Questions tagged [stability]

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

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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 ...
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53 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 ...
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132 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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35 views

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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1answer
197 views

Method to solve linear, first order ODE of generalized matrix matrix form

The equation and its meaning: Consider two sets $(A)_{l=0,...,m_a},$,$(B)_{l=0,...,m_b}$ of hermitian matrices and a set of positive semidefinite matrices $(C)_{l=0,...,m_c}$. Each matrix has the ...
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94 views

Method of Lines Runge-Kutta nonlinear stability and behavior

I have a system of 4 nonlinear 1st-order PDEs. I want to solve them numerically by method of lines, first discretizing space. This leads to the system of $N\times 4$ coupled ODEs. $$ \mathbf{u}_{i} =...
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65 views

Why is my numerical solution to a set of ODEs infinite?

I am trying to solve the following linear PDEs $$\frac{\partial u_x}{\partial x}=-[i\omega b_{||}+\nabla_\perp u_\perp],$$ $$\frac{\partial b_{||}}{\partial x}=-\frac{i}{\omega}\mathcal{L}u_x,$$ $$\...
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27 views

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 ...
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1answer
45 views

Reinch's modification to the Clenshaw recurrence gives no improvement

The classical Clenshaw recurrence (see Algorithm 3.1 here) is less accurate as $x\to \pm 1$. So Reich proposed a modification to it, which is discussed as Algorithm 3.2 as well as by Oliver. While ...
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von Neumann analysis: computation of maximum value of amplification factor

In this question I adress the stability property of a numerical scheme (a FDM described below) to the problem $$ \begin{align} u_{t} &= \alpha u_{xx}+\beta u_{xxxx}, & &x\in\mathbb{R},\;t&...
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Convergent Finite Difference Scheme for Parabolic Equation

Consider the PDE $$u_t = b_{11}u_{xx} + 2b_{12}u_{xy} + b_{22}u_{yy},$$ where $b_{11}, b_{22} > 0$, and $b_{12}^2 < b_1b_2$. In Strikwerda's book, the ADI scehme \begin{align*} \left( 1 - \frac{...
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37 views

Unconditionally stable numerical method for 1st order non-linear coupled ODEs?

I am attempting to numerically solve the following system of ODEs: $$\begin{gather}\frac{dT_1}{dt} = f_1(T_1,T_2), \quad T_1(t=0)=T_{1,0} \\[3pt] \frac{dT_2}{dt} = f_2(T_1,T_2), \quad T_2(t=0)=T_{2,0}...
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42 views

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, ...
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93 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 ...
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856 views

Numerical Sensitivity in Density of States of Tight-binding model

I'm working with the tight-binding model, and I'm trying to learn the basics of how to compute the Density of States (DOS) $N(E)$ numerically. The DOS is given by $$N(E) = \frac{1}{N}\sum_k \delta(...
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213 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, ...
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1answer
132 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 ...
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1answer
132 views

Modified Equation and Stability for Centred Finite Differences for Wave Equation

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{...
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1answer
148 views

Step size and stability of Euler forward method

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 ...
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3answers
153 views

What is the flaw in my stability analysis?

The ODE $${d^2x\over dt^2}=-kx; k>0$$can be converted in the system of linear equations as $$\begin{align} {dx\over dt} & =v\\ {dv\over dt} &= -kx\\ \end{align}$$ Using Euler’s method, ...
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1answer
56 views

Stability of a finite-difference scheme for the reaction-diffusion equation

I currently need to solve numerically the following reaction-diffusion equation: $$\partial_tu=\partial^2_xu+u-u^2$$ For this purpose, I use the following numerical scheme (Crank-Nicolson??): $$ \...
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1answer
56 views

Bounding error of float32 matrix multiplication

Some numerical debugging led me to the minimal example below. I'm observing relative error of 0.75 on individual elements. Is there a way to estimate/bound this error without resorting to higher ...
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28 views

Finite difference/element method : time step and spatial resolution close to a finite singularity

I'm using the finite element method (FEM), but my question is quite a global question. It's related to this question but it is not the same. Let's assume we have this equation : $$\partial_t c - u\...
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1answer
753 views

Finite Difference Method Limitations/Stability Criteria

Is it possible to solve an equation with only a single derivative such as: $$\frac{\partial U(x,t)}{\partial t} = A - BU(x,t)$$ with finite difference methods? I ask as I am trying to solve the ...
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260 views

Stability of hyperbolic PDE and DG-FEM

In the book of Hesthaven and Warburton on discontinuous Galerlkin methods in example 2.3 (regarding solutions of the wave equation), the authors regard the following PDE: $$\frac{\partial u }{\...
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41 views

Testing the SUPG method and other methods for hyperbolic equations

I am interesting in integrating the simple equation $$ \frac{\partial \phi}{\partial t} + \mathbf{u}\cdot\nabla \phi = 0 $$ with a Dirichlet boundary condition at the influx boundary ($\mathbf{u} \...
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1answer
133 views

Stability region of explicit midpoint method

Consider the explicit midpoint method, i.e $$y_{n+1}-y_{n-1} = 2hf(y_n).$$ I'm asked to apply this method to the linear test equation, $f(y_n) = \lambda y_n,$ in order to find the method's stability ...
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81 views

Numerically Approximating the Jacobian and Comparing the Eigenvalues With Analytical Form

I am trying to study the stability of numerical discretization schemes using the Jacobian matrix of the residues with respect to the vector of conserved variables. For a simple diffusion equation ...
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48 views

Stability of SVD, Eigendecompositions, and pseudoinverse procedures in modern LAPACK routines

I have proposed an optimisation algorithm which I claim has improved upon the previous algorithm in a number of ways. One of these claims is that my proposed solution requires no explicit SVD and ...
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2answers
94 views

Stability of Crank-Nicolson for $u_t = iu_{xx}+2iu$

I want to use the Crank-Nicolson scheme to solve the equation $$u_t = iu_{xx}+2iu$$ Here's the analysis: Suppose we make a grid, with $k = dt$ and $h = dx$, the usual notation, and also $u_j^n = u(...
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1answer
77 views

Numerical Stability of a Generalized Spatial Discretization Scheme

After reading the matrix stability chapter (10) of Hirsch [1], I decided to dive in the reference list of the chapter. One of the papers [2], which is cited as reference shows an very interesting ...
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1answer
79 views

Stability of PDEs

I am currently trying to solve some PDEs with FiPy. At page 56, the manual mentions (https://www.ctcms.nist.gov/fipy/download/fipy-3.0.pdf). The largest stable timestep that can be taken for this ...
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68 views

The final Boundary Condition is Unknown, Is Backward Euler is still valid to be implemented?

I am working on conductive polymer modeling and supposed to do one-dimensional diffusion model in the thickness of the polymer, however, due to the small value of thickness in micro, when I use the ...
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What is the meaning of stability in numerical analysis? How to deterimne the stability of a numerical method?

My question may be so general and simple but I'm really confused about the meaning of the "stability". I look that up in the Internet but there was no general answer to this question. Can anyone help ...
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1answer
131 views

Why the numerical solution of advection-dominant problem is challenging

In many CFD text books, usually there is a dedicated chapter for advection term discretization. Why discretization of such term in advection-dominated problems and near the discontinuities is ...
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1answer
90 views

Assessing numerical error in solving a least squares problem

I have a linear system of the type $$Ax = b$$ I want to minimise $|b - Ax|^2$. I know there are different approaches to directly solve the system (Normal equation + Cholesky, QR decomposition, SVD ...
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2answers
400 views

Generate high n quantum harmonic oscillator states numerically

How can I generate the higher $n$ quantum harmonic oscillator wavefunction (in position space) numerically? Here, higher means around $n=500$, or say $n=2000$, where $n$ is the $n$th oscillator ...
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277 views

Any way to avoid catastrophic cancellation when computing the discriminant of a quadratic function?

Homework disclaimer... The task: We are using the following algorithm to solve the quadratic equation $x^2+2px+q=0$: $x_1=|p|+\sqrt{p^2-q}\mathtt{;}$ $\mathtt{if}\,p>0\,\mathtt{then}\,x_1=-x_1\...
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111 views

1+x not backwards stable?

If you compute 1+x for x less than the machine precision, the answer will be 1 which is the ...
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1k views

Fast and Numerically Stable Pairwise Distance Algorithms

I'm looking for resources on fast, numerically stable pairwise euclidean distance algorithms. In particular, suppose $A \in \mathbb{R}^{M \times D}$ and $B \in \mathbb{R}^{N \times D}$ are two sets of ...
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1answer
91 views

Stability Analysis

The partial differential equation, \begin{align} \dfrac{\partial f}{\partial x} + a(x)\dfrac{\partial f}{\partial y} = 0 \qquad & f(0,y) = f(L_1,y) = c_0e^{-y} \\ & f(x,0) = c_0 \;,\; f(x,L_2) ...
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71 views

Automatic differentiation of barycentric rational functions

By a barycentric rational interpolant we understand a function of the form \begin{align*} r(t) := \frac{\sum_{i=0}^{n-1} \frac{w_i y_i}{t-t_i} }{ \sum_{i=0}^{n-1} \frac{w_i}{t-t_i}} \end{align*} In ...
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251 views

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

According to the equation $$ \frac{\partial y}{\partial t} = -a\frac{\partial y}{\partial x} $$ I simulated this in python. I used center differentiation, and I determined step size based on Von-...
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1answer
102 views

Stable computation of ratio of sums of large numbers

I have two sets of large positive numbers $a_1,\ldots,a_n$ and $b_1,\ldots,b_n$. By 'large' I mean of the order of $10^{10}$. I want to calculate the ratio $$R = \frac{a_1 - a_2 + \cdots +(-1)^{n+1}...
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1answer
569 views

Von Neumann's stability analysis on non linear and coupled equations

I'd like to know if is possible to make a Von Neumann's stability analysis on an system of coupled equations, featuring quadratics: $$\begin{aligned} \frac{\partial u_1}{\partial t}&=D_1\Delta ...
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1answer
19 views

Implementation of a ratio with a well-defined limit

I'm implementing a function $f(x_1, x_2, \dots x_n)$ given by the ratio of two closed form equations. $$ f(x_1, x_2, \dots, x_n) = \frac{g(x_1, x_2, \dots, x_n)}{h(x_1, x_2, \dots, x_n)} $$ ...
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639 views

How much regularization to add to make SVD stable?

I've been using Intel MKL's SVD (dgesvd through SciPy) and noticed that results are are significantly different when I change precision between ...
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40 views

Stability of different quadrature rules in 1st-kind Volterra integral equation

I am dealing with a integral equation $$ f'(t) = -\int_0^t K(s) f(t-s)\quad t\in [0,t_\max] \tag{1}$$ in which $f(t)$ and $f'(t)$ are known, well-behaved functions of $t$ and $K(t)$ is the unknown. In ...
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143 views

Crank-Nicolson integrator: oscillations with complex matrix

I'm working on a Real-Time TDDFT implementation and I am currently comparing different propagation schemes for the propagation of the Kohn-Sham wave function, $$ \phi(t+\Delta t) = \hat{\mathcal{U}}\...
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1answer
409 views

How to derive an Implicit Runge-Kutta method from Pade approximation

I was reading some work by Butcher and I came across Pade approximations and the correlation between them and stability functions for some Implicit Runge-Kutta methods. For example, in this Pade table ...