A priori error estimates for numerical methods for scalar conservation laws. Part II : flux-splitting monotone schemes on irregular Cartesian grids

Abstract

Abstract. This paper is the second of a series in which a general theory of a priori error estimates for scalar conservation laws is constructed. In this paper, we focus on how the lack of consistency introduced by the nonuniformity of the grids influences the convergence of flux-splitting monotone schemes to the entropy solution. We obtain the optimal rate of convergence of (∆x) 1/2 in L ∞ (L 1 ) for consistent schemes in arbitrary grids without the use of any regularity property of the approximate solution. We then extend this result to less consistent schemes, called p−consistent schemes, and prove that they converge to the entropy solution with the rate of (∆x) min{1/2,p} in L ∞ (L 1 ); again, no regularity property of the approximate solution is used. Finally, we propose a new explanation of the fact that even inconsistent schemes converge with the rate of (∆x) 1/2 in L ∞ (L 1 ). We show that this well-known supraconvergence phenomenon takes place because the consistency of the numerical flux and the fact that the scheme is written in conservation form allows the regularity properties of its approximate solution (total variation boundedness) to compensate for its lack of consistency; the nonlinear nature of the problem does not play any role in this mechanism. All the above results hold in the multidimensional case, provided the grids are Cartesian products of one-dimensional nonuniform grids.

Adaptive multiscale scheme based on numerical density of entropy production for conservation laws

“…Let us also note that the apparent loss of consistency on the local truncation error seems not to affect the actual error of the scheme (as we will see in subsection 5.1.1, see also [4,22] for interesting issues). On the contrary, Puppo and Semplice [20] do not lose consistency and conservativity because their procedure does not produce the mass and the fluxes are well computed at the cell interface between the two levels of refinement.…”

Analysis of space–time discontinuous Galerkin method for nonlinear convection–diffusion problems

“…The technique applied in this paper can be extended to the case the discontinuous Galerkin time semidiscretization combined with the hp discontinuous Galerkin space discretization.The error estimates have been obtained with the aid of a "parabolic machinery" for problems with "dominating diffusion". This means that our technique is not applicable to conservation laws and does not allow to obtain results similar to [14][15][16], where the finite difference or finite volume methods were used. There are the following subjects for further work:…”

Convergence of an explicit finite volume scheme for first order symmetric systems

“…Generally, as noticed before, it can just be proved a weak BV estimate, which in our case corresponds to the estimate (3.9). This non existence of an a-priori uniform BV estimate is invoked to explain the O(h 1/4 ) convergence rate (see [5,7,14,24]), instead of O(h 1/2 ) estimate, as in the one-dimensional case. Note, however, that in a more recent work, Cockburn and Gremaud [7] have proved that it is possible to get a O(h 1/2 ) for a class of generalized finite difference schemes, without using a strong BV estimate.…”

“…Note, however, that in a recent work of Cockburn and Gremaud (see [7] and subsequent papers), a O(h 1/2 ) convergence rate has been established under some particular assumptions on the mesh and the scheme.…”