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

@article{NavarroFernandez2022ErrorEF, title={Error estimates for a finite volume scheme for advection-diffusion equations with rough coefficients}, author={V'ictor Navarro-Fern'andez and Andr{\'e} Schlichting}, journal={ArXiv}, year={2022}, volume={abs/2201.10411} }

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 concerned with advecting velocity fields that are spatially Sobolev regular and data that are merely integrable. We prove that on unstructured regular meshes the rate of convergence of approximate solutions generated by the upwind scheme towards the unique solution of the continuous model is at least one…

## Figures from this paper

## One Citation

### Optimal stability estimates and a new uniqueness result for advection-diffusion equations

- Mathematics
- 2021

This paper contains two main contributions. First, it provides optimal stability estimates for advection-diffusion equations in a setting in which the velocity field is Sobolev regular in the spatial…

## References

SHOWING 1-10 OF 60 REFERENCES

### 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.

### A CELL-CENTERED SECOND-ORDER ACCURATE FINITE VOLUME METHOD FOR CONVECTION–DIFFUSION PROBLEMS ON UNSTRUCTURED MESHES

- Computer Science
- 2004

A MUSCL-like cell-centered finite volume method to approximate the solution of multi-dimensional steady advection–diffusion equations and the slope limiter is designed to guarantee that the discrete solution of the nonlinear scheme exists.

### Convergence Rates for Upwind Schemes with Rough Coefficients

- MathematicsSIAM J. Numer. Anal.
- 2017

The case where the advecting velocity field has spatial Sobolev regularity and initial data are merely integrable is interested, and the rate of weak convergence is at least 1/2 in the mesh size.

### L-ERROR ESTIMATE FOR A FINITE VOLUME APPROXIMATION OF LINEAR ADVECTION

- Mathematics
- 2006

Abstract. We study the convergence of the upwind Finite Volume scheme a pplied to the linear advection equation with a Lipschitz divergence-free speed in Rd. We prove ah1/2−ε-error estimate in…

### Convergence order of upwind type schemes for transport equations with discontinuous coefficients

- Mathematics, Computer Science
- 2016

### Optimal stability estimates and a new uniqueness result for advection-diffusion equations

- Mathematics
- 2021

This paper contains two main contributions. First, it provides optimal stability estimates for advection-diffusion equations in a setting in which the velocity field is Sobolev regular in the spatial…

### A high-order-accurate unstructured mesh finite-volume scheme for the advection-diffusion equation

- Computer Science
- 2002

This paper presents and analyze a new approach for high-order-accurate finite-volume discretization for diffusive fluxes that is based on the gradients computed during solution reconstruction, and introduces a technique for constraining the least-squares reconstruction in boundary control volumes.

### On the finite volume element method

- Mathematics
- 1990

SummaryThe finite volume element method (FVE) is a discretization technique for partial differential equations. It uses a volume integral formulation of the problem with a finite partitioning set of…

### The numerical solution of second-order boundary value problems on nonuniform meshes

- Mathematics, Computer Science
- 1986

It is shown that certain commonly used difference schemes yield second-order accurate solutions despite the fact that their truncation error is of lower order, which illuminates a limitation of the standard stability, consistency proof of convergence for difference schemes defined on nonuniform meshes.

### Analysis of the upwind finite volume method for general initial- and boundary-value transport problems

- Mathematics
- 2012

This paper is devoted to the convergence analysis of the upwind finite volume scheme for the initial and boundary value problem associated to the linear transport equation in any dimension, on…