# Convergence Rates for Upwind Schemes with Rough Coefficients

@article{Schlichting2017ConvergenceRF, title={Convergence Rates for Upwind Schemes with Rough Coefficients}, author={Andr{\'e} Schlichting and Christian Seis}, journal={SIAM J. Numer. Anal.}, year={2017}, volume={55}, pages={812-840} }

This paper is concerned with the numerical analysis of the explicit upwind finite volume scheme for numerically solving continuity equations. We are interested in the case where the advecting velocity field has spatial Sobolev regularity and initial data are merely integrable. We estimate the error between approximate solutions constructed by the upwind scheme and distributional solutions of the continuous problem in a Kantorovich--Rubinstein distance, which was recently used for stability…

## 19 Citations

Analysis of the implicit upwind finite volume scheme with rough coefficients

- MathematicsNumerische Mathematik
- 2018

It is proved that on unstructured regular meshes the rate of convergence of approximate solutions generated by the upwind scheme towards the unique distributional solution of the continuous model is at least 1/2.

Error estimates for a finite volume scheme for advection-diffusion equations with rough coefficients

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- 2022

We study the implicit upwind finite volume scheme for numerically approximating the advection-diffusion equation with a vector field in the low regularity DiPerna-Lions setting. That is, we are…

Large time behaviors of upwind schemes and B-schemes for Fokker-Planck equations on ℝ by jump processes

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- 2020

This work revisits some standard schemes, including upwind schemes and some B-schemes, for linear conservation laws from the viewpoint of jump processes, and establishes the uniform exponential convergence to the steady states of these schemes.

Convergence analysis of upwind type schemes for the aggregation equation with pointy potential

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- 2017

A numerical analysis of upwind type schemes for the nonlinear nonlocal aggregation equation is provided. In this approach, the aggregation equation is interpreted as a conservative transport equation…

Implicit MAC scheme for compressible Navier–Stokes equations: low Mach asymptotic error estimates

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- 2020

We investigate error between any discrete solution of the implicit Marker and Cell (MAC) numerical scheme for compressible Navier-Stokes equations in low Mach number regime and an exact strong…

Eulerian and Lagrangian Solutions to the Continuity and Euler Equations with L1 Vorticity

- MathematicsSIAM J. Math. Anal.
- 2017

This paper addresses a question that arose in \cite{FilhoMazzucatoNussenzveig06}, namely whether 2D Euler solutions obtained via vanishing viscosity are renormalized when the initial data has low integrability.

Optimal stability estimates for continuity equations

- MathematicsProceedings of the Royal Society of Edinburgh: Section A Mathematics
- 2018

This review paper is concerned with the stability analysis of the continuity equation in the DiPerna–Lions setting in which the advecting velocity field is Sobolev regular. Quantitative estimates for…

Convergence of Numerical Approximations to Non-linear Continuity Equations with Rough Force Fields

- Mathematics, Computer ScienceArchive for Rational Mechanics and Analysis
- 2019

We prove quantitative regularity estimates for the solutions to non-linear continuity equations and their discretized numerical approximations on Cartesian grids when advected by a rough force field.…

Convergence of a multidimensional Glimm-like scheme for the transport of fronts

- Mathematics
- 2020

In the present paper a convergence result for this scheme is provided for a particular class of multi-dimensional problems.

Second-order finite difference approximations of the upper-convected time derivative

- MathematicsSIAM J. Numer. Anal.
- 2021

New finite difference schemes are presented for dealing with the upper-convected time derivative in the context of the generalized Lie derivative to obtain approximations of second-order in time for solving a simplified constitutive equation in one and two dimensions.

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