Error Estimates of Approximate Solutions for Nonlinear Scalar Conservation Laws

There has been an enormous amount of work on error estimates for approximate solutions to scalar conservation laws. The methods of analysis include matching the traveling wave solutions, [8, 24]; matching the Green function of the linearized problem [21]; weak W convergence theory [32]; the Kruzkov-functional method [19]; and the energy-like method [34]. The results on error estimates include: For BV entropy solutions, an O(√ ) convergence rate in L obtained by Kuznetsov [19], Lucier [25] etc; For BV entropy solutions, an O( ) convergence rate in W obtained by Tadmor-Nessyahu [27, 32], Liu-Wang-Warnecke [22] etc; For piecewise smooth solutions, an O( ) convergence rate in L obtained by Bakhvalov [1], Harabetian [9], Teng-Tang [39, 42], Fan [7] etc; For piecewise smooth solutions, an O( ) convergence rate in smooth region of the entropy solution obtained by Liu [21], Goodman-Xin [8], Engquist-Sjogreen [6], Tadmor-Tang [34, 36] etc. In this paper, we will review some known results on error estimates and will discuss some recent development on the convergence rates. The corresponding methods in obtaining these results will be briefly illustrated.