# Questions tagged [stability]

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

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### Comparison between stability and accuracy of various Finite Difference schemes

Im Analyzing the 1D convection equation (PDE) for stability, consistency, and accuracy. I know Both upwind schemes (explicit and implicit) show better stability with a higher number of waves for ...
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### Stability plot of upward difference implicit time

I am analyzing the stability of 1D convection (advection) equation as shown in the picture. When I derive the equations as shown I want to get rid of the complex number. Im asking if those stability ...
1answer
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### Accuracy gap for apparently stable solution

I was reasoning about the behaviour of the methods I'm using for my simulation and I noticed that, considering $h_s$ as the timestep over which I have unstable solutions and $h_a$ as the timestep ...
2answers
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### Exponent log to compute reciprocal power?

A MATLAB library seems to overcomplicate a computation: exp( (log(a) - log(b))/b ) which is mathematically equivalent (assuming real & positive ...
1answer
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### 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 ...
2answers
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### Using the BDF and RK4 methods to solve this coupled system of ODEs in C++

I'm trying to solve a system of ODEs using the BDF order 4 method. I find the first 3 points using RK4, then for the implicit part of the BDF, I use Newton-Raphson iteration. Unfortunately my solution ...
1answer
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### Numerical stability of taking the mean of outputs from the simulation of a discrete stochastic dynamical system

I am writing a simulation for a discrete stochastic dynamical system. Since the simulation is stochastic, I need to run the simulation multiple times and then average the values of each timestep. I ...
1answer
71 views

### How to include negative number in the log-sum-exp?

I want to know summation of some small numbers, such as {e^-1000, -e^1001, e^1002...} If all numbers are positive, I can use log-sum-exp algorithm. But unfortunately, negative numbers are also ...
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### 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 ...
0answers
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### Convergent Finite Difference Scheme for Parabolic Equation

Consider the PDE $$u_t = b_{11}u_{xx} + 2b_{12}u_{xy} + b_{22}u_{yy},$$ where $b_{11}, b_{22} > 0$, and $b_{12}^2 < b_1b_2$. In Strikwerda's book, the ADI scehme \begin{align*} \left( 1 - \frac{...
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1answer
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### Step size and stability of Euler forward method

I'm trying to calculate the maximum step size that provides stability for the following nonlinear IVP using the Euler forward method: $u'(t) = -200tu(t)^2,\qquad u_0 = 1, \qquad t\in [0,3]$, with ...
3answers
171 views

### What is the flaw in my stability analysis?

The ODE $${d^2x\over dt^2}=-kx; k>0$$can be converted in the system of linear equations as \begin{align} {dx\over dt} & =v\\ {dv\over dt} &= -kx\\ \end{align} Using Euler’s method, ...
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