# Questions tagged [convergence]

Questions related to whether the sequence of iterates generated by an iterative method has one or more limit points, and if those limit points have the correct properties.

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### Stability of Euler forward method

I am trying to solve a linear system of ODEs of the form: $$\frac{du}{dt} = A u, \quad u(0)=k$$ where $A$ is a 2x2 matrix and $u(t)$ is a 2x1 column vector. I want to solve this numerically, using ...
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### Role of rotation's pivot point in optimization?

In this paper, the authors describe how to use locally rigid transformations (sampled on nodes in space) to deform mesh vertices. In the paper, rotations are relative to the pivot point, which ...
1 vote
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### Why is the definition of convergence different for root finding algorithms as compared to sequences?

The definition of convergence for root finding algorithms is given in a few sources as: A sequence ${x^k}$ generated by a numerical method is said to converge to the root $\alpha$ with order $p\geq 1$ ...
1 vote
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### Why do we use modified pressure in incompressible multiphase solvers with gravity?

The context of my question is two-phase incompressible solvers such as interFoam in OpenFOAM, but I have seen this trick used ...
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### Faster convergence for minimizing least squares of forward modelling problems

This specific question was raised from optimizing parameters of column experiments in the hydrogeological context. I want to optimize a parameter of interest (in this case $D$), based on experimental ...
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### Computationnal Mechanics : Three Bodies contact problem with ALM

I am facing an issue with convergence of a contact problem consisting of three successive bars, as presented below. A force is applied on the right hand of the first bar so there is contact between ...
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### Issue solving nonlinear equation containing a quotient

I have a coupled set of PDEs that need to be solved as part of a larger problem. I am currently approaching this by computing spatial derivatives with finite differences and using PETSc's nonlinear ...
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1 vote
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Consider the 1D nonstationary convection-diffusion PDE \begin{alignat}{2} \partial_t u &= -a \partial_x u + D \partial_{xx}u, &\qquad x \in (0,1), t \in (0,T), \\ f(t) &= \left.\left( a ... 2 votes 1 answer 2k views ### what is non-asymptotic convergence? I guess convergence in general means it is in asymptotic sense but what does non-asymptotic convergence mean?. Can someone please explain with an example? 2 votes 0 answers 37 views ### Convergent Finite Difference Scheme for Parabolic Equation Consider the PDEu_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{... 5 votes 1 answer 131 views ### Non-conforming bi-linear finite element The four-noded bi-linear rectangle element, which sometimes goes under the name Melosh element, is non-conforming unless the element sides are aligned. Out of curiosity I have implemented this element ... 2 votes 1 answer 322 views ### Best way to check if SOR solution has converged for 2d matrix I have written a SOR algorithm to solve the Laplace equation on a 2d grid. The outside of the grid is fixed at 0 and the central square is fixed at 10. I can obtain the fully converged solution for ... 3 votes 1 answer 118 views ### Determine stability of an algorithm? This is related to a question I answered on Stack Overflow regarding calculating the square root of a number. I was thinking about it and realized that the formula is just the first in a family of ... 2 votes 4 answers 312 views ### Testing the time dependent Schrodinger Equation with an analytical solution? I am numerically solving the Schrodinger Equation in 1D first and in higher dimension later, but I want to know the convergence rate of my numerical solver in different grid size and numerical methods.... 1 vote 1 answer 129 views ### Unexpectedly Slow Convergence Implicit Euler I'm solving the coupled ODE \left[\begin{array}{c}x^\prime(z)\\p_x^\prime(z)\end{array}\right] = C(z)\cdot\left[\begin{array}{c}x(z)\\p_x(z)\end{array}\right] = \left[\begin{array}{cc}0& A(z)\\...
I have a nonlinear solver for equation $$g= c_1f(x_1,y_1)+c_2f(x_2,y_2)$$ Note that $c_1$ is much bigger than $c_2$. After using Levenberg–Marquardt algorithm, it seemed to only optimize $x_1$ and \$...