A New High-Order SPH Method for Gas Dynamics Equations: Approximate Dispersion Relations and Sound Wave Simulation

Smoothed particle hydrodynamics (SPH) is a numerical method for approximating partial differential equations based on interpolation of a function from values known at moving irregularly located nodes. In SPH, a finite smoothing kernel is used for interpolation. The support radius of the kernel $$H$$ is much smaller than the computational domain. For classical sign-definite kernels, the second-order barrier is known. This means that using these kernels, it is impossible to approximate a solution with an order higher than the second in $$H$$ . To overcome this barrier, new schemes have been proposed for approximating gas dynamics equations. These schemes are based on a method for approximating the gradient of a function without differentiating the kernel, combining the ideas of finite differences and SPH interpolation. Approximate dispersion relations (ADR) are constructed for classical and new methods. By analyzing the ADR, it is shown that in the classical SPH approximation, when moving from second-order sign-definite kernels to fourth-order sign-alternating kernels, numerical short wave instability appears. On the contrary, schemes constructed using the new gradient approximation method retain stability for high-order kernels. In addition, it is shown that, at the same numerical resolution, the new gradient approximation method makes it possible to reduce the error in calculating the phase velocity of medium-length waves (the wavelength is comparable to the radius of the kernel) by an order of magnitude compared to the classical method, provided that $$H$$ is 4 or more times greater than the distance from the particle to its nearest neighbor. Moreover, for the new method, the fourth order of approximation in $$H$$ has been obtained in the simulation of sound wave propagation.