AUTHOR=Rendón Otto , Avendaño Gilberto D. , Klapp Jaime , Sigalotti Leonardo Di G. , Vargas Carlos A. 

TITLE=Local Consistency of Smoothed Particle Hydrodynamics (SPH) in the Context of Measure Theory

JOURNAL=Frontiers in Applied Mathematics and Statistics

VOLUME=8

YEAR=2022

URL=https://www.frontiersin.org/journals/applied-mathematics-and-statistics/articles/10.3389/fams.2022.907604

DOI=10.3389/fams.2022.907604

ISSN=2297-4687

ABSTRACT=<p>The local consistency of the method of Smoothed Particle Hydrodynamics (SPH) is proved for a multidimensional continuous mechanical system in the context of measure theory. The Wasserstein distance of the corresponding measure-valued evolutions is used to show that full convergence is achieved in the joint limit <italic>N</italic> → ∞ and <italic>h</italic> → 0, where <italic>N</italic> is the total number of particles that discretize the computational domain and <italic>h</italic> is the smoothing length. Using an initial local discrete measure given by <inline-formula><mml:math id="M1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>b</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msubsup><mml:mi>m</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>x</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>h</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>x</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>, where <italic>m</italic><sub><italic>b</italic></sub> = <italic>m</italic>(<bold>x</bold><sub><italic>b</italic></sub>, <italic>h</italic>) is the mass of particle with label <italic>b</italic> at position <bold>x</bold><sub><italic>b</italic></sub>(<italic>t</italic>) and δ<sub>0,<bold>x</bold><sub><italic>b</italic></sub>(<italic>t</italic>)</sub> is the <bold>x</bold><sub><italic>b</italic></sub>(<italic>t</italic>)-centered Dirac delta distribution, full consistency of the SPH method is demonstrated in the above joint limit if the additional limit <inline-formula><mml:math id="M2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="-tex-caligraphic">N</mml:mi></mml:math></inline-formula> → ∞ is also ensured, where <inline-formula><mml:math id="M3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="-tex-caligraphic">N</mml:mi></mml:math></inline-formula> is the number of neighbors per particle within the compact support of the interpolating kernel.</p>