Convergence of the Godunov scheme for straight line systems

Here A(u) is a map from a domain U ⊂ Rn into Rn×n, and (x, t) ∈ R×R+. We assume strict hyperbolicity, i. e. the the matrix A(u) has n real and strictly different eigenvalues for each u ∈ U . In the conservative case when A(u) = Df(u) for some map f : U → Rn, Glimm [12] proved global existence of weak entropy solutions of (1) when the data has small total variation and each characteristic field is genuinely nonlinear or linearly degenerate. The recent papers [4], [6], [7], [8] have established the uniqueness and L1 stability of solutions obtained by the Glimm scheme or by wave-front tracking. A major open question is whether other methods of approximation, such as vanishing viscosity, relaxation and finite difference schemes, yield the same solutions. In the scalar case all these methods give the unique entropy solution. Some results have been proved for 2×2 systems [11] and n×n Temple class systems [15] using compensated compactness. For general conservative systems, convergence results for vanishing viscosity approximations are known only in the case where the limit solution is piecewise smooth [13, 16]. In general, if one can establish a priori estimates on the total variation of approximate solutions, then Helly’s theorem guarantees convergence. Using the uniqueness results in [5] one can then show that this limit is the appropriate entropy weak solution. Recently such estimates have been established for systems (1) under the assumption that the integral curves of the eigenvector fields of A(u) are straight lines in state space. In [3] it was proved that vanishing viscosity yield an L1 Lipschitz continuous semigroup which is consistent with the Riemann solver. See [2], [10] for results in the case of relaxation. In the present note we show that, under this straight-line assumption, also the approximate solutions constructed by the Godunov scheme converge to a unique limit, the same obtained by vanishing viscosity [3]. The main reason for considering these systems is that the effects of nonlinear coupling are somewhat easier to estimate. Indeed, under this straight-line