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Projection methods typically split up the solution of transient Stokes or Navier-Stokes methods into the solution of two separate problems - one for the velocity, one for the pressure. For Stokes, the simplest version (by Chorin/Temam) goes as follows for a timestep $k$: Compute an intermediate velocity $u^*$ by solving $$\frac{(u^*-u^k)}{dt} - \nu\Delta u^*...


4

What you are asking is not possible, because you cannot order all cells in such a way that neighbors are always in a continuous range. Suppose we try to construct continuous neighbor lists in 3D. Each interior cell $C_i$ has six neighbors, so it should appear six times in a neighbor list. The only way to construct six continuous neighbor list is as follows: ...


4

My experience has been in visual effects where simulation time is not as important as in games, but is still more important than in engineering. The methods I'm familiar with are: The material point method: A continuum description of materials The discrete element method: A discrete description Peridynamics: Nonlocal continuum description if fracture is ...


4

To give an easier example, consider the ODE for rotation \begin{align} \dot x=-ay\\ \dot y=ax \end{align} If one solves that with the Euler method, the next point is found in the direction of the tangent of the circle, increasing the radius. \begin{align} x(t+h)^2+y(t+h)^2&=(x(t)-ahy(t))^2+(y(t)+ahx(t))^2\\ &=(1+a^2h^2)(x(t)^2+y(t)^2) \end{align} ...


4

I just found out about Voronoi particle tracking so i'm definitively not an expert. I just want to share what I have found to help others on there journey. The author of the posted Shadertoy examples has a blog where he talks about it: https://michaelmoroz.github.io/Reintegration-Tracking/ Some papers that are talking about it are not behind a paywall (...


3

Every iterative solver -- Jacobi, SSOR, CG, etc -- starts with an initial approximation. One often just uses the zero vector, but there is nothing wrong with using the solution of the previous time step. In fact, extrapolating from previous time steps to the current one is an even better idea -- one the authors apparently missed! For some iterative solvers, ...


3

The recent survey published in SIAM News is a good starting point to study the work done in this area.


3

Do you wish to model the two-way coupled solid-fluid flows or you wish to carry out PIV or tracking on particle images? If you wish to model the two-way coupled solid-fluid flows, there are numerous type of models are that valid, most notably Euler-Euler approaches or Euler-Lagrange (CFD-DEM unresolved or resolved) approaches. Euler-Euler approaches are ...


2

If you're confused why this operation is called "projection", note that the equation $$ \text{div} \, \mathbf u = 0 $$ means that we only consider velocity fields that are divergence free. This is no different than asking to find a solution of some problem subject to a constraint of the form $Bx=0$ where $B$ is a matrix with fewer rows than columns. The ...


2

Use multistep methods in this case. See the Adams-Bashforth method if it's nonstiff, or Adams-Bashforth-Moulton methods, or if its stiff BDF methods. These use past timepoints like you want in order to increase the order of accuracy. I would highly recommend checking out something like Sundials which has variable timestep plus variable order methods. This ...


2

You cannot take deterministic methods and slap a kind of Milstein correction on them to get higher order. Instead, you should be looking at methods which are specifically designed for integrating SDEs at high order if you want good accuracy. If pure efficiency is what matters and the noise is additive/diagonal, then the adaptive Rossler methods mentioned by @...


2

First of all, it is impossible to intertwine a multi-step Runge–Kutta method and the Milstein–Itō methods for a multitude of reasons that go beyond the scope of this question¹. So the best you can possibly do in is: Make a deterministic Runge–Kutta step, ignoring the noise term. Apply noise (in a Milstein–Itō fashion). Go to 1. This has the problem that ...


2

It seems like you are inventing Barnes-Hut-type algorithm, which is a fundamental accelerated algorithm for n-body simulations. It follows a similar logic: you combine masses on a grid. But, it is done in a more elaborate fashion: when particles are close, you still use direct calculations (no combination), otherwise, you would lose accuracy. multipole ...


1

If you make the assumption that the force the two objects exert on each other is along the connecting line (this is necessary for the conservation of angular momentum, but assumes an infinite speed with which forces penetrate space), then you have that the differential equation for particle 2 is $$ \ddot x_2(t) = f(\|x_2(t)-x_1(t)\|) (x_2(t)-x_1(t)) $$ and ...


1

You ought to be interested in this preprint in which we discuss exactly the sort of questions you seem to be having: https://arxiv.org/abs/1612.03369


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