# Tag Info

Accepted

• 1,154

### Why is my simulation of a first-order wave equation not stable?

Numerical solution of the advection equation with centered differences in space and forward Euler in time is unconditionally unstable. So the behavior you are seeing is expected. Here is a nice ...
• 6,144

### Numerically stable computation of $x^T A x$

You don't need a temporary vector. Instead, you loop over the elements of the matrix $(i,j)$ and update a single counter: ...
• 55.8k
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### More Smearing with decreasing timestep in advection problems

You solve the 1D-advection equation with $c$ a constant velocity: $$\frac{\partial u}{\partial t} + c\frac{\partial u}{\partial x}=0~~~~~~~~(1)$$ When you discretize this equation (with an explicit ...
• 629
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### Method to solve linear, first order ODE of generalized matrix matrix form

There are a few ways you can do this. Even though RK4 fails unless stepsizes are really small, one thing that can happen a lot is that the model can start out more stiff than it really is in the full ...
• 12.3k
Accepted

### What is the flaw in my stability analysis?

But clearly, this is not the case as my programs do come up with (an approximate) solution though. I believe you did not continue the integration until you see that your integration is not convergent ...
Accepted

### What is the meaning of stability in numerical analysis? How to deterimne the stability of a numerical method?

You can define a method/algorithm stable if during the various steps it not amplifies excessively the errors on the data. So you introduce some bounded estimate. Note that this kind of errors, ...
• 1,340
Accepted

### Von Neumann stability analysis with a constant term

From your link, consider the definition of the round off error and the statement "Since the exact solution must satisfy the discretized equation exactly, the error must also satisfy the discretized ...
• 2,249
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### Why is subtraction a stable operation?

(My apologies for the terrible typesetting) Let me start with the definition of backward stability of an algorithm: An algorithm $A$ to solve a problem $P$ is called backward stable, if for all $x$ in ...
• 2,821
Accepted

### How to interpret if $\displaystyle \sum_{j = 0}^{n} \frac {1}{j!}$ is a stable algorithm for computing $e$?

As hardmath mentioned in the comments above, if data space $D$ is empty, then the first half of the condition "for all $X\in D$ there exist $Y\in D$ such that $||X-Y||/||X||\approx\epsilon_m$ and ...
• 2,821
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### Python bifurcation diagram of seasonally forced epidemiological models

Recently in https://math.stackexchange.com/questions/4542008/how-to-loop-parameter-a-in-henon-map I came into contact with the idea of an "adiabatically gliding" parameter where one gets the ...
• 6,109
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### Can this finite difference dispersion be eliminated somehow?

It isn't the spike that's causing the dispersion. The scheme you use has a dispersion relationship whereby waves of different frequency travel at different speeds. Every numerical scheme has such a ...
• 55.8k

### Stability of Euler forward method

The solution of $\frac{du}{dt} = Au$ is $u(t) = \exp(tA)u(0)$, and explicit Euler approximates $\exp(tA)$ using $\lim_{n\to\infty} \left(I+\frac{t}{n}A\right)^n$. Of course in practice you cannot ...
• 2,197
Accepted

### Improve numeric stability of subtraction in C++

This is comment, but too long for comment so I write here. I think is better you edit the post with more information, now I explain why. Here you can find some general advice to improve precision. ...
• 1,340

### Lagrange multipliers space is too rich in a mathematical view

Each Lagrange multiplier corresponds to a constraint. So if the space of Lagrange multipliers is too large, then you have too many constraints that can no longer be all satisfied at the same time ...
• 55.8k
Accepted

### Lagrange multipliers space is too rich in a mathematical view

The saddle point matrix you must solve takes the following form: $$\begin{bmatrix} A & B^T \\ B \end{bmatrix},$$ where $A$ is the unconstrained matrix and $B$ is the matrix that observes the ...
• 3,143

### Numerical stability in the product of many matrices

Orthogonal matrices are about as well-conditioned as you can get, but numerical errors still occur. One common error is loss of orthogonality. A fix for this could be to re-orthogonalize your columns ...
• 2,636

### Advantage of diagonal "jitter" for numerical stability?

Look up something on Tikhonov regularization, also known as ridge regression in machine learning. This is a standard technique (but I agree that the explanation in that notebook is somewhat poor). ...
• 11.6k
Accepted

### Advantage of diagonal "jitter" for numerical stability?

So, you want to invert your matrix $A=\Phi^T\Phi$. For $A$ to be invertible it must not have zero eigenvalues. We can show that $A$ is positive semi-definite as follows. Positive semi-definite means ...
• 760
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### Franco Brezzi's didactic paper on the Inf-Sup condition

I believe you are thinking of "A discourse on the stability conditions for mixed finite element formulations" by Franco Brezzi and Klaus-Jürgen Bathe, published in Computer Methods in ...
• 2,821
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### CFL condition in polar coordinates

I was going to write a comment, but the equation seems to view better in answers.. I assume Von Neumann analysis is the proper approach to derive this equation, but a coordinate transformation from ...
• 619
Accepted

• 1,058

### Why the numerical solution of advection-dominant problem is challenging

The difficulty is relative to something, in this case it is relative to diffusion dominated problems. Diffusion dominated aren't "easy" either, they have their own set of problems. I'll start with ...
• 3,283

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