@article{(Open Science Index):https://publications.waset.org/pdf/14882,
	  title     = {On Discretization of Second-order Derivatives in Smoothed Particle Hydrodynamics},
	  author    = {R. Fatehi and  M.A. Fayazbakhsh and  M.T. Manzari},
	  country	= {},
	  institution	= {},
	  abstract     = {Discretization of spatial derivatives is an important
issue in meshfree methods especially when the derivative terms
contain non-linear coefficients. In this paper, various methods used
for discretization of second-order spatial derivatives are investigated
in the context of Smoothed Particle Hydrodynamics. Three popular
forms (i.e. "double summation", "second-order kernel derivation",
and "difference scheme") are studied using one-dimensional unsteady
heat conduction equation. To assess these schemes, transient response
to a step function initial condition is considered. Due to parabolic
nature of the heat equation, one can expect smooth and monotone
solutions. It is shown, however in this paper, that regardless of
the type of kernel function used and the size of smoothing radius,
the double summation discretization form leads to non-physical
oscillations which persist in the solution. Also, results show that when
a second-order kernel derivative is used, a high-order kernel function
shall be employed in such a way that the distance of inflection
point from origin in the kernel function be less than the nearest
particle distance. Otherwise, solutions may exhibit oscillations near
discontinuities unlike the "difference scheme" which unconditionally
produces monotone results.},
	    journal   = {International Journal of Mechanical and Mechatronics Engineering},
	  volume    = {2},
	  number    = {4},
	  year      = {2008},
	  pages     = {428 - 431},
	  ee        = {https://publications.waset.org/pdf/14882},
	  url   	= {https://publications.waset.org/vol/16},
	  bibsource = {https://publications.waset.org/},
	  issn  	= {eISSN: 1307-6892},
	  publisher = {World Academy of Science, Engineering and Technology},
	  index 	= {Open Science Index 16, 2008},
	}