# Questions tagged [stability]

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

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### Stability of the Forward-Time Central Space method, section 9.3 in LeVeque

I am reading section 9.3 in Leveque about stability. The explanations are very short and brief so I am asking for some help to understand and elaborations. The result of the section is that for FTCS ...
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### Positive feedback instability

I have two black boxes (representing some multi-physics but that does not matter), where box #1 takes some quantity T as input and produces some quantity P as output; and box #2 takes P as input and ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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### 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 ...
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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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### 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 ...
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### 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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In the book of Hesthaven and Warburton on discontinuous Galerlkin methods in example 2.3 (regarding solutions of the wave equation), the authors regard the following PDE: $$\frac{\partial u }{\... 0answers 47 views ### Testing the SUPG method and other methods for hyperbolic equations I am interesting in integrating the simple equation$$ \frac{\partial \phi}{\partial t} + \mathbf{u}\cdot\nabla \phi = 0 $$with a Dirichlet boundary condition at the influx boundary (\mathbf{u} \... 0answers 138 views ### Numerically Approximating the Jacobian and Comparing the Eigenvalues With Analytical Form I am trying to study the stability of numerical discretization schemes using the Jacobian matrix of the residues with respect to the vector of conserved variables. For a simple diffusion equation ... 0answers 54 views ### Stability of SVD, Eigendecompositions, and pseudoinverse procedures in modern LAPACK routines I have proposed an optimisation algorithm which I claim has improved upon the previous algorithm in a number of ways. One of these claims is that my proposed solution requires no explicit SVD and ... 1answer 84 views ### Numerical Stability of a Generalized Spatial Discretization Scheme After reading the matrix stability chapter (10) of Hirsch [1], I decided to dive in the reference list of the chapter. One of the papers [2], which is cited as reference shows an very interesting ... 2answers 101 views ### Stability of Crank-Nicolson for u_t = iu_{xx}+2iu I want to use the Crank-Nicolson scheme to solve the equation$$u_t = iu_{xx}+2iu$$Here's the analysis: Suppose we make a grid, with k = dt and h = dx, the usual notation, and also u_j^n = u(... 0answers 82 views ### The final Boundary Condition is Unknown, Is Backward Euler is still valid to be implemented? I am working on conductive polymer modeling and supposed to do one-dimensional diffusion model in the thickness of the polymer, however, due to the small value of thickness in micro, when I use the ... 1answer 541 views ### Stability region of explicit midpoint method Consider the explicit midpoint method, i.e$$y_{n+1}-y_{n-1} = 2hf(y_n).$$I'm asked to apply this method to the linear test equation, f(y_n) = \lambda y_n, in order to find the method's stability ... 1answer 147 views ### Why the numerical solution of advection-dominant problem is challenging In many CFD text books, usually there is a dedicated chapter for advection term discretization. Why discretization of such term in advection-dominated problems and near the discontinuities is ... 1answer 89 views ### Stability of PDEs I am currently trying to solve some PDEs with FiPy. At page 56, the manual mentions (https://www.ctcms.nist.gov/fipy/download/fipy-3.0.pdf). The largest stable timestep that can be taken for this ... 1answer 91 views ### Assessing numerical error in solving a least squares problem I have a linear system of the type$$Ax = b I want to minimise $|b - Ax|^2$. I know there are different approaches to directly solve the system (Normal equation + Cholesky, QR decomposition, SVD ...
How can I generate the higher $n$ quantum harmonic oscillator wavefunction (in position space) numerically? Here, higher means around $n=500$, or say $n=2000$, where $n$ is the $n$th oscillator ...
Homework disclaimer... The task: We are using the following algorithm to solve the quadratic equation $x^2+2px+q=0$: $x_1=|p|+\sqrt{p^2-q}\mathtt{;}$ \$\mathtt{if}\,p>0\,\mathtt{then}\,x_1=-x_1\...