# Questions tagged [stability]

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

153 questions
Filter by
Sorted by
Tagged with
2k views

### How to determine if a numerical solution to a PDE is converging to a continuum solution?

The Lax equivalence theorem states that consistency and stability of a numerical scheme for a linear initial value problem is a necessary and sufficient condition for convergence. But for nonlinear ...
366 views

### Catastrophic cancellation in logsum

I'm trying to implement the following function in double-precision floating point with low relative error: $$\mathrm{logsum}(x,y) = \log(\exp(x) + \exp(y))$$ This is used extensively in statistical ...
1k views

### 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 ...
442 views

### Puzzling remark about stability region of fifth-order Runge-Kutta method

I came across a puzzling remark in the paper P. J. van der Houwen, The development of Runge-Kutta methods for partial differential equations, Appl. Num. Math. 20:261, 1996 On lines 8ff on page 264, ...
216 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, ...
1k views

475 views

### Linearized implicit time stepping

Consider the general FD implicit time stepping scheme $\frac{x_{t+1} - x_t}{\Delta t} = f(x_{t+1})$, where $x$ is the vector variable of interest and $f$ is some function, generally non-linear. ...
393 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 $$C = 2c\frac{\Delta t}{\Delta r \Delta \phi} \leq C_\max \enspace ,$$ ...
376 views

### How to derive an Implicit Runge-Kutta method from Pade approximation

I was reading some work by Butcher and I came across Pade approximations and the correlation between them and stability functions for some Implicit Runge-Kutta methods. For example, in this Pade table ...
190 views

### Is there a backward stable $\tilde{O}(n \log(1/\epsilon))$ algorithm to factor a complex polynomial?

Finding the roots of a complex polynomial is in general extremely numerically unstable, as discussed in (1). According to Pan ((2), (3)), this produces a cubic complexity lower bound, and he presents ...
748 views

### Stability analysis of Heun's method

I am using Heun's method with a third order upwind spatial scheme, which is suggested by Shao (2008) to be used for solving the horizontal advection part of the advection-diffusion equation. This is ...
4k views

### What is the meaning of stability in numerical analysis? How to deterimne the stability of a numerical method?

My question may be so general and simple but I'm really confused about the meaning of the "stability". I look that up in the Internet but there was no general answer to this question. Can anyone help ...
214 views

### interpolation combined with methods of characteristics can cause oscillations for the transport equation?

I would like to know about the effect of using a higher order interpolator for the methods of characteristics. I am solving $$u_t+a(x,t)u_x=0$$ with some nonsmooth initial data $u_0(x)$ by the method ...
96 views

1k views

315 views

### Clenshaw-type recurrence for derivative of Chebyshev series

The naive summation of a Chebyshev series \begin{align*} f(x) = \frac{c_0}{2} + \sum_{k=1}^{n-1} c_{k}T_{k}(x) \end{align*} which employs the three-term recurrence for evaluation of the Chebyshev ...