Sharp convergence rate of the Glimm scheme for general nonlinear hyperbolic systems
F. Ancona and A. Marson
Abstract: Consider a general strictly hyperbolic, quasilinear system, in one space dimension
where u↦A(u), u∈Ω⊂RN, is a smooth matrix-valued map. Given an initial datum u(0,·) with small total variation, let u(t,·) be the corresponding (unique) vanishing viscosity solution of (1) obtained as limit of solutions to the viscous parabolic approximation ut+A(u)ux=μ uxx, as μ→0. We prove the a-priori bound
‖uε(T,·)−u(T,·)‖=o(1)·√(ε)|log ε| (2)
for an approximate solution uε of (1) constructed by the Glimm scheme, with mesh size Δx=Δt=ε, and with a suitable choice of the sampling sequence. This result provides for general hyperbolic systems the same type of error estimates valid for Glimm approximate solutions of hyperbolic systems of conservation laws ut+F(u)x =0 satisfying the classical Lax or Liu assumptions on the eigenvalues λk(u) and on the eigenvectors rk(u) of the Jacobian matrix A(u)=DF(u).
The estimate (2) is obtained introducing a new wave interaction functional with a cubic term that controls the nonlinear coupling of waves of the same family and at the same time decreases at interactions by a quantity that is of the same order of the product of the wave strength times the change in the wave speeds. This is precisely the type of errors arising in a wave tracing analysis of the Glimm scheme, which is crucial to control in order to achieve an accurate estimate of the convergence rate as (2).