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Linear and angular momentum are conserved by semidiscrete SPH
Statement
Every C¹ solution of Fixed-width regularized conservative SPH particle system conserves total linear momentum P=Σ_im_iv_i and the antisymmetric angular-momentum tensor J=Σ_im_i(q_i⊗v_i−v_i⊗q_i), where (a⊗b)_kl=a_kb_l.
Facts & Assumptions
Given: A C¹ trajectory solving the fixed-width SPH system.
Fixed-width regularized conservative SPH particle system: q̇_i=v_i and v̇_i=−Σ_jm_j(A_i+A_j)∇W_ε(q_i−q_j).
SPH pressure forces are equal-opposite and central: F_ij=−m_im_j(A_i+A_j)∇W_ε(q_i−q_j); F_ji=−F_ij, F_ij is parallel to q_i−q_j, and ∇W_ε(0)=0.
Sums, scalar multiples, products and quotients: , , , and when : finite sums and coordinatewise products of differentiable scalar functions are differentiated by the sum and product rules.
The mean value inequality: if is continuous and differentiable on with , then : a continuously differentiable vector function with derivative zero on an interval is constant there, by applying the inequality on each closed subinterval.
Proof
Differentiate P using the finite-sum rule [F3]. The self terms vanish because ∇W_ε(0)=0 by [F2], and grouping the remaining terms by unordered pairs gives Ṗ=Σ_{i<j}(F_ij+F_ji)=0.
Differentiate J using the coordinatewise product rule [F3]; the terms v_i⊗v_i cancel, leaving Σ_i(q_i⊗m_iq̈_i−m_iq̈_i⊗q_i). Grouping each pair gives (q_i−q_j)⊗F_ij−F_ij⊗(q_i−q_j)=0 because [F2] makes F_ij parallel to q_i−q_j. Thus J̇=0. Applying [F4] to P and to each component of J on every closed subinterval shows both are constant.
Depends on
- Fixed-width regularized conservative SPH particle system
- SPH pressure forces are equal-opposite and central
- Sums, scalar multiples, products and quotients: $(f+g)'(c) = f'(c) + g'(c)$, $(\alpha f)'(c) = \alpha f'(c)$, $(fg)'(c) = f'(c)g(c) + f(c)g'(c)$, and $(f/g)'(c) = \bigl(f'(c)g(c) - f(c)g'(c)\bigr)/g(c)^{2}$ when $g(c) \ne 0$
- The mean value inequality: if $f : [a,b] \to \mathbb{R}^m$ is continuous and differentiable on $(a,b)$ with $\lVert f'\rVert_2 \le M$, then $\lVert f(b)-f(a)\rVert_2 \le M(b-a)$
Used by
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Sources
- Volker Springel, Smoothed Particle Hydrodynamics in Astrophysics, Annual Review of Astronomy and Astrophysics 48 (2010), author full text (standard reference, not scraped)
- Benjamin Seibold, MIT OpenCourseWare 18.336, Numerical Methods for Partial Differential Equations, Lecture 25: Particle Methods (Spring 2009) (standard reference, not scraped)