The physical interactions between two immiscible fluids in porous media may lead to the formation of a shock front at the interface between the two fluid phases. Numerical methods for modeling fluid flow with shocks can be classified into two categories: the shock-capturing and the shock-fitting methods. The shock-fitting method can in general provide more accurate solution of the shock front than the shock-capturing method, but suffers from the extra complexity in accommodating the moving shock front inside the background mesh. In this study, we explore the possibility of integrating the mesh-free methods for solving partial-differential equations with the shock-fitting method. In this integrated method, which we call “mesh-free shock-fitting (MFSF)”, the nodes needed for mesh-free calculations of spatial derivatives are generated on the fly to adapt to the moving shock front. We demonstrate the implementation of MFSF using numerical experiments in one and two spatial dimensions. Preliminary results show that MFSF has the potential to provide more accurate solutions with lower computational cost than conventional shock-capturing methods and can simplify many of the operations in shock-fitting methods with unstructured meshes.
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