Entropy Balance and Dispersive Oscillations in Lattice Boltzmann Models

We conduct an investigation into the dispersive post-shock oscillations in the entropic lattice-Boltzmann method (ELBM). To this end we use a root finding algorithm to implement the ELBM which displays fast cubic convergence and guaranties the proper sign of dissipation. The resulting simulation on the one-dimensional shock tube shows no benefit in terms of regularization from using the ELBM over the standard LBGK method. We also conduct an experiment investigating of the LBGK method using median filtering at a single point per time step. Here we observe that significant regularization can be achieved.Comment: 18 pages, 4 figures; 13/07/2009 Matlab code added to appendi

Computing the common zeros of two bivariate functions via Bezout resultants

The common zeros of two bivariate functions can be computed by finding the common zeros of their polynomial interpolants expressed in a tensor Chebyshev basis. From here we develop a bivariate rootfinding algorithm based on the hidden variable resultant method and B�ezout matrices with polynomial entries. Using techniques including domain subdivision, B�ezoutian regularization and local refinement we are able to reliably and accurately compute the simple common zeros of two smooth functions with polynomial interpolants of very high degree (�$\ge$ 1000). We analyze the resultant method and its conditioning by noting that the B�ezout matrices are matrix polynomials. Our robust algorithm is implemented in the roots command in Chebfun2, a software package written in object-oriented MATLAB for computing with bivariate functions