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

@article{Boyer2012AnalysisOT,
title={Analysis of the upwind finite volume method for general initial- and boundary-value transport problems},
author={Franck Boyer},
journal={Ima Journal of Numerical Analysis},
year={2012},
volume={32},
pages={1404-1439}
}
• F. Boyer
• Published 1 October 2012
• Mathematics
• Ima Journal of Numerical Analysis
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 general unstructured meshes. We are particularly interested in the case where the initial and boundary data are in $L^\infty$ and the advection vector field $v$ has low regularity properties, namely $v\in L^1(]0,T[,(W^{1,1}(\O))^d)$, with suitable assumptions on its divergence. In this general framework…

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## References

SHOWING 1-10 OF 31 REFERENCES
Trace theorems and spatial continuity properties for the solutions of the transport equation
This paper is first concerned with the trace problem for the transport equation. We prove the existence and the uniqueness of the traces as well as the well-posedness of the initial and boundary
Error Estimate and the Geometric Corrector for the Upwind Finite Volume Method Applied to the Linear Advection Equation
• Mathematics, Computer Science
SIAM J. Numer. Anal.
• 2005
It is proved that if the continuous solution is regular enough and if the norm of this corrector is bounded by the mesh size, then an order one error estimate for the finite volume scheme occurs.
Optimal Convergence of the Original DG Method on Special Meshes for Variable Transport Velocity
• Computer Science
SIAM J. Numer. Anal.
• 2010
We prove optimal convergence rates for the approximation provided by the original discontinuous Galerkin method for the transport-reaction problem. This is achieved in any dimension on meshes related
Lax theorem and finite volume schemes
A non-consistent model problem posed in an abstract Banach space is proved to be convergent and various examples show that the functional framework is non-empty.
Probabilistic Analysis of the Upwind Scheme for Transport Equations
• Mathematics
• 2007
We provide a probabilistic analysis of the upwind scheme for d-dimensional transport equations. We associate a Markov chain with the numerical scheme and then obtain a backward representation formula
Convergence of an explicit finite volume scheme for first order symmetric systems
• Mathematics
Numerische Mathematik
• 2003
A new technique is proposed, which takes advantage of the linearity of the problem, and consists in controlling the approximation error ∥u−uh∥L2 by an expression of the form −2, where u is the exact solution, g is a particular smooth function, and μh, νh are some linear forms depending on the approximate solution uh.
Convergence of an upstream finite volume scheme for a nonlinear hyperbolic equation on a triangular mesh
• Mathematics
• 1993
SummaryWe study here the discretisation of the nonlinear hyperbolic equationut+div(vf(u))=0 in ℝ3 × ℝ+, with given initial conditionu(.,0)=u0(.) in ℝ2, wherev is a function from ℝ2 × ℝ+ to ℝ2 such
Convergence of the Discontinuous Galerkin Method for Discontinuous Solutions
It is shown that approximate solutions computed using the discontinuous Galerkin method will converge in $\LtwoLtwo$ when the coefficients v and a and data f satisfy the minimal assumptions required to establish existence and uniqueness of solutions.
A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation
A numerical example is presented which shows that the known $L^2$ error estimate $\left\| {u - u^h } \right\| \leqq Ch^{k + {1 / 2}} \left\| u \right\|_{k + 1}$ for the discontinuous Galerkin method