@misc{Bornemann, author = {Bornemann, Folkmar A.}, title = {Adaptive Solution of One-Dimensional Scalar Conservation Laws with Convex Flux.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-880}, number = {SC-92-18}, abstract = {A new adaptive approach for one-dimensional scalar conservation laws with convex flux is proposed. The initial data are approximated on an adaptive grid by a problem dependent, monotone interpolation procedure in such a way, that the multivalued problem of characteristic transport can be easily and explicitly solved. The unique entropy solution is chosen by means of a selection criterion due to LAX. For arbitrary times, the solutions is represented by an adaptive monotone spline interpolation. The spatial approximation is controlled by local \$L^1\$-error estimated. As a distinctive feature of the approach, there is no discretization in time. The method is monotone on fixed grids. Numerical examples are included, to demonstrate the predicted behavior. {\bf Key words.} method of characteristics, adaptive grids, monotone interpolation, \$L^1\$-error estimates {\bf AMS(MOS) subject classification.} 65M15, 65M25, 65M50.}, language = {en} }