Abstract:We extend the definition of h-blossoming for polynomials in one variable to the polynomials in two variables, and we use this bivariate (h 1 , h 2 )-blossoming to study various properties, identities, and algorithms associated with (h 1 , h 2 )-Bézier surfaces. We construct a recursive (h 1 , h 2 )-midpoint subdivision algorithm for the (h 1 , h 2 )-Bézier surfaces and we prove its geometric rate of convergence.
We introduce the (q, h)-blossom of bivariate polynomials, and we define the bivariate (q, h)-Bernstein polynomials and (q, h)-Bézier surfaces on rectangular domains using the tensor product. Using the (q, h)-blossom, we construct recursive evaluation algorithms for (q, h)-Bézier surfaces and we derive a dual functional property, a Marsden identity, and a number of other properties for bivariate (q, h)-Bernstein polynomials and (q, h)-Bézier surfaces. We develop a subdivision algorithm for (q, h)-Bézier surfaces with a geometric rate of convergence. Recursive evaluation algorithms for quantum (q, h)-partial derivatives of bivariate polynomials are also derived.
We consider two systems of one-dimensional conservation laws that describe the process of column chromatography in chemistry in isolating a single compound from a mixture. We use recently proposed large time step and overlapping grids finite volume numerical methods in approximating solutions to a variety of initial value problems resulting in classical solutions as well as in singular and δ-shock solutions. The novel idea presented in the large time step method reduces the number of time steps needed to reach the final time and, thus, allows faster marching in the time direction. The overlapping grids methods are crucial when considering problems where an object is covered with multiple grids. It is important to ensure that the numerical method is constructed in such a way that it is conservative on the overlap and, moreover, that the approximate solutions converge to the weak solution, and in the case of a scalar equation, to the entropy solution. The main contribution of this paper is to show effectiveness of the proposed numerical methods for approximating solutions to systems of equations since their convergence was rigorously proved so far only in case of a scalar equation.
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