# All Questions

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### Understanding the Eisenstat-Walker method for choosing the tolerance of a linear solver when solving a non-linear PDE

We are working on the solution of large non-linear PDE (say the Navier-Stokes equation) which we solve using Newton's method with an analytical formulation of the jacobian. For very large systems, we ...
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### 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 ...
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### Calculating the jacobian of norm and square root terms in the Finite Element Method

In the code that my group is writing (Lethe) we use a stabilized approach to solve the Navier-Stokes equation. The GLS stabilized method we use has a stabilisation term which contains a stabilization ...
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### Generate a non-equispaced grid with higher points in the middle

I know it's a really simple problem, but I'm quite stuck on it. I have $M+1$ points $x_0, \ldots, x_M$ frand want to create a non equispaced grid where the points have higher density in the middle. ...
37 views

### 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 ...
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### Integrators for Nonlinear/Stiff PDE

It was suggested I ask this question in this section. Anyway: I have a particular nonlinear PDE of the form $$u_t(x,t)=iu_{xx}(x,t)+f(x,u(x,t)) \tag{1}$$ Where f is some nonlinear function. With ...
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### Implications of thermodynamic inconsistency in CFD calculations

During my PhD work, I had to use tabulated values of thermodynamic properties of gases in some Computational Fluid Dynamics (CFD in short) simulations. My tables are discretized in temperature and ...
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### FEniCS, refinement not 'respecting' domain boundary

Short question: how to ensure that extra points are not included as 'boundary' points after calling the refine function. More details. I am working with a hexahedral mesh in $3$d. Let $X$ be the set ...
54 views

### 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 ...
6k views

### How to prove the 2-norm of an invertible matrix is exactly the reciprocal of its minimum singular value?

If a matrix $A_{n\times n}$ is invertible, then $\left\|A^{-1}\right\|_2 = \dfrac{1}{\min\limits_{i} \sigma_i}$ where $\sigma_i$ is the $i$-th singular value of $A$
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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 $\Omega _1$ $$\begin{bmatrix}4&-1\\-1&4\end{... 1answer 55 views ### Differences between Discrete Fourier Transform and Continuous Fourier Transform? I am trying to visualize the time dependence of a free particle given an initial wave-function using Python and I just wanted to know if I could use the in built FFT implementation from NumPy to find ... 2answers 70 views ### Chebyshev differentiation via FFT with a domain [a,b] I want to ask something about Chebyshev differentiation via FFT, which can be used to obtain with spectral accuracy the derivative of a smooth function. See for instance this code in python, which ... 1answer 247 views ### ODE Solving in Scilab I have a certain ODE problem that needs to be solved using Scilab. dx(1)/dt=k*x(1)-x(5) dx(2)/dt=k2*x(2)-k1*x(1) dx(3)/dt=k1*[x(2)-x(3)] dx(4)/dt=k1*[x(3)-x(4)] <... 1answer 43 views ### Matrix multiplication not working in Scilab I entered an instruction to calculate the coordinates of a vector after a change of basis in order to repeat it many times with various vectors. X0=[1;1/2] is a ... 1answer 330 views ### General case Kutta condition I'm working on a 2D inviscid fluid simulation using a "panel method", with Potential being used to enforce the no-through boundary condition. I'm trying to incorporate the Kutta condition, which says ... 1answer 119 views ### Numerically estimating expected value of f(x) when x is normally distributed I need to estimate$$ \mathbb{E}_x[f_i(x)] = \int_{\mathbb{R}^n} f_i(x) p(x) dx $$for many functions f_i(x), where p(x) is the density of a normal distribution. The evaluation of all the ... 2answers 134 views ### Lattice Boltzmann methods vs Navier stokes/ other eulerian methods for *water* simulation Note, there is already a question here, however the answers don't answer the original question, let alone specific considerations when dealing with nearly in-compressible fluids (water). Another ... 1answer 76 views ### Eigenfaces Algorithm This might be a silly quesntion but recently I've been trying to program the eigenface algorithm using PCA, so I arranged the face vectors vertically in a matrix X such as: X = [x1,x2,x3,...,xn]; In ... 3answers 93 views ### How the gmres method iteration behaves for this **enfant terrible** matrix? Recently, I have been studied my lessons about gmres iteration, probably the most popular iteration method for general large sparse linear system of equations Ax=b. And the convergence is obtained ... 2answers 62 views ### Can a direct method like Thomas be used in a multigrid method as a smoother? As far as I know, multigrid uses stationary iterative methods as smoothers (i.e GS), but can we use a direct method also? For example, in case we have a tridiagonal system (for example 1D heat ... 1answer 47 views ### Partially Banded Matrix I have a somewhat peculiar Jx=R system that I need to solve. The matrix J is 2N -by- 2N. The first N rows have all entries filled. The next N rows are banded in two places, i.e. for the (N+k)th row, ... 1answer 31 views ### Accelerating Conjugate Gradients fitting for small localized kernel (like cubic B-spline) Question: Is there some pre-conditioner for Conjugate-Gradient (CG) cheap enough, that it is worth using even if my operator is very local (i.e. already has a low number of non-zero elements), as it ... 1answer 76 views ### Simulating advection - diffusion problem in a network of 1D pipe I'm interested in solving the following advection-diffusion system in a 1D network of pipes.$$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2} - v\frac{\partial C}{\partial x}$$... 1answer 101 views ### Solving an SDE with time-dependent parameter in R I am trying to solve a system of SDEs in R using the Diffeqr package. Let's reduce the system to a simple ODE: ... 2answers 382 views ### Correct use of scipy's sparse.linalg.spilu I'm attempting to use scipy's spilu routine as a preconditioner and I'm finding bad performance for my application (solving a global linear system arising from a DG ... 1answer 139 views ### A Bound for the inverse of the sum of identity and triangular matrix I wonder if there are any theorems which can help me to calculate an upper bound for the spectral norm of:$$\left\| \left[ I + \sum_{i=1}^{\overline{n}\in\mathbb{N}} \big( C_i - I\big)\right]^{-1}\...
I need to find the smallest eigenvalue and the corresponding eigenvector of a sparse matrix $M$ whose dimension is $\approx 10^4$. Within Matlab enviroment, I use the command ...