This paper is concerned with the global stability of strong rarefaction waves for a class of 2 × 2 hyperbolic conservation laws with artificial viscosity, i.e., the p-system with artificial viscosity { v t − u x = ε 1 v x x , u t + p ( v ) x = ε 2 u x x , ( v , u ) | t = 0 = ( v 0 , u 0 ) ( x ) → ( v ± , u ± ) as x → ± ∞ , where ε i ( i = 1 , 2 ) are positive constants and p ( v ) is a smooth function defined on v > 0 satisfying p ′ ( v ) < 0 , p ″ ( v ) > 0 for v > 0 . Let ( V ( t , x ) , U ( t , x ) ) be the smooth approximation of the rarefaction wave profile constructed similar to that of [A. Matsumura, K. Nishihara, Global stability of the rarefaction waves of a one-dimensional model system for compressible viscous gas, Comm. Math. Phys. 144 (1992) 325–335], if the H 1 -norm of the initial perturbation ( v 0 ( x ) − V ( 0 , x ) , u 0 ( x ) − U ( 0 , x ) ) is small, the nonlinear stability of strong rarefaction waves is well-understood, but for the case when ‖ ( v 0 ( x ) − V ( 0 , x ) , u 0 ( x ) − U ( 0 , x ) ) ‖ H 1 is large, to our knowledge, fewer results have been obtained and in this paper, we obtain two types of results in this direction. Roughly speaking, if ε 1 ≠ ε 2 , we can get the nonlinear stability result provided that ‖ ( v 0 ( x ) − V ( 0 , x ) , u 0 ( x ) − U ( 0 , x ) ) ‖ L 2 is small. In some sense it is a generalization of the result obtained in [D. Hoff, J.A. Smoller, Solutions in the large for certain nonlinear parabolic systems, Ann. Inst. H. Poincaré 2 (1985) 213–235] for the case ( v − , u − ) = ( v + , u + ) to the case ( v − , u − ) ≠ ( v + , u + ) and the method developed by Y. Kanel' in [Y. Kanel', On a model system of equations of one-dimensional gas motion (in Russian), Differ. Uravn. 4 (1968) 374–380] plays an essential role in obtaining the uniform lower bound for v ( t , x ) . While for the case when ε 1 = ε 2 , the above system admits positively invariant regions which yields the uniform lower bound for v ( t , x ) and based on this, two types of global stability results are obtained: first, for general flux function p ( v ) , if ‖ ( v 0 ( x ) − V ( 0 , x ) , u 0 ( x ) − U ( 0 , x ) ) ‖ H 1 depends on t 0 , a sufficiently large positive constant introduced in constructing the smooth approximation to the rarefaction wave solution, some restrictions on its growth rate as t 0 → + ∞ must be imposed. While for some special flux functions p ( v ) which contain p ( v ) = v − γ ( γ ⩾ 1 ) as a special case, similar result holds for any ( v 0 ( x ) − V ( 0 , x ) , u 0 ( x ) − U ( 0 , x ) ) ∈ H 1 ( R ) .