Questions tagged [stability]
The study of the propagation of errors in a numerical algorithm.
173
questions
1
vote
0answers
42 views
Error analysis and the Model Problem [closed]
In numerical methods for ODE's, the model problem
y' = cy
where c is complex is regarded as sufficient in performing error analysis for different methods in ...
1
vote
0answers
98 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):
$$\frac{u^{...
1
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0answers
36 views
Simulation of asymmetric structures (occupancy = 0.5) unstable
I am trying to simulate a metal-organic framework in LAMMPS using the UFF potential. It's working quite well for some structures where all molecules have an occupancy of 1.
However, when I have a ...
0
votes
1answer
319 views
Runge-Kutta Stability Regions
Based on this link, in particular Figure 1, what is the exact meaning of the plot?
To my understanding, it implies that for a given differential equation:
$$ \frac {dy}{dt} = \lambda y $$
that the ...
0
votes
1answer
77 views
How do we find the condition?
Suppose that we are given a numerical scheme.
In order to find the CFL condition , we set $U_j^n= \lambda ^ne^{ik x_j}$ and put it into the numerical scheme.
I have shown that the given method is ...
0
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1answer
59 views
Technique to find the CFL condition using the Galerkin method in space and finite-difference in time?
I am using the Galerkin method (Discontinuous to be precise) to discretize in space the scalar linear wave equation and the explicit second order centered finite difference scheme to discretize in ...
0
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1answer
100 views
Method of Lines Runge-Kutta nonlinear stability and behavior
I have a system of 4 nonlinear 1st-order PDEs. I want to solve them numerically by method of lines, first discretizing space. This leads to the system of $N\times 4$ coupled ODEs.
$$
\mathbf{u}_{i} =...
0
votes
1answer
45 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 ...
0
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1answer
42 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, ...
0
votes
1answer
78 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 ...
0
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1answer
84 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 ...
0
votes
1answer
798 views
Finite Difference Method Limitations/Stability Criteria
Is it possible to solve an equation with only a single derivative such as:
$$\frac{\partial U(x,t)}{\partial t} = A - BU(x,t)$$
with finite difference methods?
I ask as I am trying to solve the ...
0
votes
0answers
36 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
...
0
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0answers
65 views
Why is my numerical solution to a set of ODEs infinite?
I am trying to solve the following linear PDEs
$$\frac{\partial u_x}{\partial x}=-[i\omega b_{||}+\nabla_\perp u_\perp],$$
$$\frac{\partial b_{||}}{\partial x}=-\frac{i}{\omega}\mathcal{L}u_x,$$
$$\...
0
votes
0answers
30 views
Compute the sum of probabilities when they are given as logits
Say I have a set of numerous probabilities given by their logarithm : $\{\ln p_i, 1 \leq i \leq N\}$.
I want to compute $\sum p_i$, if possible without exponentiating $\ln p_i$, since some of those ...
0
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0answers
37 views
Unconditionally stable numerical method for 1st order non-linear coupled ODEs?
I am attempting to numerically solve the following system of ODEs:
$$\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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0answers
43 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} \...
0
votes
0answers
76 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 ...
0
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0answers
392 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
votes
1answer
1k 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 ...
0
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0answers
47 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 ...
-1
votes
2answers
414 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 ...
-1
votes
1answer
604 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 ...
