework extends classical semi-Lagrangian methods beyond the DiPerna–Lions setting and suggests future work on fully nonlinear second-order equations and coupled systems.

Abstract

The aim of this work is to investigate semi-Lagrangian approximation schemes on unstructured grids for viscous transport and conservative equations with measurable coefficients that satisfy a one-sided Lipschitz condition. To establish the convergence of the schemes, we exploit the characterization of the solution for these equations expressed in terms of measurable time-dependent viscosity solution and, respectively, duality solution. We supplement our theoretical analysis with various numerical examples to illustrate the features of the schemes.
Paper Structure (11 sections, 14 theorems, 114 equations, 9 figures)

This paper contains 11 sections, 14 theorems, 114 equations, 9 figures.

Table of Contents

  1. Introduction
  2. The model problem
  3. The viscous transport equation
  4. The conservative equation
  5. The numerical scheme
  6. The numerical scheme for the viscous transport equation
  7. The numerical scheme for the conservative equation
  8. Convergence of the scheme for the viscous transport equation
  9. Convergence of the scheme for the conservative equation
  10. Numerical illustration
  11. Conclusions

Key Result

Proposition 2.1

For every $(t,x) \in [0,T] \times \mathbb{R}^d$ and $\mathbb{P}$-almost surely, there exists a unique strong solution $\Phi_{s,t}(x)$ of eq:SDE defined on $[t,T] \times \mathbb{R}^d$. For all $p \in [2,\infty)$, there exists a constant $C = C_{p} > 0$ depending only on the assumptions hyp:OSLC and h and for all $t \in [0,T]$, $s_1,s_2 \in [t,T]$, and $x \in \mathbb{R}^d$. Moreover, for all $0 \le

Figures (9)

  • Figure 1: Test 1. (Forward) Conservative equation. The discretization parameters are $\Delta = (\Delta x, h) = (0.02, 0.06)$.
  • Figure 2: Test 1. (Backward) Inviscid transport equation. The initial solution is chosen as the cumulative distribution function of the exact solution for the conservative case. Comparison between exact and numerical solutions at various times. Discretization parameters: $\Delta = (\Delta x, h) = (0.02, 0.06)$.
  • Figure 3: Test 1. Comparison of convergence rates in $L^\infty$ and Wasserstein norms for the two numerical schemes.
  • Figure 4: Test 2. Triangulation used ($\Delta x=0.08$) and vector field $b$.
  • Figure 5: Test 2. Solution of the transport equation at various moments of its evolution.
  • ...and 4 more figures

Theorems & Definitions (29)

  • Proposition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Theorem 2.4
  • Corollary 2.5
  • Remark 2.6
  • Definition 2.7
  • Theorem 2.8
  • Remark 2.9
  • Remark 3.1
  • ...and 19 more