# Questions tagged [stability]

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

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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 ...
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### 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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### 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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### 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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### 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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### Courant Friedrichs Lewy condition - how to get it?

I am interested, how can we get CFL condition for every type of PDE? It's known that for 1st order linear equation $$\frac{\partial u}{\partial t}+a\frac{\partial u}{\partial x}=0$$ CFL is get from ...
556 views

### In what regime do the continuous and discontinuous Galerkin method become unstable for advection-diffusion systems?

I know that the finite volume method (based around a central different stencil) is unstable for advection dominated advection-diffusion problems. This leads to different adaptive schemes to can be ...
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### 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 ...
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### Is stabilization of energy equation needed when momentum equation needs it?

When SUPG/PSPG stabilization is added to momentum equation of flow problem, is needed stabilization for energy equation also? I would guess that when stabilization for velocity works fine so one gets ...
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### 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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### 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 ...
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### Von Newman stability analysis for 2D acoustic wave equation explicit

Von Newman stability analysis for acoustic wave equation explicit centered differences: 2nd order time and space (N 2)'th order: \begin{eqnarray} U_{jk}^{n+1} = \left( \frac{\Delta t V_{jk} }{\...
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 ...