Boundary condition handling, not just kernel choice, largely determines whether SPH simulations of Burgers' equation stay stable, and particles can cross through each other near boundaries when smoothing length far exceeds particle spacing.
Benz, Smooth Particle Hydrodynamics - a Review NATO SAI Series C: Mathematical and Physical Sciences (Kluwer Academic Publishers, 1990)
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On the boundary condition and related instability in the Smoothed Particle Hydrodynamics
Boundary condition handling, not just kernel choice, largely determines whether SPH simulations of Burgers' equation stay stable, and particles can cross through each other near boundaries when smoothing length far exceeds particle spacing.