Skip to main content

Questions tagged [stability]

The study of the propagation of errors in a numerical algorithm.

Filter by
Sorted by
Tagged with
-2 votes
0 answers
83 views

stability issues - how to find and fix them? [closed]

When reading about the stability of the softmax function, I found a nice trick one can use during implementation: subtracting the maximal element from all elements. That way none of elements "...
Przemek B's user avatar
3 votes
1 answer
116 views

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 ...
Nicolas Tardieu's user avatar
4 votes
1 answer
158 views

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 $A$ ...
shuhalo's user avatar
  • 3,670
0 votes
0 answers
45 views

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 ...
Chaozhi Qiu's user avatar
2 votes
2 answers
75 views

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 ...
mikejm's user avatar
  • 123
0 votes
2 answers
2k views

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 ...
FriendlyNeighborhoodEngineer's user avatar
2 votes
0 answers
80 views

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{\...
Jacob Ivanov's user avatar
0 votes
0 answers
44 views

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 ...
FriendlyNeighborhoodEngineer's user avatar
1 vote
1 answer
119 views

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 ...
user46892's user avatar
3 votes
1 answer
185 views

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 ...
rainbow's user avatar
  • 31
2 votes
1 answer
145 views

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 \...
FriendlyNeighborhoodEngineer's user avatar
5 votes
0 answers
90 views

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 ...
jjjjjj's user avatar
  • 325
6 votes
1 answer
1k views

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)^{...
NNN's user avatar
  • 760
1 vote
0 answers
132 views

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)...
NNN's user avatar
  • 760
1 vote
1 answer
168 views

Conditioning and Stability of generalized eigenvalue problem

The (generalized) eigenvalue problems with a multiple eigenvalue are the ill-posed ones. I have two questions that should be simple for experts: (1) Is the eigenvalue problem much more sensitive to ...
alpx's user avatar
  • 113
2 votes
0 answers
68 views

Literature request for pinning the corner singularities in finite differences

Corner singularities cause instability of finite difference solutions. They occur at the intersection of boundary conditions. I was recently going through an online video course that was explaining ...
FriendlyNeighborhoodEngineer's user avatar
0 votes
1 answer
302 views

computing Lyapunov exponents numerically

I am trying to compute numerically the Lyapunov exponents of an ODE. I follow the method described in Parker, Chua "Practical Numerical Algorithms for Chaotic systems" There is also relevant ...
0x11111's user avatar
  • 101
1 vote
1 answer
92 views

Improved euler on hybrid methods where both time and space are discretized?

I am trying to understand how to use the improved euler method on MPM simulations. In the kind of MPM simulation I am doing with forward euler the order of operations is as follows: Write particle ...
Makogan's user avatar
  • 263
2 votes
1 answer
285 views

Why do we use modified pressure in incompressible multiphase solvers with gravity?

The context of my question is two-phase incompressible solvers such as interFoam in OpenFOAM, but I have seen this trick used ...
Robert Manson-Sawko's user avatar
0 votes
1 answer
54 views

RFEM: Tensile force in member in buckling analysis

Here is a simple T-shaped structure in RFEM. The structure is loaded from above, into the intersection of all members. I performed a buckling analysis, and obtained the buckling modes that I'm ...
S. Rotos's user avatar
  • 133
2 votes
1 answer
131 views

Ways to fix block Lanczos tridiagonalisation numerical instability for matrix with degenerate, closely spaced eigenvalues?

I want to run a block Lanczos block-tridiagonalization on a hermitian, sparse matrix (of relatively small size $\sim 10^2 \times 10^2$). However the matrix typically has many eigenvalues that are ...
lm1909's user avatar
  • 21
1 vote
2 answers
135 views

Numerical code to solve LLG is not preserving norm

I am new to this thread. I am trying to do a simple exercise on solving the LLG equation. The equation reads: $\frac{d\vec{m}}{dt} = -\gamma(\vec{m} \times\vec{H})$. Given a normalized input state ($...
rahman62's user avatar
0 votes
0 answers
132 views

2D wave equation is numerically unstable using Finite Difference Method

I'm working with simulating both the heat and wave equation in 2D in a Python code. When simulating the heat equation, I learned about the CFL which I used to get a numerical stable solution. I found ...
Tanamas's user avatar
  • 101
0 votes
0 answers
43 views

Instability due to different lumping dimensions in second-order linear system

I have a problem that consists of a linear hyperbolic differential equation with some boundary and initial conditions which leads after discretization in space using finite elements to the following ...
DanielRch's user avatar
  • 482
0 votes
2 answers
260 views

Practical implementation of the discrete compatibility condtion

I've recently started looking into writing a finite-differences-based solver for the Poisson equation of the form $$\nabla\left(\varepsilon\nabla\varphi\right)=\rho$$ in 2D for arbitrary geometries (...
Akhaim's user avatar
  • 83
3 votes
2 answers
786 views

Python bifurcation diagram of seasonally forced epidemiological models

TL:DR How can one implement a bifurcation diagram of a seasonally forced epidemiological model such as SEIR (susceptible, exposed, infected, recovered) in Python? I already know how to implement the ...
Jared Frazier's user avatar
2 votes
0 answers
111 views

Numerical precision on tricky coupled nonlinear boundary value problem on infinite interval

I am trying to solve with high precision the following coupled system $(f,h)$ on $[0,\infty]$: $$-h''-\frac{1}{r}h'+\lambda_c h(f^2-1)+4\lambda_h h^3=0$$ $$-f''-\frac{1}{r}f'+\frac{1}{r^2}f+ f(-\...
phyphy's user avatar
  • 33
1 vote
1 answer
203 views

Boundary value problem solver fails on trivial case

I am trying to solve a boundary value problem on $[0, \infty]$, using scipy's scipy.integrate.solve_bvp and I am seeing that the solutions are not converging even ...
phyphy's user avatar
  • 33
1 vote
0 answers
34 views

How to classify ODE equilibria that are stable but slowly changing in value with time?

I'm numerically solving a system of coupled ODEs where time is the independent variable. At each time, I can solve for the equilibrium values of my state variables where their respective derivatives ...
quantumflash's user avatar
2 votes
3 answers
171 views

Should reaction be taken into account in the CFL condition when solving advection-diffusion-reaction equations numerically?

I'm dealing with the numerical resolution of advection-diffusion-reaction equations. I've encountered references on CFL conditions for conservation laws as well as advection-diffusion equations. I'm ...
Max_89's user avatar
  • 61
1 vote
1 answer
70 views

stability of a numercial scheme for a hyperbolic system?

This is related to my question here https://math.stackexchange.com/questions/4447383/lax-wendroff-scheme-stability-analysis-for-a-linear-system-of-conservation-laws . Consider the numerical scheme ...
NotaChoice's user avatar
4 votes
0 answers
85 views

Numerical algorithms made stable by unums which are unstable on IEEE floats

For unums, there is good evidence (see figure 5) that accuracy is better than IEEE floats. (Note: I use the term "unum" broadly to refer to any of the various iterations and revisions to the ...
user14717's user avatar
  • 2,155
0 votes
0 answers
96 views

How to estimate stability and stiffness of a system of coupled ODEs?

I'm running into issues with Python/Julia ODE solvers requiring prohibitively small timesteps to evolve a system of 4 coupled ODEs (the order of magnitude of the state variables and time unit span ~40-...
quantumflash's user avatar
2 votes
0 answers
235 views

Numerical instability in the inverse Laplace transform

I have a problem with Laplace inversion and my function is not numerically stable for the Laplace inverse, but I do not understand the cause of this problem. Here is my code and graph of this problem. ...
Ali AlCapone's user avatar
1 vote
0 answers
48 views

Generate numerically stable equation automatically

I remember there is a software (and its website interface) that can generate numerically stable equation from an expression automatically. To see the problem it tries to solve, let’s take a look at a ...
Firman's user avatar
  • 181
2 votes
0 answers
59 views

robust numerical calculation with large (almost) offsetting terms

I am attempting to evaluate an expression that defines a probability of a particular event. With 3 different events, I can (under some statistical assumptions) characterize the probability of event 1 (...
eyejay's user avatar
  • 21
4 votes
1 answer
219 views

Stepping over a rapid oscillation in advection

As a model of a more complex problem, I am studying a linear advection equation with a rapidly oscillating velocity $$ \partial_t u + a \cos(\omega t) \partial_x u = 0 $$ with initial condition $u(x,0)...
Endulum's user avatar
  • 735
3 votes
1 answer
407 views

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

I am trying to solve problem $15.1$ from Numerical Linear Algebra by Trefethen and Bau, which reads Determine whether the algorithm is backward stable, stable but not backward stable, or unstable. ...
user56202's user avatar
  • 163
2 votes
1 answer
854 views

Why is subtraction a stable operation?

In Numerical Linear Algebra by Trefethen & Bau, it is claimed that subtraction is backward stable. Here is the proof: Let $f(x, y) = x-y$ and let $\tilde f(x,y)$ be the answer you get when doing $...
user56202's user avatar
  • 163
2 votes
0 answers
137 views

Floquet theory for periodic delay differential equations: current numerical routines

I would like to determine the stability of a system of periodic delay differential equations (a seasonal host-parasite model). I've tried to implement the method described in Lemma 2.5 in this paper: ...
Alnasc's user avatar
  • 31
4 votes
1 answer
545 views

Computing second derivatives with Neumann boundary condition

I am implementing a finite difference method for a PDE with a Neumann boundary condition. I will simplify my question to a single dimension. Suppose I have a PDE $$\frac{\partial u}{\partial t} = \...
user41226's user avatar
2 votes
0 answers
490 views

2d advection-diffusion: cell Péclet number and numerical stability

I am studying the numerical resolution of 2d advection-diffusion problems with finite element methods. $$\frac{\partial u}{\partial t} + \beta\cdot\nabla u = \nabla\cdot(\nabla u) \, .$$ It is said in ...
Max_89's user avatar
  • 61
2 votes
0 answers
132 views

About the the stability of using an explicit scheme on the heat equation

Before I get to the heat equation I'd like to talk about the advection equation. Descritize that with FD in time and BD in space: \begin{equation} \dfrac{u^{n+1}_i - u^{n}_i}{\Delta t} + v \dfrac{u^{n}...
CuteCompute's user avatar
4 votes
1 answer
147 views

Stability analysis simplification for PDE

I have the nonlinear PDE $$\frac{\partial U(z,t)}{\partial t} + A(U)\frac{\partial U(z,t)}{\partial z} + B(U)U(z,t) + C(z,t) = 0,$$ where $A(U)$ and $B(U)$ are guaranteed to be real and positive. I ...
mcv100's user avatar
  • 43
0 votes
0 answers
97 views

Transient advection equation with stabilized FEM

I am interested in solving the transient advection equation $\left\{\begin{array}{ll}\partial_{t} u+\beta \cdot \nabla u=f & \text { in } \Omega, t>0 \\ u=0 & \text { on } \partial \Omega^{-...
balborian's user avatar
  • 601
1 vote
2 answers
246 views

Stability condition FCTS method

The FTCS method comes from the discretization of a diffusion PDE like this: $$ a^{2} \frac{u_{i+1}^{k}-2 u_{i}^{k}+u_{i-1}^{k}}{\Delta x^{2}}=\frac{u_{i}^{k+1}-u_{i}^{k}}{\Delta t} $$ If I have the ...
LongJohn's user avatar
0 votes
1 answer
123 views

Free Electron Schrödinger Equation (Energy Method)

For the simplest atom, its wave function is described by the PDE of Schrodinger equation: $$ -i h \frac{\partial u}{\partial t }=\frac{h^{2}}{2m} \Delta u + \frac{e^{2}}{r}u$$ The potential $\frac{e^{...
user avatar
0 votes
2 answers
133 views

Stability of the Forward-Time Central Space method, section 9.3 in LeVeque

I am reading section 9.3 in Leveque about stability. The explanations are very short and brief so I am asking for some help to understand and elaborations. The result of the section is that for FTCS ...
econmajorr's user avatar
2 votes
1 answer
103 views

Positive feedback instability

I have two black boxes (representing some multi-physics but that does not matter), where box #1 takes some quantity T as input and produces some quantity P as output; and box #2 takes P as input and ...
Maxim Umansky's user avatar
1 vote
0 answers
128 views

Comparison between stability and accuracy of various Finite Difference schemes

I`m Analyzing the 1D convection equation (PDE) for stability, consistency, and accuracy. I know Both upwind schemes (explicit and implicit) show better stability with a higher number of waves for ...
Abdelrahman Mabrouk's user avatar

1
2 3 4 5