# Questions tagged [stability]

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

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### Stability of numerical schemes for non-linear equations with a Jacobian with negative eigenvalues

Let us assume I have an A-stable numerical scheme. I believe that given any linear equation $y' = Ay$, it means that the numerical scheme applied to this equation is stable (and therefore convergent ...
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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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### 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 ...
105 views

### 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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### Backward stable projection and normalization of a vector

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, ...
866 views

### Lax-Richtmyer stability analysis

I would like to get to know more in details about Lax-Richtmyer stability analysis (esp in examples), but I didn't manage to find anything except a definition. Could you advice any sources for this ...
192 views

### Problem Condition and Algorithm Stability

Consider 2 mathematical problems: $$f_1(x) = a - x \\ f_2(x) = e^x -1$$ The condition number for a function is defined as follows: $$k(f) = \left| x \cdot \frac{f'}{f} \right|$$ Lets analyze ...
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### 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 ...
332 views

### choice of the norm for Crank Nicolson stability estimate

I have a variable coefficient pde of the form $$u_t=c(t,x)u_{xx}, t\in [0,T], x\in [0,1]$$ with initial data $u_0=u(0,x)$ and $c(x,t)\in C([0,T]\times[0,1])$. I use three point discretization for 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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### 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 ...
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### Stability analysis for a hyperbolic PDE on staggered grid

I am trying to understand the stability of a finite difference equation on the staggered grid. I could understand the Von Neumann stability analysis for the collocated grid for a simple acoustic ...