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The smoothing kernel is the back bone of Smoothed Particle Hydrodynamics (SPH), a modern technique to integrate the equations of motion of a fluid. Actually, both the properties of the smoothing kernel and its smoothing length play a key role in SPH. In particular, the smoothing length is used to adjust the resolution of numerical results. Moreover, to ensure the conservation of basic conservation laws, namely conservation of mass, linear momentum, angular momentum and energy, symmetric kernels are needed. Additionally, it is found that gradient approximations are second-order accurate under some pre-defined assumptions. In this work we present higher order approximations for gradients which ensure the conservation of the relevant physical quantities. However, the corresponding smoothing kernel of these approximations is not positive definite, a customary approximation to ensure that the density never becomes negative anywhere at any time. Nevertheless, combinations of higher order approximations could ensure that the kernel is positive definite, but strictly speaking, higher order accuracy is then not reached. The use of combinatorial parameters provides a way to control the characteristics of these kernels, providing higher order accuracy, while ensuring that the kernel remains positive definite.
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