𝐿¹-convergence rate of viscosity methods for scalar conservation laws with the interaction of elementary waves and the boundary

Abstract

This paper is concerned with global error estimates for viscosity methods to initial-boundary problems for scalar conservation laws u t + f ( u ) x = 0 {u_t} + f{\left ( u \right )_x} = 0 on [ 0 , ∞ ) × [ 0 , ∞ ) \left [ {0, \infty } \right ) \times \left [ {0, \infty } \right ) , with the initial data u ( x , 0 ) = u 0 ( x ) u\left ( {x, 0} \right ) = {u_0}\left ( x \right ) and the boundary data u ( 0 , t ) = u − u\left ( {0, t} \right ) = {u_ - } , where u − {u_ - } is a constant, u 0 ( x ) {u_0}\left ( x \right ) is a step function with a discontinuous point, and f ∈ C 2 f \in {C^2} satisfies f > 0 , f ( 0 ) = f ′ ( 0 ) = 0 f > 0, f\left ( 0 \right ) = f’\left ( 0 \right ) = 0 . The structure of global weak entropy solution of the inviscid problem in the sense of Bardos-Leroux-Nedelec [11] is clarified. If the inviscid solution includes the interaction that the central rarefaction wave collides with the boundary x = 0 x = 0 and the boundary reflects a shock wave, then the error of the viscosity solution to the inviscid solution is bounded by O ( ϵ 1 / 2 + ϵ | I n ϵ | + ϵ ) O\left ( {\epsilon ^{1/2}} + \epsilon \left | {In\epsilon } \right | + \epsilon \right ) in L 1 {L^1} -norm. If the inviscid solution includes no interaction of the central rarefaction wave and the boundary or the interaction that the rarefaction wave collides with the boundary and is absorbed completely or partially by the boundary, then the error bound is O ( ϵ | I n ϵ | + ϵ ) O\left ( \epsilon \left | {In\epsilon } \right | + \epsilon \right ) . In particular, if there is no central rarefaction wave included in the inviscid solution, the error bound is improved to O ( ϵ ) O\left ( \epsilon \right ) . The proof is given by a matching method and the traveling wave solutions.