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The matched internal-energy equation conserves total semidiscrete energy
Statement
Every C¹ solution of Fixed-width regularized conservative SPH particle system conserves H=Σ_im_i(|v_i|²/2+e_i). The identity uses exactly the displayed matched internal-energy equation.
Facts & Assumptions
Given: A C¹ trajectory solving the fixed-width system, with A_i=p_i/ρ_i² and g_ij=∇W_ε(q_i−q_j).
Fixed-width regularized conservative SPH particle system: v̇_i=−Σ_jm_j(A_i+A_j)g_ij and ė_i=A_iΣ_jm_j(v_i−v_j)·g_ij.
SPH pressure forces are equal-opposite and central: g_ji=−g_ij. The coefficient A_i+A_j is symmetric by commutativity of addition.
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 differentiable scalar function with derivative zero on an interval is constant there.
Proof
Differentiate K=Σ_im_i|v_i|²/2 using [F3] coordinatewise and substitute [F1]. Pairing the (i,j) and (j,i) terms using [F2] yields K̇=−Σ_{i<j}m_im_j(A_i+A_j)(v_i−v_j)·g_ij.
Differentiate U=Σ_im_ie_i using [F3] and substitute [F1]. The (i,j) term is m_im_jA_i(v_i−v_j)·g_ij, while the (j,i) term equals m_im_jA_j(v_i−v_j)·g_ij by [F2]. Hence U̇=Σ_{i<j}m_im_j(A_i+A_j)(v_i−v_j)·g_ij.
Steps 1.1 and 1.2 cancel term by term, so Ḣ=K̇+U̇=0. Applying [F4] with derivative bound M=0 on any closed subinterval shows H is constant on each solution interval.
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)