# Questions tagged [stability]

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

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### Linear stability in gravity driven Flow numerical solution

I am trying to solve a gravity driven flow problem for thin films and I am having some difficulties solving the PDE resulting from the linear stability analysis for the steady state solution. This has ...
1 vote
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### 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 ...
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### 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 ...
1 vote
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### 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 ...
74 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 ...
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### 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-...
77 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. ...
1 vote
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### 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 ...
57 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 (...
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### 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: ...
167 views

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### Finite Difference for Advection Equation With Source

I'm trying to find a convergent finite difference scheme for the PDE \begin{equation} \begin{split} u_t + (x-1) u_x &= (x-1)u, \hspace{.5cm} x \in [0,1] \\ u(x,0) &= 1 \\ u(1,t) &= 1. \\ \...
49 views

### In sights into why higher order finite differencing method leads faster to instability

I was playing around with numerically solving the 1D wave equation with density and stiffness varying with position using central differencing methods and noticed that for certain discretization steps ...
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### discretizing advection equation with variable wave speed + stability

I currently have a code that solves $u_t+ cu_x=0$ with periodic boundary conditions, and constant $c$ (using an upwind method). I'm wondering how I would alter this code to solve something of the form ...
188 views

### How do I apply BDF2 in a STRANG splitting

I have a 3D diffusion equation that I want to solve using a time splitting (2D+1D). Assume that $A$ is the 2D discrete diffusion operator and $B$ is the 1D discrete diffusion operator. I want to use a ...
384 views

### How to use numerical integration to calculate the surface area of a superellipsoid?

I am working in an application in which I need to calculate the surface area of a superellipsoid. I have read that there is no closed form solution (see here), so I am trying to compute it using ...
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### How to increase the stability of a DAE solver?

I am trying to solve a set of linear PDEs of the form $$F\left(\vec{y},\frac{\partial \vec{y}}{\partial x},\frac{\partial^2 \vec{y}}{\partial x^2},\frac{\partial \vec{y}}{\partial t}\right)=0.$$ To ...
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### Compute the sum of probabilities when they are given as logits

Say I have a set of numerous probabilities given by their logarithm : $\{\ln p_i, 1 \leq i \leq N\}$. I want to compute $\sum p_i$, if possible without exponentiating $\ln p_i$, since some of those ...
1 vote
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### Showing stability of numerical scheme: Decaying norm implies stability?

I have a little trouble formulating my question since I am not really sure what conclusions I am supposed to draw from the results I have obtained. I am sorry for the long problem formulation below, ...
112 views

### Determine stability of an algorithm?

This is related to a question I answered on Stack Overflow regarding calculating the square root of a number. I was thinking about it and realized that the formula is just the first in a family of ...
335 views

### log-sum-exp trick for signed/complex numbers

I need to evaluate a sum of values that are on many different orders of magnitude in scale but might be signed. I’ve had great luck with the “log-sum-exp” trick for an unsigned version of my problem, ...
359 views

### Numerical stability in the product of many matrices

I have to calculate in numpy the matrix-product of many matrices (~400). Are there common practices to increase numerical stability? If this is relevant, the matrices are $300\times 300$ orthogonal ...
I am trying to use the modified equation to derive the stability condition for the finite difference approximation  \frac{u(x,t+\Delta t) - 2 u(x, t) + u(x, t -\Delta t)}{\Delta t^2} = c^2 \frac{...
I'm trying to calculate the maximum step size that provides stability for the following nonlinear IVP using the Euler forward method: $u'(t) = -200tu(t)^2,\qquad u_0 = 1, \qquad t\in [0,3]$, with ...