Questions tagged [stability]

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

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922 views

How to choose the relaxation time in the Lattice Boltzmann Method?

We know that the relaxation time is very important in LBM. I have searched lost of papers, but can't find some systematic introductions about the choice of relaxation time in SRT LBM. Could you give ...
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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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1answer
269 views

Runge-Kutta Stability Regions

Based on this link, in particular Figure 1, what is the exact meaning of the plot? To my understanding, it implies that for a given differential equation: $$ \frac {dy}{dt} = \lambda y $$ that the ...
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81 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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2answers
110 views

How to support or contradict a hypothesis on unconditional stability using numerical optimization

The main motivation behind my next question is that I think I derived a higher order numerical scheme for linear advection equation that is unconditionally stable using Von Neumann stability analysis. ...
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1answer
187 views

What does it mean for a finite difference scheme to be $L^1$-stable?

I am trying to answer a question about a finite difference scheme. I need to show that the method is stable in the L1-norm. I can't find a single definition of what that means, so does anyone have a ...
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1answer
131 views

Stability condition for explicit/implicit via non negative coefficients

To make stability proofs simpler, I can consider an explicit scheme written as $$V(n+1,i)=aV(n,i-1)+bV(n,i)+cV(n,i+1)$$ and one can show that if $a,b,c\ge 0$ and $a+b+c\le1$, then the explicit method ...
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1answer
434 views

More Smearing with decreasing timestep in advection problems

I find it kind of counter intuitive, that the result of an advection gets more smeared out at the borders when decreasing the timestep (which should make it more accurate). Let there be a equally ...
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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 ...
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2answers
375 views

Stability of matrix equations in MATLAB

Is there a method to check for unconditional stability or positive-definiteness of large matrices in MATLAB? For example, I know that matrices with property M (positive main diagonal elements and ...
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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} - \...
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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^...
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1answer
548 views

Stability analysis for coupled nonlinear system of partial differential equations

I'm trying to solve a nonlinear partial differential equation \begin{equation} L(u_{xxtt},u_{xx}u_{tt},u_{xt}^2,u_{xt},u_{tt})=0 \end{equation} using finite difference methods. In order to remove the ...
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1answer
240 views

Calculate inverse of dense matrix with entries of very different magnitude

I need to calculate the inverse of a dense matrix, with some elements taking values as high as 1e9 and some around 1e2. What would be the best method to do it? Note: I am more concerned about the ...
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2answers
1k views

What does the Von Neumann's stability analysis tell us about non-linear finite difference equations?

I am reading a paper [1] where they solve the following non-linear equation \begin{equation} u_t + u_x + uu_x - u_{xxt} = 0 \end{equation} using finite difference methods. They also analyse the ...
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1answer
216 views

Usefulness of elements with mesh-dependent stability

After doing some mathematics related to the stability of elements in 3D Stokes problem I was slightly shocked to realize that $P_2-P_1$ is not stable for an arbitrary tetrahedral mesh. More precisely, ...
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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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1answer
194 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{\...
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1answer
467 views

The condition for stability using the leapfrog method

I have the ODE below $$\frac{d}{dt}\pmatrix{x\\ y} = \pmatrix{0 &1\\-a &0}\pmatrix{x\\ y} \enspace .$$ The $m=1$ leapfrog method is defined as: $$y_{n+1} = y_{n-1} + 2f_nh \enspace .$$ For ...
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1answer
69 views

How do we find the condition?

Suppose that we are given a numerical scheme. In order to find the CFL condition , we set $U_j^n= \lambda ^ne^{ik x_j}$ and put it into the numerical scheme. I have shown that the given method is ...
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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^{-...
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2answers
375 views

significance of energy equation and its conservation

I am interested to know the significance of the energy conservation laws when modelling fluids (or other materials). Am I correct in saying that if energy is conserved then stability is achieved. In ...
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1answer
175 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 ...
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334 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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2answers
284 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 ...
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1answer
69 views

How do I add some floating point numbers, keeping numerical accuracy in mind?

I am solving a problem involving the line with the set of points $(x_3,y_3)$ that are equidistant to two given points $(x_1,y_1)$ and $(x_2,y_2)$. The equation for this line is $$(x_3 - x_1)^2 + (y_3 ...
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1answer
376 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 ...
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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 ...
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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 ...
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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 ...
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144 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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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?
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458 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 ...
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1answer
314 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 ...
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1answer
151 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 ...
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417 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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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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1answer
225 views

Forcing an ODE solver to preserve the norm

I have an ODE of the form $$ \frac{dy}{dt} = -i H y \enspace .$$ where $y$ is a complex vector and $H$ is a time dependent Hermitian matrix. The norm of the solution $y(t)$ at any point in time ...
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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 ...
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1answer
748 views

Stability analysis of Heun's method

I am using Heun's method with a third order upwind spatial scheme, which is suggested by Shao (2008) to be used for solving the horizontal advection part of the advection-diffusion equation. This is ...
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1answer
238 views

Order of operations, numerical algorithms

I have read that (1) Ill conditioned operations should be performed before well conditioned ones. As an example, one should calculate $xz-yz$ as $(x-y)z$ since subtraction is ill conditioned while ...
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1answer
522 views

Numerically stable approach for calculating x in Ax=b

I have an equation $Ax=b$ for which I need to solve for numerous $x$ matrices given $b$. Both $x$ and $b$ are nx1 matrices. Unfortunately, $A$ is a 32x32 matrix and inversion gives highly unstable ...
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1answer
1k views

Can an approximated Jacobian with finite differences cause instability in the Newton method?

I have implemented a backward-Euler solver in python 3 (using numpy). For my own convenience and as an exercise, I also wrote a small function that computes a finite difference approximation of the ...
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4answers
2k views

Why are upwind schemes stable in convection flow calculation?

It is well known that upwind schemes are stable when calculating convection flows with $|\text{Pe}|>2$, $\text{Pe}$ is the Peclet number. Why is that, and why is central difference unstable? Is ...
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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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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 ...
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1answer
61 views

Locally evaluate nonlinear dynamic system's stability using eigenvalues

I'm working with Computational Neuroscience. I have a large Synaptic Matrix (x axis: presynaptic NeuronID, y axis: postsynaptic NeuronID) in a Modular network. This matrix is close to a random one and ...
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1answer
829 views

How to avoid negative values of numerical solution of transport equation using FEM scheme?

The transport equation is actually an advection-diffussion-reaction equation, which has the form as $$\frac{\partial C}{\partial t} + v_1 \frac{\partial C}{\partial x} + v_2 \frac{\partial C}{\...
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1answer
495 views

Avoid arithmetic overflow in matrix multiplication

I am solving the following matrix equation for $\mathbf{x}$: $$(J^{\mathbf{T}}J)\mathbf{x}=J^{\mathbf{T}}\mathbf{r}$$ $J$ is $m\times n$ matrix $\mathbf{x}$ is vector of size $n$ $\mathbf{r}$ is ...
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0answers
306 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 ...