Error estimates for numerical approximations to scalar conservation laws (Q2781532)

From the conclusion: We prove a couple of results. First, we prove in Chapter 3 that a linear first order finite volume scheme on a regular triangular grid is total variation bounded. The idea is to use a larger stencil than the direct neighbors. This result indicates that one may use a larger stencil as well, when trying to prove a strong \(BV\) estimate on arbitrary unstructured grids. From such an estimate it is standard to deduce optimal error estimates using Kruzkov techniques as we point out in Chapter 2.NEWLINENEWLINENEWLINEThen, we introduce a new technique to prove a priori error estimates. Remarkable about this technique is that the consistency error in the sense of mean values of the entropy weak solution is involved. This consistency error is measured in some negative norm. It is possible to exploit the conservation form of the schemes under consideration in this negative norm. The stability of the schemes is measured, as usual, in \(BV\). By interpolating between these two spaces we have been able to recover the same error estimates proven with Kruzkov/Kuznetsov techniques. However, it is the first time that a consistency error has come up in the error analysis of finite volume schemes approximating scalar hyperbolic problems. Using these guidelines of proof one may try to prove error estimates for higher order schemes like \(MUSCL\)- or \(ENO\)-schemes since the consistency error for such schemes is indeed of higher order (than for first order schemes) if one has assumed some piecewise regularity of the entropy weak solution. However, it is difficult to prove the stability estimates, that are needed in this technique, for these schemes.NEWLINENEWLINENEWLINEBy using a perturbation argument one can get the same rate of convergence for certain \(MUSCL\) schemes as for first order approximations. We prove this for a second order \(MUSCL\) scheme combined with a second order \(TVD\)-Runge-Kutta time evolution in Chapter 6. For this result we employ Kruzkov techniques. Of course, this result is not optimal. However, it tells us that the schemes under consideration converge at least as fast as first order schemes. NEWLINENEWLINENEWLINEA different technique for constructing higher order schemes is the use of staggered grids. The advantage of using staggered grids is that some regularity of solutions to Riemann problems can be exploited. However, before analyzing those higher order schemes, one has to analyze first order schemes on such staggered grids which has been an open problem. In Chapter 7 we give a complete a priori and a posteriori error analysis for the staggered Lax-Friedrichs scheme on a very general class of unstructured grids in any spatial dimension. The idea is to reinterpret this scheme and to link it to a scheme on a fixed grid. The a posteriori error estimate for the staggered Lax-Friedrichs scheme should be tested numerically. NEWLINENEWLINENEWLINEFinally, we study implicit finite volume discretizations. It turns out that it is problematic to use them for the approximation of non-stationary solutions. In many situations the amount of numerical work needed for the calculation is at most as large as the work needed when using an explicit scheme while the accuracy is even worse compared to explicit schemes.