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Local well-posedness of the regularized finite-particle SPH system
Statement
For the system of Fixed-width regularized conservative SPH particle system, U={z:ρ_i(q)>0 for every i} is open and its finite-dimensional vector field F is continuous and locally Lipschitz on U. From each initial state z₀∈U there is a unique local C¹ solution.
Facts & Assumptions
Given: Fixed finite N, positive masses, W∈C², P∈C¹, and an initial state with every ρ_i>0.
Fixed-width regularized conservative SPH particle system: each ρ_i is a finite sum of W_ε evaluated at position differences, and F consists of finite sums of W_ε, ∇W_ε and the smooth ratio P(ρ,e)/ρ² on the set ρ_i>0.
Explicit local Lipschitz bound for the regularized SPH vector field: on every compact convex positive-density tube, F has an explicit finite state-Lipschitz constant.
Picard-Lindelöf local existence and uniqueness for first-order systems: a continuous vector field locally Lipschitz in state on an open time-state domain has a unique local solution through every initial point.
Heine-Borel in : with the Euclidean metric a subset of is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line: nondegenerate closed boxes in finite-dimensional state space are compact. A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value: continuous real-valued functions on a nonempty compact metric space attain their minima.
Proof
Each ρ_i is continuous by [F1], so U is open as the finite intersection of the inverse images of (0,∞). On U, denominators ρ_i² are nonzero, P is C¹, W_ε is C², and the finite sums defining F are continuous.
Fix z₀∈U. Since U is open, choose a nonempty nondegenerate closed box K about z₀ small enough to lie in U. It is convex and compact by [F4]. Each ρ_i is continuous and positive on K, so [F4] gives a positive minimum for each; taking the minimum over finitely many i gives a positive lower density bound on K. [F2] makes F Lipschitz there. Regard F as time-independent on R×U and apply [F3] to obtain a unique local C¹ solution. No choice principle is used.
Depends on
- Fixed-width regularized conservative SPH particle system
- Explicit local Lipschitz bound for the regularized SPH vector field
- Picard-Lindelöf local existence and uniqueness for first-order systems
- Heine-Borel in $\mathbb{R}^n$: with the Euclidean metric a subset of $\mathbb{R}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line
- A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value
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)