Questions tagged [stability]

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

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1answer
610 views

Stability of numerical schemes for non-linear equations with a Jacobian with negative eigenvalues

Let us assume I have an A-stable numerical scheme. I believe that given any linear equation $y' = Ay$, it means that the numerical scheme applied to this equation is stable (and therefore convergent ...
4
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3answers
158 views

Numerical solution of IVP for linear ODE with variable coefficient blows up

Cross posted in Mathematica.SE, I'll try to rephrase it in a more general way here. A friend of mine showed me this initial value problem (IVP) for a linear ordinary differential equation (ODE) with ...
4
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2answers
2k views

Numerical instability of spherical pendulum

Problem statement I am trying to simulate a spherical pendulum, with rod length $r$ south-polar angle $\theta$ and azimuthal angle $\phi$ initial values $(\theta_0,\phi_0)= (0,0)$ My particular ...
4
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1answer
105 views

Is stabilization of energy equation needed when momentum equation needs it?

When SUPG/PSPG stabilization is added to momentum equation of flow problem, is needed stabilization for energy equation also? I would guess that when stabilization for velocity works fine so one gets ...
4
votes
3answers
524 views

Backward stable projection and normalization of a vector

Given a machine precision unit vector $n$, and an arbitrary vector $v$, I want an unconditionally backward stable method to compute $$f(v) = \frac{v-nn'v}{\left|v-nn'v\right|}$$ In other words, ...
4
votes
2answers
866 views

Lax-Richtmyer stability analysis

I would like to get to know more in details about Lax-Richtmyer stability analysis (esp in examples), but I didn't manage to find anything except a definition. Could you advice any sources for this ...
4
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2answers
192 views

Problem Condition and Algorithm Stability

Consider 2 mathematical problems: $$ f_1(x) = a - x \\ f_2(x) = e^x -1 $$ The condition number for a function is defined as follows: $$ k(f) = \left| x \cdot \frac{f'}{f} \right| $$ Lets analyze ...
4
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2answers
1k views

Solution oscillations with a small timestep in backward Euler

I am using backward Euler in a FEM scheme for a convection-diffusion problem. On a given mesh, I can take arbitrarily large time steps, as expected. But if I decrease time step, at some point it will ...
4
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2answers
332 views

choice of the norm for Crank Nicolson stability estimate

I have a variable coefficient pde of the form $$u_t=c(t,x)u_{xx}, t\in [0,T], x\in [0,1]$$ with initial data $u_0=u(0,x)$ and $c(x,t)\in C([0,T]\times[0,1])$. I use three point discretization for the ...
4
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1answer
2k views

Von Newman stability analysis for 2D acoustic wave equation explicit

Von Newman stability analysis for acoustic wave equation explicit centered differences: 2nd order time and space (N 2)'th order: \begin{eqnarray} U_{jk}^{n+1} = \left( \frac{\Delta t V_{jk} }{\...
4
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0answers
67 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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219 views

Stability analysis for a hyperbolic PDE on staggered grid

I am trying to understand the stability of a finite difference equation on the staggered grid. I could understand the Von Neumann stability analysis for the collocated grid for a simple acoustic ...
4
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0answers
115 views

Using entropy functions for increasing numerical stability

Regarding the numerical stabilization of two-dimensional advection equation, \begin{equation} \dfrac{\partial f}{\partial t} + \Big(\dfrac{d\varepsilon_1(k)}{dk}\Big)\dfrac{\partial f}{\partial z} - \...
4
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0answers
1k views

CFL Condition and Convection Diffusion Equation in 2D

I am solving the convection-diffusion equation in 2D using Finite Differences with the $\theta$ scheme. The velocity of the fluid and the diffusion coefficient is low in my case (in the range of $10^{-...
4
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0answers
309 views

Frozen coefficient method (von Neumann stability analysis)

Earlier it was considered that frozen coefficients method for Neumann stability analysis for finite difference scheme is more heuristic than rigorous. But I have read some information in a book by ...
3
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4answers
10k views

Estimating the Courant number for the Navier-Stokes Equations under differing Reynolds number regimes

I am familiar with the Courant-Friedrich-Lewy Condition in as far as it applies to the stability of explicit finite difference schemes for standard parabolic and hyperbolic PDEs. However, when ...
3
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2answers
269 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 ...
3
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2answers
150 views

what do zero real parts of eigenvalues mean? Any good references?

I am solving a 1D advection problem of the the form $$dQ/dt=[A]Q$$ where {Q} is the vector of unknowns and [A] is the matrix of coefficients of spatial discretisation. I have worked out the ...
3
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1answer
983 views

Courant Friedrichs Lewy condition - how to get it?

I am interested, how can we get CFL condition for every type of PDE? It's known that for 1st order linear equation $$\frac{\partial u}{\partial t}+a\frac{\partial u}{\partial x}=0 $$ CFL is get from ...
3
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3answers
2k views

Caculating the mean of vector accurately

I am having trouble with calculating a mean of vector with sufficient accuracy. My current solution which works but it quite slow and has unpredictable performance: mean_sum = mean = math.fsum(values)...
3
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2answers
544 views

In what regime do the continuous and discontinuous Galerkin method become unstable for advection-diffusion systems?

I know that the finite volume method (based around a central different stencil) is unstable for advection dominated advection-diffusion problems. This leads to different adaptive schemes to can be ...
3
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2answers
288 views

Does artifical dissipation term makes scheme inconsistent?

Central schemes like JST uses artificial dissipation for the stabilization. This modification is an artificial one. Does this additional term makes system inconsistent? Can we expect this term to be ...
3
votes
1answer
154 views

stabilizing advection-diffusion with multi-grid?

If one chooses to discetize the advection-diffusion (AD) equation using the standard Galerkin finite element method, stability issues may arise in cases of high Peclet number (i.e., high advection to ...
3
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1answer
189 views

intuition behind the different discrete norms for Crank Nicolson

I am solving a heat equation $u_t=Au$ with Crank-Nicolson finite-difference method and $A$ is a usual discretization matrix for $u_{xx}$ term. I want to tell something about the whole error vector ...
3
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1answer
81 views

Finding Numerical Stability of Simple System with Integral Term Due to Low Pass Filter

As a toy problem, I was looking at a classical second order system of the following form: \begin{align} \ddot{x}(t) + c \dot{x}(t) + k x(t) = 0 \end{align} Instead of the basic system above, I want ...
3
votes
1answer
176 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, ...
3
votes
1answer
55 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 ...
3
votes
1answer
57 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 ...
3
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2answers
86 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 ...
3
votes
1answer
140 views

Stability of PDE Discretizations with Multistep Time Discretizations

Let's pretend we have a spatially discretized PDE of the following form: \begin{align} \frac{\partial^2 \boldsymbol{u}^k}{\partial t^2} = D\boldsymbol{u}^k \end{align} where $D$ can be any form for ...
3
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1answer
673 views

Von Neumann stability analysis in 3d

I need to get a stability criterion for the numerical scheme for equation $$\frac{\partial u}{\partial t}-\frac{\partial^2 u}{\partial x^2}-\frac{\partial^2 u}{\partial y^2}-\frac{\partial^2 u}{\...
3
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2answers
120 views

In numerical methods, eg, finite differencing approaches, does there exist convergent schemes that are not both consistent and stable?

In a book that our course is following this semester, the theorem given is only in one direction: if the scheme is both consistent and stable, then the scheme is convergent. However, since this ...
3
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1answer
326 views

linear stability analysis using spectral radius

I am analysing the stability of a series of 1D linear equations of the form \begin{equation} \frac{d}{dt} x = A x \end{equation} discretised using upwind and central finite volume methods, etc, with ...
3
votes
2answers
122 views

How does constraint resolution affect the stability/accuracy of numerical integration?

I understand some basic analysis techniques (local truncation error, global error, zero-stable, absolute stable, etc.) of numerical integration. But I find it hard to apply these techniques in ...
3
votes
1answer
106 views

Numerically stable real solution(s) to a system of bivariate quadratics

I have a a system of bivariate polynomials as follows: $ E(u,v): e_2(u) v^2 + e_1(u) v + e_0(v) = 0 \\ F(u,v): f_2(u) v^2 + f_1(u) v + f_0(v) = 0$ where $e_n(u) = e_{n_2}u^2 + e_{n_1} u + e_{n_0}$ ...
3
votes
1answer
90 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 ...
3
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2answers
663 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(...
3
votes
1answer
216 views

Stability Criterion for this Explicit Scheme

I am solving an unsteady flow using the dual-time Navier-Stokes equation in which I write my momentum equation as: $$\frac{\partial u}{\partial \tau} + \frac{\partial u}{\partial t} + \frac{\partial u^...
3
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0answers
131 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}}\...
3
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0answers
474 views

Difference between fast and normal Givens Rotations?

would someone be so kind as to explain me the difference between the ordinary givens-rotation and the fast givens-rotation? I know that the fast givens Rotation reduces the Count of operations to ...
3
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0answers
175 views

Stability analysis for explicit time discretization in the Finite Element Method

I have been looking for stability analysis of general reaction-diffusion problems, of the form $\frac{\partial u}{\partial t}=\nabla\cdot D\nabla u-k\,u$ , to be solved using the standard Finite ...
2
votes
3answers
140 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, ...
2
votes
2answers
170 views

Why have specialised upwind schemes been developed to solve hyperbolic equations?

Are upwind schemes such as Godunov type methods superior to central differencing schemes? Do the reasons include superiority in modelling hyperbolic problems with Dirichlet BC's?
2
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1answer
97 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 ...
2
votes
1answer
118 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 ...
2
votes
1answer
87 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) ...
2
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1answer
198 views

Stability in discretization of a PDE

Suppose I want to numerically solve for $f(x,k)$ the one-dimensional Boltzmann equation for electrons in steady-state condition, given as: \begin{equation} \left( \dfrac{\hbar k}{m} \right) \dfrac{\...
2
votes
1answer
190 views

How can I analyze the stability of a PDE discretization at a boundary?

I have a numerical discretization of a partial differential equation that seems to be unstable or stable at a boundary point, depending on what finite difference scheme I am using. Are there standard ...
2
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1answer
65 views

Model Spinup time

While I look up programming and technical stuff on Stack Overflow all the time, I figured that would probably be the wrong forum for a question like this. Some background: I'm a climate modeler ...
2
votes
1answer
114 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{...