Local error estimates for discontinuous solutions of nonlinear hyperbolic equations.

*(English)*Zbl 0732.65084Author’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.

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.

Reviewer: U. GĂ¶hner (Stuttgart)

##### MSC:

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |

35L65 | Hyperbolic conservation laws |