In this paper the problem of consistency of smoothed particle hydrodynamics ( SPH ) is solved .
A novel error analysis is developed in n -dimensional space using the Poisson summation formula , which enables the treatment of the kernel and particle approximation errors in combined fashion .
New consistency integral relations are derived for the particle approximation which correspond to the cosine Fourier transform of the classically known consistency conditions for the kernel approximation .
The functional dependence of the error bounds on the SPH interpolation parameters , namely the smoothing length h and the number of particles within the kernel support { \cal { N } } is demonstrated explicitly from which consistency conditions are seen to follow naturally .
As { \cal { N } } \to \infty , the particle approximation converges to the kernel approximation independently of h provided that the particle mass scales with h as m \propto h ^ { \beta } , with \beta > n .
This implies that as h \to 0 , the joint limit m \to 0 , { \cal { N } } \to \infty , and N \to \infty is necessary for complete convergence to the continuum , where N is the total number of particles .
The analysis also reveals the presence of a dominant error term of the form ( \ln { \cal { N } } ) ^ { n } / { \cal { N } } , which tends asymptotically to 1 / { \cal { N } } when { \cal { N } } \gg 1 , as it has long been conjectured based on the similarity between the SPH and the quasi-Monte Carlo estimates .