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Questions tagged [stability]

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

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19 votes
2 answers
3k views

How to determine if a numerical solution to a PDE is converging to a continuum solution?

The Lax equivalence theorem states that consistency and stability of a numerical scheme for a linear initial value problem is a necessary and sufficient condition for convergence. But for nonlinear ...
Jed Brown's user avatar
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19 votes
1 answer
537 views

Catastrophic cancellation in logsum

I'm trying to implement the following function in double-precision floating point with low relative error: $$\mathrm{logsum}(x,y) = \log(\exp(x) + \exp(y))$$ This is used extensively in statistical ...
Neil Toronto's user avatar
17 votes
1 answer
1k views

When should implicit methods be used in the integration of hyperbolic PDEs?

Numerical methods for solving PDEs (or ODEs) fall into two broad categories: explicit and implicit methods. Implicit methods allow larger stable timesteps but require more work per step. For ...
David Ketcheson's user avatar
16 votes
1 answer
273 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, ...
knl's user avatar
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15 votes
1 answer
672 views

Puzzling remark about stability region of fifth-order Runge-Kutta method

I came across a puzzling remark in the paper P. J. van der Houwen, The development of Runge-Kutta methods for partial differential equations, Appl. Num. Math. 20:261, 1996 On lines 8ff on page 264, ...
Brian Zatapatique's user avatar
14 votes
1 answer
460 views

What spatial discretizations work for incompressible flow with anisotropic boundary meshes?

High Reynolds number flows produce very thin boundary layers. If wall resolution is used in Large Eddy Simulation, the aspect ratio may be on the order of $10^6$. Many methods become unstable in this ...
Jed Brown's user avatar
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13 votes
2 answers
2k views

Alternatives to von neumann stability analysis for finite difference methods

I'm working on solving the coupled one-dimensional poroelasticity equations (biot's model), given as: $$-(\lambda+ 2\mu) \frac{\partial^2 u}{\partial x^2} + \frac{\partial p}{\partial x} = 0$$ $$\...
Paul's user avatar
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13 votes
1 answer
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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 ...
Stephen Bosch's user avatar
12 votes
3 answers
622 views

Heuristic check of numerical stability

Assume I have a real valued function $f(x_1,\ldots ,x_N)$ of some variables $x_i$ which I want to evaluate numerically. In general the formula for $f$ can contain products, rationals, trancendental ...
highsciguy's user avatar
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11 votes
1 answer
646 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 ...
Matthew Cassell's user avatar
10 votes
2 answers
525 views

Where can I find a good reference for the stability properties of several methods of solving parabolic PDEs?

Right now I have a code that uses the Crank-Nicholson algorithm, but I think that I would like to move to a higher-order algorithm for timestepping. I know that the Crank-Nicholson algorithm is stable ...
Dan's user avatar
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10 votes
2 answers
1k 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 ...
Yaroslav Bulatov's user avatar
10 votes
1 answer
443 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 ...
Bananach's user avatar
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9 votes
1 answer
451 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 ...
Krastanov's user avatar
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9 votes
1 answer
1k views

Linearized implicit time stepping

Consider the general FD implicit time stepping scheme $\frac{x_{t+1} - x_t}{\Delta t} = f(x_{t+1})$, where $x$ is the vector variable of interest and $f$ is some function, generally non-linear. ...
Patrick's user avatar
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9 votes
1 answer
890 views

CFL condition in polar coordinates

In this question, I suggested that the Couran-Friedrichs-Lewy (CFL) condition for the wave equation in polar coordinates reads $$C = 2c\frac{\Delta t}{\Delta r \Delta \phi} \leq C_\max \enspace ,$$ ...
nicoguaro's user avatar
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9 votes
2 answers
2k 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 ...
Hunter's user avatar
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8 votes
2 answers
743 views

How to decide stability of Runge-Kutta method for non-linear ODE?

I'm working on a parameter study of Duffing's equation $\ddot x + \delta \dot x + \alpha x + \beta x^3 = \gamma \cos{\omega t},$ where $\delta, \alpha, \beta, \gamma$ and $\omega$ are real ...
trolle3000's user avatar
8 votes
1 answer
1k views

Clenshaw-type recurrence for derivative of Chebyshev series

The naive summation of a Chebyshev series \begin{align*} f(x) = \frac{c_0}{2} + \sum_{k=1}^{n-1} c_{k}T_{k}(x) \end{align*} which employs the three-term recurrence for evaluation of the Chebyshev ...
user14717's user avatar
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8 votes
2 answers
397 views

Lagrange multipliers space is too rich in a mathematical view

Background: Lagrange multiplier method has been employed in numerous fields, such as contact problems, material interfaces, phase transformation, stiff constraints or sliding along interfaces. It is ...
Wenjin Xing's user avatar
8 votes
2 answers
364 views

Stabilization of convection-dominated flow and turbulence modeling

Are stabilization techniques for convection-dominated flows like SUPG+PSPG, interior penalty methods, etc. able to handle turbulent flows without tubulence model being employed, at least up to some ...
Jan Blechta's user avatar
8 votes
0 answers
800 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, ...
rcpinto's user avatar
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7 votes
3 answers
960 views

analyze stability on a nonuniform grid

Assume you have a stability constraint between the space distance in time and space, for example, with an explicit Euler method for $u_t=u_{xx}$ we know $\tau\leq h^2/2$. That is, one can do stability ...
Kamil's user avatar
  • 1,176
7 votes
4 answers
4k views

Finite Difference Method Stability

The diffusion equation is: $ \frac{\partial T}{\partial t} = \alpha \left( \frac{\partial^2 T}{\partial x^2} \right) $ An explicit finite difference approach can be used to solve this, forward in ...
zed111's user avatar
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7 votes
1 answer
3k views

What is the origin of the spurious oscillations in the Crank-Nicolson scheme?

I was reading about the Crank-Nicolson method, and it is often said that it can produce "spurious oscillations" or that this method is prone to "ringing", especially for large time step and stiff ...
Matthieu's user avatar
7 votes
1 answer
3k views

Stable time step limits for Velocity-Verlet integration

I'm implementing a mass-spring solid mechanics solver and I'd like to use the Velocity-Verlet time integration scheme. However, I cannot find anything about the maximum stable time step -- either ...
tpg2114's user avatar
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7 votes
1 answer
2k views

method of frozen coefficients and its relation to von Neumann stability analysis

I am considering two equations $$u_t=a(x)u_{xx}$$ and $$v_t=b(x)v_x$$ as classical representatives of the parabolic and hyperbolic family of equations. If $a(x)=a$ and $b(x)=b$ were constants, to show ...
Kamil's user avatar
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7 votes
2 answers
345 views

Numerical investigation of stability of motion (confinement)

I am trying to find the required specifications of a RF trap, in which a proton can be confined.(trap dimensions, voltage frequency and amplitude used, etc). I have to solve the equations of motion ...
Mostafa's user avatar
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7 votes
1 answer
1k 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}{\...
winterfly's user avatar
7 votes
1 answer
172 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 ...
user14717's user avatar
  • 2,155
7 votes
1 answer
214 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}...
EpsilonDelta's user avatar
6 votes
1 answer
1k views

Why is matrix inversion unstable when svd is stable?

I've heard that matrix is inversion is unstable whereas the SVD is stable. Now, if $A$ is an invertible matrix, then its SVD is $$ A = USV^T $$ Then wouldn't it's inverse just be $$ A^{-1} = (USV^T)^{...
NNN's user avatar
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6 votes
1 answer
353 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 }{\...
dba's user avatar
  • 295
6 votes
3 answers
10k views

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 ...
MohammadSh's user avatar
6 votes
2 answers
230 views

Is there a backward stable $\tilde{O}(n \log(1/\epsilon))$ algorithm to factor a complex polynomial?

Finding the roots of a complex polynomial is in general extremely numerically unstable, as discussed in (1). According to Pan ((2), (3)), this produces a cubic complexity lower bound, and he presents ...
Geoffrey Irving's user avatar
6 votes
1 answer
1k 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 ...
Eliad's user avatar
  • 249
6 votes
3 answers
779 views

What are the negatives of using higher order finite diference schemes?

I was looking at this wikipedia page: http://en.wikipedia.org/wiki/Finite_difference_coefficient It is a lists of higher order finite difference approximations, is there any negatives in using these ...
user1750289's user avatar
6 votes
2 answers
2k views

Finite elements for Stokes with traction boundary conditions

Suppose we are given the Stokes equations with Neumann conditions on part of the boundary: $-\nabla\cdot\boldsymbol{\sigma} = \mathbf{f}, \quad \text{and} \quad \nabla\cdot \mathbf{u} = 0 \quad \...
Christian Waluga's user avatar
6 votes
1 answer
1k views

Stability analysis of coupled ordinary differential equations

Given a forward-in-time approximation I have the coupled equations: $$ \frac{T^{(n+1)} - T^{(n)}}{\Delta t} = x T^{(n)} - y h^{(n)} \\ \frac{h^{(n+1)} - h^{(n)}}{\Delta t} = -z h^{(n)} - \alpha T^{(n)}...
hertzsprung's user avatar
6 votes
2 answers
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 ...
Rody Oldenhuis's user avatar
6 votes
1 answer
278 views

interpolation combined with methods of characteristics can cause oscillations for the transport equation?

I would like to know about the effect of using a higher order interpolator for the methods of characteristics. I am solving $$u_t+a(x,t)u_x=0$$ with some nonsmooth initial data $u_0(x)$ by the method ...
Kamil's user avatar
  • 1,176
6 votes
2 answers
285 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 ...
gTcV's user avatar
  • 445
6 votes
2 answers
2k 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 ...
Chris Swierczewski's user avatar
5 votes
3 answers
511 views

Advantage of diagonal "jitter" for numerical stability?

In a machine learning code, that computes optimum parameters $\theta _{MLE}$ of a linear regression model, by maximum likelihood estimation: $$ \boldsymbol \theta^\text{ML} = (\boldsymbol\Phi^T\...
Algo's user avatar
  • 304
5 votes
2 answers
169 views

Exponent log to compute reciprocal power?

A MATLAB library seems to overcomplicate a computation: exp( (log(a) - log(b))/b ) which is mathematically equivalent (assuming real & positive ...
OverLordGoldDragon's user avatar
5 votes
2 answers
405 views

Levenberg optimizer halts quickly when given more variables, or fewer constraints

I'm using the g2o C++ optimization library to refine a GPS trajectory using accelerometer data. The program uses a Levenberg-Marquardt optimizer over data points representing the position and ...
Potatoswatter's user avatar
5 votes
1 answer
4k views

Stability of numerical method for 1D Burger's equation

I am trying to solve 1D viscous Burger's equation numerically and I cannot apply von Neumann analysis because the equation is non-linear. How do I predict the stability criteria for my system? I also ...
tumchaaditya's user avatar
5 votes
2 answers
791 views

Stability of the first-order exponential integrator method

The question is about the first-order exponential integration method described in this article. Consider a system of ordinary differential equations $$y'(t) = -A\,y(t) + \mathcal{N}(t, y), \qquad y(...
Ivan's user avatar
  • 297
5 votes
2 answers
2k 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 ...
Dominik Lark's user avatar
5 votes
1 answer
183 views

Stabilization of solution to one-dimensional system of PDE

I am trying to solve numerically next PDE system: $$\frac{\partial c}{\partial t}=\epsilon\frac{\partial}{\partial x}(\frac{\partial c}{\partial x}+\rho\frac{\partial \varphi}{\partial x}+\frac{vc}{1-...
oleksandrv's user avatar

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