# Preprint 2009-012

# 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

*u*_{t}+*A*(*u*)*u*_{x}=0, (1)

where
*u*↦*A*(*u*),
*u*∈Ω⊂**R**^{N},
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
*u*_{t}+*A*(*u*)*u*_{x}=*μ* *u*_{xx},
as μ→0.
We prove the a-priori bound

‖*u*^{ε}(*T*,·)−*u*(T,·)‖=o(1)·√(ε)|log ε| (2)

for an approximate solution
*u*^{ε}
of (1) constructed by the Glimm scheme, with mesh size
Δ*x*=Δ*t*=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
*u*_{t}+*F*(*u*)_{x} =0
satisfying the classical Lax or Liu assumptions on the eigenvalues
λ_{k}(*u*)
and on the eigenvectors
*r*_{k}(*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).