Questions tagged [stability]

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

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416 views

What is the source of the error in the Sherman-Morrison formula application?

The Sherman-Morrison formula $$ (A+uv^T)^{-1} = A^{-1} - \frac{A^{-1}uv^TA^{-1}}{1+v^TA^{-1}u} $$ results in small errors in relation to the standard matrix inverse operation after each application, ...
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200 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}\,...
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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 ...
4
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0answers
216 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
112 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
305 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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1answer
587 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
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0answers
126 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
453 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
170 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
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222 views

Von Neumann analysis for coupled PDE's

I am interested in understanding how to perform stability analysis for coupled (to keep things do-able, lets say linear) PDE's In the case of a single PDE, i understand the logic behind the VN ...
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52 views

How to classify chaotic systems from a stability perspective

I am wondering what chaotic systems are from the perspective of numerical analysis. I am talking about 'deterministic chaos' such as for instance the 'logistic map' exhibits it. That is, the solution ...
2
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0answers
143 views

what do positive real parts of eigenvalues mean?

I am solving a 1D advection problem of the the form $$d{Q}/dt = [A]{Q}$$ where {Q} is the vector of unknowns and [A] is the matrix of coefficients of spatial discretisation. I have worked out the ...
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25 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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56 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 ...
1
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0answers
44 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 ...
1
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0answers
37 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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0answers
83 views

Kahan Summation for Three-Term Recurrences

Kahan summation applies to summation problems, but not to three-term recurrence relations. However, a three-term recurrence shares many of the features of a summation-albeit with a rescaling step at ...
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0answers
138 views

Stability of the explicit MacCormack Scheme to solve the Navier Stokes equations with Wilcox's K-Omega Turbulence Model

I am solving turbulent pipe flow with an explicit MacCormack scheme and Wilcox k-omega model. The laminar version of the code had three distinct stability criteria which worked fine after ...
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0answers
80 views

Methods to approximate discretized derivatives in PDEs

When solving a general PDE such as $$ \frac {\partial ^2 E}{\partial t ^2} = \frac {\partial ^2 E}{\partial z ^2} - \frac {\partial E}{\partial z} $$ this equation can be solved by the method of ...
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0answers
40 views

Stability in Discretization of 1D Stationary Boltzmann equation

I want to discretize and numerically solve the following PDE: \begin{equation} v(k)\dfrac{\partial f}{\partial x} + E(x)\dfrac{\partial f}{\partial k} = S\{f\} \end{equation} using finite volume (box ...
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0answers
333 views

Computing solutions with singularities using MATLAB ODE45

I am new to solving numerically ODES and thus it is difficult for me to judge the reliability/trustworthiness of the results that I have produced for the following problem: I am dealing with a 2nd ...
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0answers
109 views

Reducing oscillations a 3D Alternating direction explicit scheme for the diffusion equation?

Hi I have made a 3D alternating direction explicit scheme for solving the diffusion equation, which will eventually replace a FTCS scheme in model of bubble dynamics in tissue. I have been testing it ...
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0answers
167 views

Tips on improving stability in numerical scheme for non-linear PDE

I am solving a non-linear second order system of PDEs in two variables. The equations are too complicated to write out here, but an essential feature is that there is a propagating wave which then ...
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0answers
89 views

Stability question (finite difference): dealing with corner nodes

Consider one initial boundary value problem for sphere. $$\frac{\partial u}{\partial t}=\operatorname{div}A\nabla u +f$$ Here is explicit numerical scheme (we consider that it is stable): $$\frac{u^{...
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35 views

Simulation of asymmetric structures (occupancy = 0.5) unstable

I am trying to simulate a metal-organic framework in LAMMPS using the UFF potential. It's working quite well for some structures where all molecules have an occupancy of 1. However, when I have a ...
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33 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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57 views

Bound for Expectation of Singular Value

In my case, $X_{\boldsymbol{\delta}}\in\mathbb{R}^{d\times M}$ is a function of Rademacher variables $\boldsymbol{\delta}\in\{1,-1\}^M$ with $\delta_i$ independent uniform random variables taking ...
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0answers
58 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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0answers
337 views

Stability of nonlinear partial differential equation

I want to find an expression for the stability of the nonlinear Poisson equation. I know about von Neumann stability analysis which applies to linear equations as far as I know. Any suggestion how to ...
0
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0answers
46 views

comparison of stability of two non-linear methods

I have solved a numerical problem using two different sets of non-linear governing equations. I want to get an understanding of the stability of the methods relative to each other. To do so, I solving ...