The Reconstruction Problem Revisited

AbstractTile role of reconstruction in avoiding oscillations in upwind sc|_emes is reexamined,with the aim of providing simple, concise proofs. In one dimension, it is shown that if thereconstruction is any arbitrary function bounded by neighboring ('ell averages and increasingwithin a ('ell for increasing data, the resulting scheme is monotonicity preserving, even thoughthe reconstructed function may havo overshoots and undershoots a.t the cell edges and is ingeneral not. a monotone fimction, in the special case of linear reconstruction, it is shownthat merely bounding the reconstruction between neighboring cell averages is sutficient toobtain a monotonicity preserving scheme.In two dimensions, it. is shown that some ID TVI) limiters applied in each directionresult in schemes that are no! positivity preserving, i.e. do not give positive updates whenthe data are positive. A simple proof is given to show that if the reconstruction inside the cellis bounded by the neighboring cell averages (including corner neighbors), t.hell the schemeis positivity preserving. A new limiter that enforces this condition but is not a.s dissipativeas the Minmod limiter is also presented.IntroductionIn this paper, we begin by reexamining l.[le reconstruction step in upwind schemes 1-'_with the aim of deriving slightly more general results and providing simple, concise proofs.NASA/TM-- 1999-209082 1