nwhm/limits

Many nonlinear conservation laws develop shocks. In one dimension vanishing viscosity limits are often used to develop unique weak solutions after shock formation. For conservation laws which come from mechanics, it is often assumed that at a shock, energy is dissipated into heat. For multidimensional conservation laws, the correct way to do this is unclear. Viscosity solutions can be obtained numerically by ensuring that the error in the approximation of the numerical solutions is dissipative. Other solutions may however be possible, and so the choice of numerical method seems important when the solution exists but is not unique.

One might attempt to obtain other solutions of the conservation law after shock formation by considering vanishing dispersion or vanishing capillarity limits. For one dimensional conservation laws, this has been investigated by Schonbek and by Bona and Schonbek:

J.L. Bona and M.E. Schonbek “Travelling-wave solutions to the Korteweg-de Vries-Burgers equation.” Proc. Roy. Soc. Edinburgh Sect. A 101 (1985), no. 3-4, 207–226. Mathscinet review

M.E. Schonbek “Convergence of solutions to nonlinear dispersive equations.” Comm. Partial Differential Equations 7 (1982), no. 8, 959–1000. Mathscinet review

M.E.Schonbek “Second-order conservative schemes and the entropy condition.” Math. Comp. 44 (1985), no. 169, 31–38. Mathscinet review

It is possible to construct numerical schemes without dissipation. Some of the theory for this is in:

B. Leimkuhler and S. Reich “Simulating Hamiltonian dynamics.” Cambridge Monographs on Applied and Computational Mathematics, 14. Cambridge University Press, Cambridge, 2004 Mathscinet review Ebook

Typically numerical schemes without dissipation have dispersive errors. Lax:

P.D. Lax “Weak solutions of nonlinear hyperbolic equations and their numerical computation” Comm. Pure Appl. Math. 7, (1954). 159–193. Mathscinet review

has shown that well chosen finite difference schemes can converge to viscosity solutions for scalar conservation laws by using the maximum principle. The study above is in both Eulerian and Lagrangian coordinates but is limited to one dimensional systems. A related question is what occurs when there is dispersive regularization or a dispersive error in the numerical approximation of a hyperbolic conservation law. This question has been studied by among others,

P.D. Lax “On dispersive difference schemes” Phys. D 18 (1986), no. 1-3, 250–254. Mathscinet review

J. Goodman and P.D. Lax “On dispersive difference schemes. I.” Comm. Pure Appl. Math. 41 (1988), no. 5, 591–613. Mathscinet review

In these studies it is shown that a dispersive regularization of Burger’s equation does not tend to a weak solution after the shock is formed. At present, extension of this work to multidimensional systems of hyperbolic conservation laws still requires work. Studies of weak dispersion limits for dispersive regularizations of Burgers equation are in:

P.D. Lax and C. Levermore “The small dispersion limit of the Korteweg-de Vries equation”. I, II, III, Comm. Pure Appl. Math. 36 (1983), 253-290, 571-593, 809-829. Mathscinet review

and

S. Venakides, “The zero dispersion limit of the Korteweg-de Vries equation with periodic initial data” Trans. Amer. Math. Soc. 301 (1987), no. 1, 189–226. Mathscinet review

In many physical systems, there is both dispersion and dissipation. Recent work for the one dimensional shallow water system seems to show that a well chosen combination of the two seems to give better numerical results for ocean/lake dynamics. A numerical implementation and a summary of a model are in:

Bonneton, P.; Chazel, F.; Lannes, D.; Marche, F.; Tissier, M. A splitting approach for the fully nonlinear and weakly dispersive Green-Naghdi model. J. Comput. Phys. 230 (2011), no. 4, 1479–1498. Mathscinet review

Some example simulations can be found on David Lannes webpage under simulation numerique. There has not yet been a two dimensional extension of this work.