Your reasoning for vector type equation is correct when you wrote it down for each individual component. What occurs is that you need to write your interpolation basis function and your test function ...

I second the idea of using Eigen, which is pretty efficient, but also very simple to include. If you need a lot more performance, you could try to use PETSc or Trilinos. They are very powerful ...

Well, the first obvious test is to check if the determinant of the Jacobian matrix is positive. If it not, it means that your element has inverted and your answer is going to be invalid. Otherwise, I ...

You appear (and correct me if I am wrong, i do not wish to presume) to be confused on the notion of what is an order of convergence. The order of convergence is the order at which your approximate ...

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 ...

A good way to achieve this is to use the Warp By Vector filter to warp your field in the Z direction using the value of your scalar. Just to give a brief overview, The warp by vector warps your field ...

This is one of those scenarios where assessing the convergence order of your scheme is difficult, because, as you explained, your time-step is "linked" to your spatial discretization. What ...

It depends on the equation you are solving. For example, consider the following heat equation with a source term: $$\frac{dT}{dt} - \nabla^2 T = S$$ If $S \neq 0$, then you have a source term that ...

A good way to obtain to verify the CFD portion of your code is to proceed using the method of manufactured solution (MMS). You can look at the book by Oberkampf & Roy or even this article by me. ...

The Lattice Boltzmann is not a meshless method. Actually, when looking at it, it is a finite difference method on an homogenous structured Cartesian grid (dx = dy = dz). However, the variables ...

I will try to answer as best as I can your three questions. 1) Your approach is quite classical in that you are considering the particles to be an active scalar. What most people would usually do is ...

The answer is that you cannot increase the time step past the limit: $$\Delta t < \frac{1}{2Re} \Delta y^2$$ This is because you are using an explicit scheme. Explicit scheme have a stability ...

Let's assume the following heat transient heat transfer equation in 1D : $$\frac{\partial T} {\partial t} = \alpha \frac{\partial^2 T}{\partial x^2}$$ If we take its finite difference ...

I have some experience working in particle trajectory simulations using interpolated flow field information. In our case, we needed to simulate the motion of numerous particle using a single time ...

If you include the boundary condition directly in the matrix, you will only get the g value at the points where the boundary is prescribed. If we use 5 nodes with the following BCS: $$u_1=g_1$$ and ...

EDIT I had forgotten about this, but a while ago a friend of mine did his whole thesis work on the Rayleigh-Taylor instability in the context of statistical analysis of turbulence. You can find it ...

It seems your questions is more on "How to debug that error" than on the error itself. If you wish me to look into the error specifically, I will do this, I promise :). Otherwise, it is great to ...

I would highly suggest you go with an available FEM open-source library (say deal.II, FENICS, MFEM, etc.) instead of writing your own FEM code and then using PETSC as the underlying parallel algebra ...

The packing of spheres (or non-spherical particles) within a geometry can be achieved quite reliably using the Discrete Element Method. I refer you to the following introduction article (written by me ...

I think that both can be considered to be equally valid. In fact, asymptotically (with an infinite number of lattice) you will tend towards the incompressible athermal Navier-Stokes equations. In this ...

I believe these issues arise because you are solving the transient form of the heat equation which, being parabolic, can lead to some instability for continuous galerkin method. There are ways to ...

As mentioned in a previous comment, this depends on what your goal is. Modern visualization software such as Paraview allow a representation of data at cell centers or at cell vertices. Furthermore, ...

I will answer your points 1 (still trying to find an answer for 2) since these seem to be the key of your question 1) This depends. If your analytical solution is a steady solution, then when you ...

I believe Mathematica has some functionality for basic automatons. See for instance this reference: http://reference.wolfram.com/language/tutorial/CellularAutomata.html However, if you wish to ...

Following the comments from origimbo, the solution is actually very simple. You just need to generate a sufficient script using the trace option of paraview, if this script is functional (load your ...

Your outlet boundary condition is not correct. In your outlet BCs, you have to impose the value of $f_6$, $f_3$ and $f_7$ since these populations are not updated by the streaming step. This paper ...

I believe this paper by Breuer et al. which uses both the lattice Boltzmann and the finite volume method could be of interest for you. There is tremendous information therein and I have used it before ...

The displacement of the car is given as the integral of the car velocity + the initial displacement. Consequently, if you do not have data points covering the 0 to 30 minutes range (velocity at 0min ...