Space-time CE/SE method for solving repulsive chemotaxis model

Abstract

This work is concerned with the numerical investigation of nonlinear mathematical systems that describe repulsive chemotaxis phenomena in biology. A space-time conservation element (CE) and solution element (SE) method-based splitting procedure is proposed in one space dimension. In chemotaxis, the orientation of the cells changes. They move to the location which are chemically more favorable or move away from repellents or toxin. Mathematically, these are the systems of nonlinear coupled partial differential equations. The analytical solutions for this set of equations is not possible. A traditional less order accurate solution may fail to describe the underlying phenomena. Therefore, the state of the art numerical procedures is the need for such systems to get physically reliable solution in acceptable computational cost. Unlike tradition numerical procedures, the CE/SE-method has distinct attributes like it treats time and space in a unified fashion. Both conserved quantities and corresponding derivatives are considered to be unknowns due to which inherited numerical diffusion is reduced. Several benchmark numerical test problems are simulated for validation of the scheme. The numerical solutions of case studies are obtained for one space dimension. Moreover, one more central numerical scheme introduced by Nessyahu–Tadmor (NT) is also adapted on staggard grids which is considered to be a black box solver for such models for comparison. It was observed that both schemes perform well for the current mathematical model. However, the CE/SE scheme is more capable of capturing the peaks produced in the solution profile. Further, convergence study is also carried out from both schemes which reveal that the proposed method is fast as compared to NT central scheme.