# Questions tagged [stability]

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

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### 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 ...
482 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 ...
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 ...
1k 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 ...
239 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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### 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 ...
10k views

### Estimating the Courant number for the Navier-Stokes Equations under differing Reynolds number regimes

I am familiar with the Courant-Friedrich-Lewy Condition in as far as it applies to the stability of explicit finite difference schemes for standard parabolic and hyperbolic PDEs. However, when ...
392 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 $$C = 2c\frac{\Delta t}{\Delta r \Delta \phi} \leq C_\max \enspace ,$$ ...
294 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 ...
215 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, ...
403 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 ...
3k views

### Meaning of CFL condition on parabolic problems

I've been studying this FEM theory and for the parabolic problems, there's the analysis of stability of the $\theta$-method. I followed the analysis and they get this CFL (Courant-Friedrich-Lewy) ...
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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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### 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 ...
1k views

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

I am reading a paper  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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### 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} }{\...
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### 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 ...
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### 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 ...
Given a machine precision unit vector $n$, and an arbitrary vector $v$, I want an unconditionally backward stable method to compute $$f(v) = \frac{v-nn'v}{\left|v-nn'v\right|}$$ In other words, ...