Piecewise Polynomial Schemes

The only solution is k = 1, so that the only binary schemes with piecewise polynomial limit curves are the B-splines. 

Recently, Kothari et al. [56] have proposed a fully polynomial-time approximation scheme (FPTAS) for a variation on this price-schedule problem in which the cost functions are piecewise and marginal-decreasing and each supplier has a capacity constraint. The approach is to construct a 2-approximation to a generalized knapsack problem, which can then be used to scale a dynamicprogramming algorithm and compute an (1 + e) approximation in worst-case time T = 0 nc) /e), for n bidders and with a maximum of c pieces in each bid. ... 

Figure 18 shows the results (on the five-spot model problem) obtained by an adaptive ADER4 scheme. The shocks are resolved sharper in this case compared to the ADERl results of Figure 16. Moreover, the rarefaction appears to be much smoother, there is no structure of an underl3ung triangular mesh visible. However, if we look at the corresponding adaptive mesh in Figure 19, we see that there is even a coarser mesh in the areas of the rarefactions. In fact, behind the shocks the error indicator allows the recoarsening of the mesh to its coarsest level. This is due to the increased approximation quality of the higher order ADER4 scheme, which uses piecewise cubic polynomials instead of piecewise constant functions to reconstruct the water saturation...

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