Numerical Solutions

N-Body Quotes
From p.89 of Atmospheric and Space Flight Dynamics: Modeling and Simulation with MATLAB by Professor Ashish Tewari (bio) we read:



On p.2 of a Master's thesis Evaluation of mass loss in the simulation of stellar clusters using a new multiphysics software environment by Guillermo Kardolus we see:





A Princeton University programming assignment says:



From On the Reliability of N-body Simulations:



The paper Global Error Measures for Large N-Body Simulations describes:



General Quotes
The book Nuclear Astrophysics: A Course of Lectures tells us on p.259:



The abstract of a medical research paper Simulation and air-conditioning in the nose says:



From a question posted on researchgate.net:




 * Mohammad Firoz Khan, Ph.D. responds:



Jason Brownlee, Ph.D., tells us on machinelearningmastery.com:



http://www.math.pitt.edu/~sussmanm/2071Spring09/lab02/index.html



Two Body Approximations
https://www.askamathematician.com/2011/10/q-what-is-the-three-body-problem/

https://academic.oup.com/mnras/article/440/1/719/1747624

We developed a Keplerian-based Hamiltonian splitting for solving the gravitational N-body problem. This splitting allows us to approximate the solution of a general N-body problem by a composition of multiple, independently evolved two-body problems. While the Hamiltonian splitting is exact, we show that the composition of independent two-body problems results in a non-symplectic non-time-symmetric first-order map. A time-symmetric second-order map is then constructed by composing this basic first-order map with its self-adjoint. The resulting method is precise for each individual two-body solution and produces quick and accurate results for near-Keplerian N-body systems, like planetary systems or a cluster of stars that orbit a supermassive black hole.

https://hanspeterschaub.info/Papers/UnderGradStudents/ConicReport.pdf

The patched-conic approximation has thus been developed as a more accurate solution to interplanetary transfer description. It involves partitioning the overall transfer into distinct conic solutions. For instance, as a spacecraft travels from Earth to Mars, its orbit is approximated as a hyperbolic departure, an elliptic transfer, and a hyperbolic arrival. The patched-conic approximation breaks the entire orbit down into several two-body problems. In other words, only one celestial body’s influence is considered to be acting upon the spacecraft at all times.

https://academic.oup.com/mnras/article/452/2/1934/1069988

In this paper, we present a new symplectic integrator for collisional gravitational N-body dynamics. The integrator is inspired by the non-symplectic and non-reversible integrator in Gonçalves Ferrari et al. (2014), SAKURA, and makes use of Kepler solvers. Like SAKURA we decompose the N-body problem into two-body problems. In contrast to SAKURA, our two-body problems are not independent. The integrator is reversible and symplectic and conserves nine integrals of motion of the N-body problem to machine precision.

https://academic.oup.com/mnras/article/440/1/719/1747624

It seems that a truly rigorous and elegant solution will be achieved only by finding a mathematical transformation that reduces the many-body problem to a one-body problem. In such a formulation each atom, nucleus or electron can be treated alone with the contributions of all the others summed together. In Alder's opinion such a development will really allow working on the deep-lying problems of the quantum-mechanical structure of matter. Physics has a long history of reducing many-body problems to one-or two-body problems in order to find more powerful solutions, and Alder and his colleagues have high hopes of doing it for this one.

Galaxy Simulator
The following was provided to us as an example of the numerical solution of multiple bodies - The Numerical Solution of the N-Body Problem. Read the below quotes and decide whether the methods are describing a full simulation of gravity.

From the introduction:

The Barnes-Hut Galaxy Simulator

Looked up Barnes Hut: https://beltoforion.de/en/barnes-hut-galaxy-simulator/

The Barnes-Hut Galaxy Simulator

The above shows that there could be a numerical solution that doesn't use gravity fully, discrediting the "numerical solutions exist" idea. Like with the previous quotes and examples, liberal assumptions are made, rather than a true simulation of the laws involved.