Questions tagged [stability]
The study of the propagation of errors in a numerical algorithm.
180
questions
0
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0answers
72 views
Transient advection equation with stabilized FEM
I am interested in solving the transient advection equation
$\left\{\begin{array}{ll}\partial_{t} u+\beta \cdot \nabla u=f & \text { in } \Omega, t>0 \\ u=0 & \text { on } \partial \Omega^{-...
1
vote
2answers
89 views
Stability condition FCTS method
The FTCS method comes from the discretization of a diffusion PDE like this:
$$
a^{2} \frac{u_{i+1}^{k}-2 u_{i}^{k}+u_{i-1}^{k}}{\Delta x^{2}}=\frac{u_{i}^{k+1}-u_{i}^{k}}{\Delta t}
$$
If I have the ...
0
votes
1answer
86 views
Free Electron Schrödinger Equation (Energy Method)
For the simplest atom, its wave function is described by the PDE of Schrodinger equation:
$$
-i h \frac{\partial u}{\partial t }=\frac{h^{2}}{2m} \Delta u + \frac{e^{2}}{r}u$$
The potential $\frac{e^{...
0
votes
2answers
83 views
Stability of the Forward-Time Central Space method, section 9.3 in LeVeque
I am reading section 9.3 in Leveque about stability. The explanations are very short and brief so I am asking for some help to understand and elaborations. The result of the section is that for FTCS ...
2
votes
1answer
89 views
Positive feedback instability
I have two black boxes (representing some multi-physics but that does not matter), where box #1 takes some quantity T as input and produces some quantity P as output; and box #2 takes P as input and ...
1
vote
0answers
47 views
Comparison between stability and accuracy of various Finite Difference schemes
I`m Analyzing the 1D convection equation (PDE) for stability, consistency, and accuracy.
I know Both upwind schemes (explicit and implicit) show better stability with a higher number of waves for ...
1
vote
0answers
51 views
Stability plot of upward difference implicit time
I am analyzing the stability of 1D convection (advection) equation as shown in the picture. When I derive the equations as shown I want to get rid of the complex number.
I`m asking if those stability ...
1
vote
1answer
107 views
Accuracy gap for apparently stable solution
I was reasoning about the behaviour of the methods I'm using for my simulation and I noticed that, considering $h_s$ as the timestep over which I have unstable solutions and $h_a$ as the timestep ...
5
votes
2answers
123 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 ...
0
votes
1answer
78 views
Technique to find the CFL condition using the Galerkin method in space and finite-difference in time?
I am using the Galerkin method (Discontinuous to be precise) to discretize in space the scalar linear wave equation and the explicit second order centered finite difference scheme to discretize in ...
3
votes
2answers
171 views
Using the BDF and RK4 methods to solve this coupled system of ODEs in C++
I'm trying to solve a system of ODEs using the BDF order 4 method. I find the first 3 points using RK4, then for the implicit part of the BDF, I use Newton-Raphson iteration. Unfortunately my solution ...
1
vote
1answer
36 views
Numerical stability of taking the `mean` of outputs from the simulation of a discrete stochastic dynamical system
I am writing a simulation for a discrete stochastic dynamical system. Since the simulation is stochastic, I need to run the simulation multiple times and then average the values of each timestep. I ...
2
votes
1answer
104 views
How to include negative number in the log-sum-exp?
I want to know summation of some small numbers, such as {e^-1000, -e^1001, e^1002...}
If all numbers are positive, I can use log-sum-exp algorithm. But unfortunately, negative numbers are also ...
4
votes
3answers
213 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:
$$ \boldsymbol \theta^\text{ML} = (\boldsymbol\Phi^T\...
1
vote
1answer
82 views
Finite Difference for Advection Equation With Source
I'm trying to find a convergent finite difference scheme for the PDE
\begin{equation}
\begin{split}
u_t + (x-1) u_x &= (x-1)u, \hspace{.5cm} x \in [0,1] \\
u(x,0) &= 1 \\
u(1,t) &= 1. \\
\...
2
votes
0answers
41 views
In sights into why higher order finite differencing method leads faster to instability
I was playing around with numerically solving the 1D wave equation with density and stiffness varying with position using central differencing methods and noticed that for certain discretization steps ...
0
votes
0answers
63 views
discretizing advection equation with variable wave speed + stability
I currently have a code that solves $u_t+ cu_x=0$ with periodic boundary conditions, and constant $c$ (using an upwind method). I'm wondering how I would alter this code to solve something of the form
...
1
vote
1answer
137 views
How do I apply BDF2 in a STRANG splitting
I have a 3D diffusion equation that I want to solve using a time splitting (2D+1D). Assume that $A$ is the 2D discrete diffusion operator and $B$ is the 1D discrete diffusion operator.
I want to use a ...
3
votes
1answer
242 views
How to use numerical integration to calculate the surface area of a superellipsoid?
I am working in an application in which I need to calculate the surface area of a superellipsoid. I have read that there is no closed form solution (see here), so I am trying to compute it using ...
2
votes
0answers
44 views
How to increase the stability of a DAE solver?
I am trying to solve a set of linear PDEs of the form
$$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.$$
To ...
0
votes
1answer
115 views
Method of Lines Runge-Kutta nonlinear stability and behavior
I have a system of 4 nonlinear 1st-order PDEs. I want to solve them numerically by method of lines, first discretizing space. This leads to the system of $N\times 4$ coupled ODEs.
$$
\mathbf{u}_{i} =...
0
votes
0answers
66 views
Why is my numerical solution to a set of ODEs infinite?
I am trying to solve the following linear PDEs
$$\frac{\partial u_x}{\partial x}=-[i\omega b_{||}+\nabla_\perp u_\perp],$$
$$\frac{\partial b_{||}}{\partial x}=-\frac{i}{\omega}\mathcal{L}u_x,$$
$$\...
0
votes
0answers
59 views
Compute the sum of probabilities when they are given as logits
Say I have a set of numerous probabilities given by their logarithm : $\{\ln p_i, 1 \leq i \leq N\}$.
I want to compute $\sum p_i$, if possible without exponentiating $\ln p_i$, since some of those ...
1
vote
0answers
42 views
von Neumann analysis: computation of maximum value of amplification factor
In this question I adress the stability property of a numerical scheme (a FDM described below) to the problem
$$
\begin{align}
u_{t} &= \alpha u_{xx}+\beta u_{xxxx}, & &x\in\mathbb{R},\;t&...
0
votes
1answer
47 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 ...
2
votes
0answers
35 views
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{...
0
votes
0answers
42 views
Unconditionally stable numerical method for 1st order non-linear coupled ODEs?
I am attempting to numerically solve the following system of ODEs:
$$\begin{gather}\frac{dT_1}{dt} = f_1(T_1,T_2), \quad T_1(t=0)=T_{1,0} \\[3pt]
\frac{dT_2}{dt} = f_2(T_1,T_2), \quad T_2(t=0)=T_{2,0}...
0
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1answer
55 views
Showing stability of numerical scheme: Decaying norm implies stability?
I have a little trouble formulating my question since I am not really sure what conclusions I am supposed to draw from the results I have obtained. I am sorry for the long problem formulation below, ...
3
votes
1answer
109 views
Determine stability of an algorithm?
This is related to a question I answered on Stack Overflow regarding calculating the square root of a number. I was thinking about it and realized that the formula is just the first in a family of ...
3
votes
1answer
268 views
log-sum-exp trick for signed/complex numbers
I need to evaluate a sum of values that are on many different orders of magnitude in scale but might be signed. I’ve had great luck with the “log-sum-exp” trick for an unsigned version of my problem, ...
2
votes
1answer
220 views
Numerical stability in the product of many matrices
I have to calculate in numpy the matrix-product of many matrices (~400). Are there common practices to increase numerical stability?
If this is relevant, the matrices are $300\times 300$ orthogonal ...
2
votes
1answer
170 views
Modified Equation and Stability for Centred Finite Differences for Wave Equation
I am trying to use the modified equation to derive the stability condition for the finite difference approximation
$$
\frac{u(x,t+\Delta t) - 2 u(x, t) + u(x, t -\Delta t)}{\Delta t^2} = c^2 \frac{...
2
votes
1answer
321 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 ...
2
votes
3answers
178 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
1answer
67 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??):
$$ \...
3
votes
1answer
60 views
Bounding error of float32 matrix multiplication
Some numerical debugging led me to the minimal example below. I'm observing relative error of 0.75 on individual elements.
Is there a way to estimate/bound this error without resorting to higher ...
1
vote
0answers
30 views
Finite difference/element method : time step and spatial resolution close to a finite singularity
I'm using the finite element method (FEM), but my question is quite a global question. It's related to this question but it is not the same.
Let's assume we have this equation : $$\partial_t c - u\...
5
votes
1answer
281 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:
$$\frac{\partial u }{\...
0
votes
0answers
49 views
Testing the SUPG method and other methods for hyperbolic equations
I am interesting in integrating the simple equation
$$
\frac{\partial \phi}{\partial t} + \mathbf{u}\cdot\nabla \phi = 0
$$
with a Dirichlet boundary condition at the influx boundary ($\mathbf{u} \...
1
vote
0answers
143 views
Numerically Approximating the Jacobian and Comparing the Eigenvalues With Analytical Form
I am trying to study the stability of numerical discretization schemes using the Jacobian matrix of the residues with respect to the vector of conserved variables.
For a simple diffusion equation ...
1
vote
0answers
57 views
Stability of SVD, Eigendecompositions, and pseudoinverse procedures in modern LAPACK routines
I have proposed an optimisation algorithm which I claim has improved upon the previous algorithm in a number of ways.
One of these claims is that my proposed solution requires no explicit SVD and ...
0
votes
1answer
84 views
Numerical Stability of a Generalized Spatial Discretization Scheme
After reading the matrix stability chapter (10) of Hirsch [1], I decided to dive in the reference list of the chapter. One of the papers [2], which is cited as reference shows an very interesting ...
2
votes
2answers
105 views
Stability of Crank-Nicolson for $u_t = iu_{xx}+2iu$
I want to use the Crank-Nicolson scheme to solve the equation
$$u_t = iu_{xx}+2iu$$
Here's the analysis: Suppose we make a grid, with $k = dt$ and $h = dx$, the usual notation, and also $u_j^n = u(...
0
votes
0answers
82 views
The final Boundary Condition is Unknown, Is Backward Euler is still valid to be implemented?
I am working on conductive polymer modeling and supposed to do one-dimensional diffusion model in the thickness of the polymer, however, due to the small value of thickness in micro, when I use the ...
3
votes
1answer
555 views
Stability region of explicit midpoint method
Consider the explicit midpoint method, i.e
$$y_{n+1}-y_{n-1} = 2hf(y_n).$$
I'm asked to apply this method to the linear test equation, $f(y_n) = \lambda y_n,$ in order to find the method's stability ...
2
votes
1answer
148 views
Why the numerical solution of advection-dominant problem is challenging
In many CFD text books, usually there is a dedicated chapter for advection term discretization.
Why discretization of such term in advection-dominated problems and near the discontinuities is ...
0
votes
1answer
89 views
Stability of PDEs
I am currently trying to solve some PDEs with FiPy. At page 56, the manual mentions (https://www.ctcms.nist.gov/fipy/download/fipy-3.0.pdf).
The largest stable timestep that can be taken for this ...
2
votes
1answer
91 views
Assessing numerical error in solving a least squares problem
I have a linear system of the type
$$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 ...
3
votes
2answers
623 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 ...
5
votes
0answers
410 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\...
