The convective transport, both pure and combined with diffusion and reaction, can be observed in a wide range of physical and industrial applications, such as, for example, heat and mass transfer, crystal growth or biomechanics. The numerical approximation of this class of problem can present a substantial difficulty due to regions of high gradients (steep fronts) of the solution, where generation of spurious oscillation or smearing should be precluded. This work is devoted to the developement of an efficient numerical technique to deal with pure linear convection and convection-dominated problems in the framework of convection-diffusion-reaction systems.

The particle transport method to be presented in the talk is based on using meshless numerical particleswhich carry out the solution along the characteristics defining the convective transport. The resolution of steep fronts of the solution is controlled by a special spacial adaptivity procedure. The semi-Lagrangian particle transport method uses an Eulerian fixed grid to represent the solution. In the case of convection-diffusion-reaction problems the methos is compined with diffusion and reaction solvers within an operator splitting approach. To transfer the solution from the particle set onto the grid a fast monotone projection technique is designed.

The performance of the proposed computational approach has been assessed in diverse tests and benchmark problems, such as wave transport, rigid-body rotation, mixing of hot and cold fronts, layered reaction. The numerical results confirm that the method has a spacial accuracy of the second order and for pure linear convection problems the method demonstrate optimal linear complexity. The method works on stuctured and unstructured meshes, demonstrating the high-resolution property in the regions of steep fronts of the solution. Moreover, the paritcle transport method can be successfully used for the numerical simulation of the real-life problems in, for example, chemical The particle transport method for convection-dominated problems.