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Error Estimates for Discontinuous Galerkin Approximations to the Vlasov-Unsteady Stokes System

Harsha Hutridurga, Krishan Kumar, Amiya K. Pani

TL;DR

This work provides a rigorous, structure-preserving numerical framework for a coupled Vlasov-unsteady Stokes system describing thin sprays. It develops a semi-discrete discontinuous Galerkin scheme in space (DG-DG) and employs a Stokes projection for the fluid part together with a specialized projection for the kinetic equation, complemented by a Lie-Trotter time-splitting. The authors prove uniqueness of strong solutions in $3$D, derive optimal $L^2$-error rates for the density and velocity fields in $2$D, and extend the discussion to $3$D with corresponding adjustments; numerical experiments corroborate the theoretical predictions. This work advances reliable, conservation-preserving simulations for kinetic-fluid spray models and informs error-controlled development of Vlasov-Navier–Stokes-type solvers.

Abstract

In the first part of this paper, uniqueness of strong solution is established for the Vlasov-unsteady Stokes problem in 3D. The second part deals with a semi discrete scheme, which is based on the coupling of discontinuous Galerkin approximations for the Vlasov and the Stokes equations for the 2D problem. The proposed method is both mass and momentum conservative. Based on a special projection and also the Stokes projection, optimal error estimates in the case of smooth compactly supported initial data are derived. Moreover, the generalization of error estimates to 3D problem is also indicated. Finally, based on time splitting algorithm, some numerical experiments are conducted whose results confirm our theoretical findings.

Error Estimates for Discontinuous Galerkin Approximations to the Vlasov-Unsteady Stokes System

TL;DR

This work provides a rigorous, structure-preserving numerical framework for a coupled Vlasov-unsteady Stokes system describing thin sprays. It develops a semi-discrete discontinuous Galerkin scheme in space (DG-DG) and employs a Stokes projection for the fluid part together with a specialized projection for the kinetic equation, complemented by a Lie-Trotter time-splitting. The authors prove uniqueness of strong solutions in D, derive optimal -error rates for the density and velocity fields in D, and extend the discussion to D with corresponding adjustments; numerical experiments corroborate the theoretical predictions. This work advances reliable, conservation-preserving simulations for kinetic-fluid spray models and informs error-controlled development of Vlasov-Navier–Stokes-type solvers.

Abstract

In the first part of this paper, uniqueness of strong solution is established for the Vlasov-unsteady Stokes problem in 3D. The second part deals with a semi discrete scheme, which is based on the coupling of discontinuous Galerkin approximations for the Vlasov and the Stokes equations for the 2D problem. The proposed method is both mass and momentum conservative. Based on a special projection and also the Stokes projection, optimal error estimates in the case of smooth compactly supported initial data are derived. Moreover, the generalization of error estimates to 3D problem is also indicated. Finally, based on time splitting algorithm, some numerical experiments are conducted whose results confirm our theoretical findings.

Paper Structure

This paper contains 15 sections, 21 theorems, 190 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Certain aspects of the continuum model
  3. Qualitative and quantitative aspects of the model problem
  4. Existence and Uniqueness result in $3$D
  5. Discontinuous Galerkin approximations
  6. Semi-discrete dG formulation
  7. Some properties of the discrete solution
  8. A priori Error Estimates
  9. Error estimates for Stokes system
  10. Error estimates for the kinetic equation
  11. Special Projection:
  12. Optimal error estimates:
  13. Comment on $3$D
  14. Numerical Experiments
  15. Conclusion

Key Result

Lemma 2.1

For $\bm{u} \in L^1(0,T;\bm{L}^\infty),$ if $\sup_{C^r_{t,v}}f_0 \in L^\infty_{loc}\left({\mathbb R}_+;L^1({\mathbb R}^d)\right)$, where $C^r_{t,v} := \Omega_x \times B(e^tv,r), \, \forall\, r > 0$ and $B(e^tv,r)$ denotes the ball centered at $e^tv$ having radius $r$, then, the following estimate ho

Figures (2)

  • Figure 1: Convergence rates for the distribution function $f$, for the velocity $\bm{u}$ and for the pressure $p$ in the Example \ref{['exm-1-vus']}.
  • Figure 2: Convergence rates for the distribution function $f$, for the velocity $\bm{u}$ and for the pressure $p$ in the Example \ref{['exm-2-vus']}.

Theorems & Definitions (39)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Remark 2.5
  • Remark 2.6
  • Lemma 2.7
  • proof
  • ...and 29 more