Meshless finite difference method with B-splines for numerical solution of coupled advection-diffusion-reaction problems

In this paper, a meshless finite difference (FD) method with B-splines is presented for numerical solution of coupled advection-diffusion-reaction (ADR) problems. The proposed method is mathematically simple to program and truly meshless. It combines the approximation capability and high resolution of B-splines and the ease of implementation of differential quadrature technique to discretize the system of equations. The presence of mesh is replaced by overlapping local domains containing of regular or scattered nodes, in which the solution inside is approximated by B-spline basis functions fashioned in local collocation. The fourth order Runge-Kutta (RK) method is employed for time integration. The effectiveness of the proposed method is shown by solving several coupled ADR problems, including cross reaction-diffusion, chemotaxis, pattern formation and tumor invasion into surrounding healthy tissue. Numerical results demonstrate high resolution, stability and robustness of the proposed method. It captures the emergence and evolution of sharp fronts in the problems well, thus depicting dynamics of the problems accurately. The method also places no restriction on the shape of computational domains. The present method's convergence rate is also elucidated empirically in this study and shown to be high. It is shown that the proposed method is an accurate and effective solver for coupled advection-diffusion-reaction problems in two-dimensional domains.