A POD-RBF-FD scheme for simulating chemotaxis models on surfaces

Abstract

The main aim of this paper is to develop a new framework of a meshless approximation for solving numerically three nonlinear partial differential equations in biology, i.e., the chemotaxis models defined on the smooth, closed manifolds embedded in R3. The radial basis function-generated finite difference scheme is considered to deal with the spatial variables, which depends only on the location of nodes and the value of normal vector at each point per spherical cap. The robust artificial hyperviscosity formulation is derived for each model, which has been used for preventing the spurious growth modes in the numerical solution. An implicit–explicit time discretization is employed to deal with the time variable. The resulting fully discrete scheme is solved via the biconjugate gradient stabilized method with a zero-fill incomplete lower upper preconditioner per time step, where a positivity-preserving filter is used to prevent the negative sign of the cell density variable. Besides, to reduce the used central processor unit (CPU) time, the proper orthogonal decomposition is considered for constructing a set of new orthogonal basis vectors based on the singular value decomposition. The developed numerical method is called a proper orthogonal decomposition-radial basis function-generated finite difference (POD-RBF-FD) scheme. Finally, the ability of the proposed method is investigated by simulation results showing the blowing-up, pattern formulation (perforated stripe) and aggregations of bacteria on some surfaces.