ASYMPTOTICALLY COMPATIBLE SPH-LIKE PARTICLE DISCRETIZATIONS OF ONE DIMENSIONAL LINEAR ADVECTION MODELS

Motivated by the smoothed particle hydrodynamics (SPH), we present nonlocal models for linear advection with a variable coefficient in one spatial dimension together with their particle based numerical discretizations. We establish that these numerical methods are robust in the sense that they are convergent as the particle spacing and the smoothing length shrink to zero independently of each other. We demonstrate the important role of nonlocal continuum models to ensure the stability of our numerical methods. The nonlocal models constructed here follow two different strategies: the first model relies on choosing an upwind kernel and the second on introducing a nonlocal viscous term. We study discrete numerical schemes for both models that are in essence particle-like quadrature based finite differences, yet the distinction is clearly drawn in the sense that the scheme for the first model is based on the first moment of the nonlocal kernel while the other is conceived on the basis of renormalized SPH.