# Computational Science Stack Exchange Community Digest

## Top new questions this week:

### "Geometry of ill-conditioning" for least-squares problems

It is an idea that dates back to Demmel, 1987 that the condition number of a problem is often related to the distance to the closest ill-posed problems. In Section 3 of the above paper, the author ...

reference-request least-squares condition-number intuition

### Calculate Transformation Matrix between two sensors

My question is if I can calculate the transformation matrix between two sensors. Each sensor provides a $4\times 4$ matrix for every timestep recorded. The sensors are moving and have some noise in ...

optimization python matrix algorithms nonlinear-equations

### How to deal with pseudo-compressibility of lattice Boltzmann method when you are calculating mass flux?

In lattice Boltzmann method, we have a concept, which is called pseudo-compressibility and it is defined based on the fact that LBM simulates incompressible flows by having small Mach number to ensure ...

incompressible lattice-boltzmann-methods

### Why wall shear stress calculated from LBM directly and the one calculated based on velocity profile are so different in some cases?

First of all, I hope you accept my apologizes if my question seems off topic here. But, I asked this question in ParaView forum and after a week still I did not receive any response yet, so I'm ...

unstructured-mesh paraview lattice-boltzmann-methods

### Attempting to perturb ODE when initial condition is equilibrium point does not work

I have the following system of differential equations: $$x' = ax- cy + e1$$ $$y' = by- dx + e2$$ for variables $x,y$ and parameters $a,b,c,d,e1,e2$. I'd like to solve this in python, which is ...

python ode
 asked by Jason Cole 1 vote

### Can the standard multigrid performance be used for time-dependent PDEs?

Consider a time dependent pde(i.e u(x,t)).I know when only space-coarsening is used the standard multigrid performance can be applied but what if instead we use only time-coarsening?Can we apply the ...

pde finite-difference iterative-method discretization multigrid

### understanding Domain Decomposition with example

I am new in Domain Decomposition method. I am started to solve $-\Delta u = f$ in $\Omega$ and $u = 0$ on $\partial\Omega$. From that I get in $\Omega _1$ ...

pde sparse-matrix domain-decomposition

## Greatest hits from previous weeks:

### What is the fastest way to calculate the largest eigenvalue of a general matrix?

EDIT: I am testing if any eigenvalues have a magnitude of one or greater. I need to find the largest absolute eigenvalue of a large sparse, non-symmetric matrix. I have been using R's eigen() ...

linear-algebra algorithms eigensystem sparse-matrix

### In FEM, why is the stiffness matrix positive definite?

In FEM classes, it's usually taken for granted that the stiffness matrix is positive definite, but I just can't understand why. Could anyone give some explanation? For instance, we can consider the ...

finite-element matrix stiffness

### Basic explanation of shape function

I just started studying FEM in a more structured basis compared to what I used to do during my undergraduate courses. I am doing this because, despite the fact that I can use the "FEM" in commercial ...

finite-element

### How to formulate lumped mass matrix in FEM

When solving time dependent PDE's using the finite element method, for example say the heat equation, if we use explicit time stepping then we have to solve a linear system because of the mass matrix. ...

finite-element time-integration navier-stokes

### Estimating the Courant number for the Navier-Stokes Equations under differing Reynolds number regimes

I am familiar with the Courant-Friedrich-Lewy Condition in as far as it applies to the stability of explicit finite difference schemes for standard parabolic and hyperbolic PDEs. However, when ...

fluid-dynamics stability

### Computational Complexity of 2D Convolution

I am using image filtering for an image processing algorithm I'm developing. I'm using a predefined Matlab function to do the convolution, but I'd like to know what the computational complexity is for ...

complexity image-processing

### Importance of the Time Step Value for the Accuracy of a Transient CFD Simulation

Overview My understanding is that one should use a time step $\Delta t < \frac{h}{v}$ (where h - smallest mesh element, v - velocity) to get an accurate result. But how important is this really ...

fluid-dynamics