Numerical solutions in chromatography using large time step and overlapping grids methods

Abstract

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 $$\delta $$ -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.