Inthispaper,anewmethodfornumericallysolvingnonlinearconvection-dominateddiffusionproblemsisdevisedandanalysed.Thediscretetimeapproximationswithtimesteppingalongcharactcristicsarecstablishedandsolvedinspacesposscssingreproducingkernelfunctions.Ateachtimestep,theexactsolutionoftheapproximateproblemisgivenbyexplicitexpression.Thecomputationaladvantageofthismethodisthattheschemesareabsolutelystable,andareexplicitlysolvableaswell.Thestabilityanderrorestimatesarederived.Somenumericalresultsaregiven.