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Local error estimates for discontinuous solutions of nonlinear hyperbolic equations. (English) Zbl 0732.65084
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.

##### 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
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