an:04211422
Zbl 0732.65084
Tadmor, Eitan
Local error estimates for discontinuous solutions of nonlinear hyperbolic equations
EN
SIAM J. Numer. Anal. 28, No. 4, 891-906 (1991).
00157466
1991
j
65M06 65M15 35L65
local error estimates; discontinuous entropy solution; nonlinear scalar conservation law; approximate viscosity regularization; post-processing; backward linear transport equation; discontinuous coefficients; E-condition; E-difference schemes
Author's summary: Let \(u(x,t)\) be the possibly discontinuous entropy solution of a nonlinear scalar conservation law with smooth initial data. Suppose \(u_{\varepsilon}(x,t)\) is the solution of an approximate viscosity regularization, where \(\varepsilon >0\) is the small viscosity amplitude. It is shown that by post-processing the small viscosity approximation \(u_{\varepsilon}\), pointwise values of \(u\) and its derivatives with an error as close to \(\varepsilon\) as desired can be recovered.
The analysis relies on the adjoint problem of the forward error equation, which in this case amounts to a backward linear transport equation with discontinuous coefficients. The novelty of our approach is to use a (generalized) E-condition of the forward problem in order to deduce a \(W^{1,\infty}\)-energy estimate for the discontinuous backward transport equation; this, in turn, leads to \(\varepsilon\)-uniform estimate on moments of the error \(u_{\varepsilon}-u\).
The approach presented does not ``follow the characteristics'' and, therefore, applies mutatis mutandis to other approximate solutions such as E-difference schemes.
U. G??hner (Stuttgart)