Adaptive Meshfree Method of Backward Characteristics for Nonlinear Transport Equations

In previous work, a new adaptive meshfree advection scheme for numerically solving linear transport equations has been proposed. The scheme, being a combination of an adaptive semi-Lagrangian method and local radial basis function interpolation, is essentially a method of backward characteristics. The adaptivity of the meshfree advection scheme relies on customized rules for the refinement and coarsening of scattered nodes. In this paper, the method is extended to nonlinear transport equations. To this end, in order to be able to model shock propagation, an artificial viscosity term is added to the scheme. Moreover, the local interpolation method and the node adaption rules are modified accordingly. The good performance of the resulting method is finally shown in the numerical examples by using two specific nonlinear model problems: Burgers equation and the Buckley-Leverett equation, the latter describing a two-phase fluid flow in a porous medium.