An adaptive sparse kernel technique in greedy algorithm framework to simulate an anomalous solute transport model

In the current work, an efficient and powerful computational technique is implemented to simulate an anomalous mobile-immobile solute transport process. The process is mathematically modelled as a time-fractional mobile-immobile diffusion equation in the sense of Riemann-Liouville derivative. Firstly, an implicit time integration procedure is used to semi-discretize the model in the time direction. The unconditional stability of the proposed time discretization scheme has been proven. Then an adaptive sparse meshless method has been formulated and implemented to fully discretize the model. In this approach, a kernel-based collocation method is equipped with a greedy sparse approximation procedure to discretize the governing problem on a convenient neighborhood of each data point with acceptable accuracy. Therefore, it leads to a sparse and well-conditioned algebraic system. Some test problems on regular and irregular computational domains are presented to verify the validity, efficiency, and accuracy of the method.