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Convergence and error estimates in finite volume schemes for general multidimensional scalar conservation laws. I. Explicite monotone schemes. (English) Zbl 0823.65087

Convergence of explicit finite volume schemes for scalar conservation laws in several space dimensions is studied. The triangulation is regular and the numerical solution is piecewise constant in space and time. It is assumed that the flows across cell faces lead to a monotone scheme when applied to a one-dimensional scalar conservation law. The main result is an error estimate for problems where the initial condition \(u^ 0\) belongs to BV. This estimate gives a \(h^{1/4}\) convergence rate in \(L_ 1\) if \(u^ 0\) has compact support. In addition uniform convergence in time is obtained if \(u^ 0 \in L^ \infty \cap L^ 1\).
These results are obtained following the approach of N. N. Kuznetsov [Zh. Vychisl. Mat. Mat. Fiz 16, 1489-1502 (1976; Zbl 0354.35021)] and do not involve measured valued solutions. It is indicated how the results could be extended to general \(E\)-schemes.

MSC:

65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
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References:

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