Error estimate for the upwind finite volume method for the nonlinear scalar conservation law
Journal of Computational and Applied Mathematics
Applied Numerical Mathematics
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We consider a general class of finite volume schemes on unstructured but quasi-uniform meshes for first-order systems of hyperbolic balance laws on unstructured meshes. Provided the system is equipped with at least one entropy-entropy flux tuple and the associated Cauchy problem allows for a classical solution $u$ we give conditions such that the finite volume approximation $u_h$ converges to $u$ if the mesh parameter $h$ tends to zero. In fact we prove an error estimate of the form ${\|u-u_h\|}_{L^2} \le C \sqrt{h}$, where $C$ is independent of $h$. The proof relies on a stability result for classical solutions in the class of entropy solutions due to Dafermos [Arch. Rational Mech. Anal., 94 (1979), pp. 373-389] and DiPerna [Indiana Univ. Math. J., 28 (1979), pp. 137-188]. Finally, we present examples such that the conditions to apply the general convergence estimate can be satisfied (at least in part). The examples cover general scalar equations, weakly coupled systems, and the system of elastodynamics in one dimension. Moreover, we generalize the concept of entropy conservative methods due to Tadmor [Math. Comp., 49 (1987), pp. 91-103] and show how this can be used to establish the convergence of finite volume methods for the system's case.