Abstract
We consider the approximation by multidimensional finite volume schemes of the transport of an initial measure by a Lipschitz flow. We first consider a scheme defined via characteristics, and we prove the convergence to the continuous solution, as the time step and the ratio of the space step to the time step tend to zero. We then consider a second finite volume scheme, obtained from the first one by addition of some uniform numerical viscosity. We prove that this scheme converges to the continuous solution, as the space step tends to zero, whereas the ratio of the space step to the time step remains bounded by below and by above, and under assumption of uniform regularity of the mesh. This is obtained via an improved discrete Sobolev inequality and a sharp weak BV estimate, under some additional assumptions on the transport flow. Examples show the optimality of these assumptions.
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References
Abia L.M., Angulo O., López-Marcos J.C.: Size-structured population dynamics models and their numerical solutions. Discrete Contin. Dyn. Syst. Ser. B 4(4), 1203–1222 (2004)
Ackleh A.S., Ito K.: An implicit finite difference scheme for the nonlinear size-structured population model. Numer. Funct. Anal. Optim., 18(9–10), 865–884 (1997)
Ambrosio L.: Transport equation and Cauchy problem for BV vector fields. Invent. Math. 158(2), 227–260 (2004)
L. Ambrosio, G. Crippa, C. De Lellis, F. Otto, and M. Westdickenberg. Transport equations and multi-D hyperbolic conservation laws, volume 5 of Lecture Notes of the Unione Matematica Italiana. Springer-Verlag, Berlin, 2008. Lecture notes from the Winter School held in Bologna, January 2005, Edited by Fabio Ancona, Stefano Bianchini, Rinaldo M. Colombo, De Lellis, Andrea Marson and Annamaria Montanari.
L. Ambrosio, A. Figalli, G. Friesecke, T. Paul, and J. Giannoulis. Semiclassical limit of quantum dynamics with rough potentials and well posedness of transport equations with measure initial data. 2010.
S. Bianchini and M. Gloyer. An estimate on the flow generated by monotone operators. preprint, 2010.
Bouchut F., James F.: One-dimensional transport equations with discontinuous coefficients. Nonlinear Anal. 32(7), 891–933 (1998)
BouchutF. James F., Mancini S.: Uniqueness and weak stability for multi-dimensional transport equations with one-sided Lipschitz coefficient. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4(1), 1–25 (2005)
Cocozza-Thivent C., Eymard R., Mercier S.: A finite-volume scheme for dynamic reliability models. IMA J. Numer. Anal. 26(3), 446–471 (2006)
Davis M.: Piecewise-deterministic markov processes: a general class of non-diffusion stochastic models. Journal of the Royal Statistical Society, Series B 46, 353–388 (1984)
Eymard R., Mercier S., Prignet A.: An implicit finite volume scheme for a scalar hyperbolic problem with measure data related to piecewise deterministic markov processes. Journal of Computational and Applied Mathematics 222(2), 293–323 (2008)
I. Fonseca and W. Gangbo. Degree theory in analysis and applications, volume 2 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press Oxford University Press, New York, 1995. Oxford Science Publications.
Gosse L., James F.: Numerical approximations of one-dimensional linear conservation equations with discontinuous coefficients. Math. Comp. 69(231), 987–1015 (2000)
Maniglia S.: Probabilistic representation and uniqueness results for measure-valued solutions of transport equations. J. Math. Pures Appl. (9) 87(6), 601–626 (2007)
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Bouchut, F., Eymard, R. & Prignet, A. Finite volume schemes for the approximation via characteristics of linear convection equations with irregular data. J. Evol. Equ. 11, 687–724 (2011). https://doi.org/10.1007/s00028-011-0106-2
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DOI: https://doi.org/10.1007/s00028-011-0106-2