(Copyright: Society for Industrial and Applied Mathematics, 2000) TADMOR, EITAN; TANG, TAO
We obtain sharp pointwise error estimates for relaxation approximation to scalar
conservation laws with piecewise smooth solutions. We first prove that the first-order partial derivatives
for the perturbation solutions are uniformly upper bounded (the so-called Lip+ stability). A
one-sided interpolation inequality between classical L1 error estimates and Lip+ stability bounds
enables us to convert a global L1 result into a (nonoptimal) local estimate. Optimal error bounds on
the weighted error then follow from the maximum principle for weakly coupled hyperbolic systems.
The main difficulties in obtaining the Lip+ stability and the optimal pointwise errors are how to
construct appropriate “difference functions” so that the maximum principle can be applied.
