Fixed-width regularized conservative SPH particle system

Fix d,N≥1, positive masses m_1,…,m_N, ε>0, the kernel W_ε from Scaled symmetric compactly supported kernel family, and a pressure map P∈C¹((0,∞)×R,R). Positions q_i and velocities v_i lie in R^d with the Euclidean inner product from The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$, and specific internal energies e_i lie in R. Define

${\rho }_i(q)={\sum }_{j=1}^N{m}_j{W}_{\epsilon }(q_i-q_j),p_i=P({\rho }_i,e_i),A_i=\frac{p_i}{{\rho }_i^2}\mathrm{.}$

The state domain is U={ (q,v,e): ρ_i(q)>0 for every i }. On U, the fixed-width semidiscrete system is

${\dot{q}}_i=v_i,{\dot{v}}_i=-{\sum }_{j=1}^N{m}_j(A_i+A_j)\mathrm{\nabla }{W}_{\epsilon }(q_i-q_j),{\dot{e}}_i=A_i{\sum }_{j=1}^N{m}_j(v_i-v_j)\cdot \mathrm{\nabla }{W}_{\epsilon }(q_i-q_j)\mathrm{.}$

A controlled tube is a compact subset of U on which ρ_i≥ρ_*>0 for every i. The system is semidiscrete: time remains continuous while the spatial state is represented by finitely many particles. It has fixed masses and fixed ε, includes the displayed matched internal-energy equation, and contains no artificial viscosity. No continuum Euler convergence is part of this definition.