Finite volume schemes for nonhomogeneous scalar conservation laws: error estimate

Abstract

In this paper, we study finite volume schemes for the nonhomogeneous scalar conservation law u t + divF (x, t, u) = q(x, t, u) with initial condition u(x, 0) = u 0 (x). The source term may be either stiff or nonstiff. In both cases, we prove error estimates between the approximate solution given by a finite volume scheme (the scheme is totally explicit in the nonstiff case, semi-implicit in the stiff case) and the entropy solution. The order of these estimates is h 1 4 in space-time L 1 -norm (h denotes the size of the mesh). Furthermore, the error estimate does not depend on the stiffness of the source term in the stiff case.

Error analysis of a dynamic model adaptation procedure for nonlinear hyperbolic equations

“…It is obviously the same when one aims at deriving error estimates, see [22,10,24,3,17,8,20,9,12,23] and Appendix A.2. The main tool we use is an estimate when considering two conservation laws with different flux functions.…”

“…Adaptive modeling. We aim now at solving the coarse model (9) in the domain where u c is close to u f , and to turn back to the resolution of the system (7) in the zones where u c is not a satisfactory approximation of u f . To do so, we will introduce a time-dependent partition of R, i.e:…”

Numerical approximation of entropy solutions for hyperbolic integro-differential equations

“…The proof relies on the seminal work of Kruzhkov and Kuznetsov which has been used to prove convergence(rates) in the homogeneous case ( [2,7,13,16,19]) and in the inhomogeneous case with local source terms ( [1,10,17]). The first main additional difficulty and novelty in our case is the global character of the integral operator T. As noted above we loose the basic property of the exact solutions of the equation (1) in the homogeneous case: the finite speed of propagation.…”

“…We use for the sake of brevity the notation n,(j,l) ≡ [1], Proposition 3. If we proceed as in the proof of this proposition using our Assumption 2, the L ∞ -estimate (25), and (26) we arrive at…”

First and second order error estimates for the Upwind Source at Interface method

“…Several results concerning the convergence analysis of numerical approximations for scalar conservation laws are inspired by this fundamental theory (see references in [29] and [3], [11], [19], [26], [27], for instance). Another approach to the global existence of weak solutions for hyperbolic systems of balance laws is presented in [8], which is also referred to [12] with some extensions.…”