An efficient implementation of a numerical method for a chemotaxis system

On some explicit non-standard methods to approximate nonnegative solutions of a weakly hyperbolic equation with logistic nonlinearity

“…Recent investigations in computational physics and numerical simulations have focused their attentions in the development of techniques to approximate propagating wave solutions in several media [9], including the wave equation itself [3]. On the other hand, several numerical methods have been developed in order to preserve the positivity of the solutions of some differential equations via non-standard techniques [2,8,[20][21][22]24,27,30]. However, the problem of designing positivity-preserving, finite-difference schemes to approximate nonnegative solutions of nonlinear, hyperbolic generalizations of the Fisher-KPP equations has not been successfully solved, yet.…”

A local radial basis function method for advection–diffusion–reaction equations on complexly shaped domains

“…This geometry is chosen either to simplify the development of the numerical method or possibly because the particular numerical method is only applicable on such simple domains. Previously used numerical methods for chemotaxis problems on rectangular domains include a discontinuous Galerkin method [11] and various finite difference and finite volume methods [8,7,43,49,50]. In [6], a finite volume scheme is developed and applied to chemotaxis problems on circular domains.…”

A mathematical model for IL6-induced differentiation of neural progenitor cells on a micropatterned polymer substrate

“…(The advection portion of the PDE yielded an explicit quantity, based on the previous timestep, and therefore casues no additional difficulty.) An efficient method for approximating this system is provided in [51], with details on its implementation provided in [52] and [55]. The following is simply a restatement of this method's derivation as provided in [51], with slight alterations due to the differing boundary conditions.…”

“…The method for solving the system Au = g presented in [51] is derived under the assumption , and the fact that F n = F T n , we get…”

“…The solution to the PDE system was numerically approximated by using a Finite Volume approach as discussed in [50]. In the case where the diffusion coefficients were constant, the resultant system was solved using an efficient Fast Fourier Transform (FFT) method as discussed in [51]- [52]. In the case where the diffusion coefficients were variable, exhibiting a dependence on the species that is diffusing (a quorumsensing mechanism), the PDE was numerically approximated using an Iterative Alternating Direction Implicit (ADI) Method discussed below.…”