Questions tagged [stability]
The study of the propagation of errors in a numerical algorithm.
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Franco Brezzi's didactic paper on the Inf-Sup condition
I am trying to recover the title of a very didactic paper by Franco Brezzi in which he explains the Inf-Sup condition with a very simple case. As I recall, it was a simple Laplacian equation in 1D ...
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Numerically stable computation of $x^T A x$
We have a large sparse symmetric positive-definite matrix $A \in \mathbb R^{N \times N}$ and a vector $x \in \mathbb R^N$. How do I practically compute the inner product $x^T A x$ when the matrix $A$ ...
- 3,680
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2nd-order backward difference approximation for temporal terms of Euler–Bernoulli beam
I am trying to solve the Euler–Bernoulli beam numerically in structural dynamics analysis:
$$\frac{\partial^2w(x,t)}{\partial t^2}= \frac {q(x,t)}{\mu(x)}-\frac {1}{\mu(x)}\frac {\partial^2}{\partial ...
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How to improve and stabilize this code simulating a damped mass-spring oscillator? Runge-Kutta?
I wrote the following function which is simulating a damped mass-spring oscillator. It is being driven by the audio sample input at 44.1 kHz sampling to create the same effect as a resonant bandpass ...
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When can I use finite differences for differentiation?
Finite differences are usually used to integrate ODE's and PDE's. However, sometimes they can be used for differentiation which I illustrated simply by using the Matlab code below to differentiate the ...
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Preventing an Overflow in Exponential Integrating Factor
The following is the Discrete Spectral Vorticity Evolution PDE for incompressible flow:
$$ \frac{\partial \Omega_{pq}}{\partial t} = \nu \left( \frac{\partial^2 \Omega_{pq}}{\partial x^2} + \frac{\...
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Which numerical method can I use to solve this system of hyperbolic PDEs?
Backround
The mathematical model I am trying to numerically solve models wave propagation inside a cylinder with specific material properties suited for dynamic loading. The cylinder's upper base is ...
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1
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Step size constraint in Euler backward
I am dealing with an assignment in MATLAB. It has to do with 'self-driving' cars which are driving in-front/behind eachother. Assuming M cars on a single-lane road, each car adjusts its speed based on ...
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Stability of Euler forward method
I am trying to solve a linear system of ODEs of the form:
$$ \frac{du}{dt} = A u, \quad u(0)=k$$
where $A$ is a 2x2 matrix and $u(t)$ is a 2x1 column vector. I want to solve this numerically, using ...
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Can this finite difference dispersion be eliminated somehow?
I am trying to solve the wave equation
$$ {\partial ^2u(t,x) \over \partial x^2} = {\partial ^2u(t,x) \over \partial t^2} \tag1 $$
with the following boundary and initial conditions:
$$ {\partial u \...
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optimization scaling techniques
Consider a convex QP of the form
$$
\min_x \bigl\{\tfrac12 x^\top Q x + q^\top x : Ax\leq b\bigr\}\tag{P}
$$
with dual
$$
\min_y \bigl\{\tfrac12 (A^\top y + q)^\top Q^{\dagger} (A^\top y+q) + b^\top y ...
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Why is matrix inversion unstable when svd is stable?
I've heard that matrix is inversion is unstable whereas the SVD is stable.
Now, if $A$ is an invertible matrix, then its SVD is
$$
A = USV^T
$$
Then wouldn't it's inverse just be
$$
A^{-1} = (USV^T)^{...
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Stability for the 2d diffusion equation
I'm reading this set of notes on numerical stability of the diffusion equation. I can't seem to derive equation (7.17) from the previous equation. I've tried using $\sin^2(\theta)=\frac{1-\cos(2\theta)...
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