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# 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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### What is the principle behind the convergence of Krylov subspace methods for solving linear systems of equations?

As I understand it, there are two major categories of iterative methods for solving linear systems of equations: Stationary Methods (Jacobi, Gauss-Seidel, SOR, Multigrid) Krylov Subspace methods (...
2answers
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### 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 ...
1answer
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### Why does iteratively solving the Hartree-Fock equations result in convergence?

In the Hartree-Fock self-consistent field method of solving the time-independent electronic Schroedinger equation, we seek to minimize the ground state energy, $E_{0}$, of a system of electrons in an ...
1answer
463 views

### How to establish that an iterative method for large linear systems is convergent in practice?

In computational science we often encounter large linear systems which we are required to solve by some (efficient) means, e.g. by either direct or iterative methods. If we focus on the latter, how ...
1answer
197 views

### Implications of thermodynamic inconsistency in CFD calculations

During my PhD work, I had to use tabulated values of thermodynamic properties of gases in some Computational Fluid Dynamics (CFD in short) simulations. My tables are discretized in temperature and ...
2answers
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### Which iterative linear solvers converge for positive semidefinite matrices?

I want to know which of the classic linear solvers (e.g Gauss-Seidel, Jacobi, SOR) are guaranteed to converge for the problem $Ax=b$ where $A$ is positive semi definite and of course $b \in im(A)$ (...
1answer
292 views

### Why do we have to rerun the CFD solver for higher Reynolds number?

I started to learn OpenFOAM from the Cavity tutorial which is provided at the web-site. When experimenting with different Reynolds numbers, in section "2.1.8.2 Running the code", tutorial says to ...
3answers
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### Basin of attraction for Newton's method

Newton's method for solving nonlinear equations is known to converge quadratically when the starting guess is "sufficiently close" to the solution. What is "sufficiently close"? Is there literature ...
1answer
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### DIIS method to accelerate SCF convergence for stretched geometries

I am implementing from scratch an Hartree-Fock calculation in the STO-3G basis set to perform Born-Oppenheimer molecular dynamics. I have a Restricted Hartree-Fock procedure that can reproduce very ...
0answers
160 views