Corrected SPH formulation

^{1}
^{2} has been introduced in order to satisfy the
so-called consistency conditions:

(1)
$$\underset{\Omega}{\int}W\left(y-x,h\right)=1,\forall x$$

(2)
$$\underset{\Omega}{\int}\left(y-x\right)W\left(y-x,h\right)=0,\forall x$$

These equations insure that the integral approximation of a
function f coincides with f for constant and linear functions of space.

CSPH is a
correction of the kernel functions:

(3)
${\widehat{W}}_{j}\left(x,h\right)={W}_{j}\left(x,h\right)\alpha \left(x\right)\left[1+\beta \left(x\right)\u2022\left(x-{x}_{j}\right)\right]$
with
${W}_{j}\left(x,h\right)=W\left(x-{x}_{j},h\right)$

Where the parameters

$\alpha \left(x\right)$
and

$\beta \left(x\right)$
are evaluated by enforcing the consistency condition, now given by the
point wise integration as:

(4)
$$\sum _{j}{V}_{j}{\widehat{W}}_{j}}\left(x,h\right)=1,\forall x$$

(5)
$$\sum _{j}{V}_{j}}\left(x-{x}_{j}\right){\widehat{W}}_{j}\left(x,h\right)=0,\forall x$$

These equations enable the explicit evaluation of the correction
parameters

$\alpha \left(x\right)$
and

$\beta \left(x\right)$
as:

(6)
$$\beta \left(x\right)={\left[{\displaystyle \sum _{j}{V}_{j}\left(x-{x}_{j}\right)\otimes}\left(x-{x}_{j}\right){W}_{j}\left(x,h\right)\right]}^{-1}{\displaystyle \sum _{j}{V}_{j}}\left({x}_{j}-x\right){W}_{j}\left(x,h\right)$$

(7)
$$\alpha \left(x\right)=\frac{1}{{\displaystyle \sum _{j}{V}_{j}{W}_{j}\left(x,h\right)\left[1+\beta \left(x\right)\u2022\left(x-{x}_{j}\right)\right]}}$$

Since the evaluation of gradients of corrected kernel (which are
used for the SPH integration of continuum equations) becomes very expensive, corrected SPH
limited to order 0 consistency has been introduced. Therefore, the kernel correction reduces
to the following equations:

(8)
$${\widehat{W}}_{j}\left(x,h\right)={W}_{j}\left(x,h\right)\alpha \left(x\right)$$

that is

(9)
$$\alpha \left(x\right)=\frac{1}{{\displaystyle \sum _{j}{V}_{j}{W}_{j}\left(x,h\right)}}$$

(10)
$$\sum _{j}{V}_{j}}{\widehat{W}}_{j}\left(x,h\right)=1,\forall x$$

Note: SPH corrections generally insure a better representation even if the
particles are not organized into a hexagonal compact net, especially close to the
integration domain frontiers. SPH corrections also allow the smoothing length
$h$
to values different to the net size
$\text{\Delta}x$
to be set.