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21 votes
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What is the state of the art in solving stiff initial value problems?

So there is a ton to say about this, and we will actually be putting a paper out that tries to summarize it a bit, but let me narrow it down to something that can be put into a quick StackOverflow ...
Chris Rackauckas's user avatar
15 votes

Why are systems with clustered eigenvalues easy to solve?

A good explanation of this phenomena with many examples is given in Iterative Methods for Linear and Nonlinear Equations by Tim Kelley. The crux of it comes down to the fact that each step of a ...
whpowell96's user avatar
  • 2,546
14 votes
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$L^2$-convergence of finite element method when right hand side is only in $H^{-1}$ (Poisson eqn)

Yes, this is the standard Aubin-Nitsche (or duality) trick. The idea is to use the fact that $L^2$ is its own dual space to write the $L^2$-norm as an operator norm $$\|u\|_{L^2} = \sup_{\phi\in L^2\...
Christian Clason's user avatar
13 votes
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How to directly compute the inverse of an ill-conditioned dense matrix

Though it is a relatively rare situation when you actually have to calculate an inverse of the matrix, not all techniques were created equally. I would use the term badly-conditioned instead of ill-...
Anton Menshov's user avatar
  • 8,672
11 votes
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How important is learning hardware/architecture for scientific computing?

I haven't worked in quantum chemistry specifically, but I've worked in other areas where high performance is a correctness requirement (along with scientific accuracy), so I think we're on the same ...
Pseudonym's user avatar
  • 351
10 votes
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Intro to DG Finite Element methods

For parabolic/elliptic PDE's, I highly recommend Beatrice Riviere's book: Discontinuous Galerkin methods for solving elliptic and parabolic equations: theory and implementation. For hyperbolic PDE'...
Paul's user avatar
  • 12k
10 votes
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4th order tensor rotation - sources to refer

There are two main ways to write stress/strain tensors as 6 components vectors: Voigt notation, that is the most common; and Mandel-Kelvin notation, that has the advantage of writing stress and ...
nicoguaro's user avatar
  • 8,525
9 votes

Robust algorithm for $2 \times 2$ SVD

I needed an algorithm that has little branching (hopefully CMOVs) no trigonometric function calls high numerical accuracy even with 32 bit floats We want to calculate $c_1, s_1, c_2, s_2, \sigma_1$ ...
petiaccja's user avatar
9 votes
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First appearance of the phrase "inverse crime"

The term inverse crime for a numerical test of a parameter identification method that uses data contained in the range of the discrete(!) forward operator used for the inversion (thus essentially ...
Christian Clason's user avatar
9 votes
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Consumer hardware for scientific computing?

One issue that you should be aware of is that NVIDIA has a market segmentation strategy in which it sells relatively inexpensive GPU's to the gaming and graphics workstation markets (GeForce and ...
Brian Borchers's user avatar
9 votes
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Searching for recent code source for "Parallel scientific computing in C++ and MPI "

First of all, the book you mention is very old. In fact, it misses the last two MPI standards (3 and 4), and every C++ standard from C++11 on. Secondly, know that MPI has officially no C++ bindings, ...
Victor Eijkhout's user avatar
9 votes
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FEM for vector valued problems: reference request

Short answer: Just replicate the vector of interpolation functions into a block-diagonal matrix, as showed e.g. on page 5 in this lecture note. Detailed answer: Mathematically oriented texts typically ...
Zoltan Csati's user avatar
8 votes
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A priori FEM estimates without $H^2$ regularity

The usual argument for error estimates in the energy norm is to first use the best-approximation property to get things back to the interpolation error. That is, $$ \| u-u_h \|_{H^1} \le C \| u-u_I \...
Wolfgang Bangerth's user avatar
8 votes

Why are systems with clustered eigenvalues easy to solve?

At the $k$th iteration, typical Krylov methods for solving $Ax=b$ (such as CG, MINRES, and GMRES) implicitly construct a $k$th order polynomial $Q(x)$ such that: $Q(0) = 1$. $|Q(\lambda_i)|$ is as ...
Nick Alger's user avatar
  • 3,143
7 votes

Physics Simulation in C++

I think you are missing a very important and crucial step that lies exactly between the physics and simulation: the mathematical model. In order to model any physics, one has to formulate the ...
Anton Menshov's user avatar
  • 8,672
7 votes
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Term for the typical "linear in the larger dimension, quadratic in the smaller" cost for linear algebra

The book "Introduction to Applied Linear Algebra" by Boyd and Vandenberghe has an appendix about complexity of basic operations in linear algebra and they call this case big-times-small-...
Dirk's user avatar
  • 1,738
7 votes
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The Formula of Explicit Runge-Kutta Fourteen order

The 14th order methods due to Feagin can be found in DifferentialEquations.jl. An example using them with 128-bit floating point arithmetic is as follows: ...
Chris Rackauckas's user avatar
6 votes

Looking for Runge-Kutta 8th order in C/C++

summarizing some points: If it's a long-term integration of a non-dissapative model, a symplectic integrator is what you're looking for. Otherwise, since it's an equation of motion, Runge-Kutta ...
Chris Rackauckas's user avatar
6 votes

What good are hard-sphere event-driven molecular dynamics simulations in the face of chaos?

Yes, it is possible to show that the statistical behavior of the approximate system will reach that of the "exact" system. (This is true even though hard-sphere dynamics do not accurately describe ...
aeismail's user avatar
  • 3,513
6 votes
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Good references for dual-weighted residual (DWR) method for goal-oriented adaptive mesh refinement (AMR)

The canonical "first" reference for the method is a paper by Becker and Rannacher that was ultimately published as an article in the ENUMATH 97 proceedings, but is often cited as the ...
Wolfgang Bangerth's user avatar
6 votes
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Scientific computing code development hands on introduction

If you're interested in the process of developing scientific software for the simulation of continua based on the finite element method, you might be interested in the deal.II library (https://www....
Wolfgang Bangerth's user avatar
6 votes
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Mesh refinement in the Finite Element Method

You really don't want to implement this yourself -- you'll spend a year or two on things others have already done, and will have done far better than you can hope for. The difficulty is generally ...
Wolfgang Bangerth's user avatar
6 votes
Accepted

Continuous vs discontinuous space-time FEM

More concretely, it can be shown that discontinuous Galerkin (dG(r)) schemes lead to strongly A-stable time stepping schemes and continuous Galerkin (cG(r)) schemes are A-stable time stepping schemes, ...
Julian Roth's user avatar
6 votes
Accepted

Literature request covering Chebyshev's pseudospectral collocation method

When I studied spectral collocation methods, I remember that the Chebyshev and Fourier Spectral Methods from Boyd was an easy read (as the Trefethen book you mentioned) with many practical details. If ...
Zoltan Csati's user avatar
5 votes

Why do Newton-Krylov iterations stagnate in this problem?

I'm not sure if this is the answer, but consider the case as $\alpha \rightarrow \infty$. In this case you're trying to solve $\iint_{\Omega} \cosh(P) dxdy = 0$ And since cosh looks like this: You ...
Charles's user avatar
  • 619
5 votes
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Preconditioner for scalar laplacian system

Preconditioning and iterative solvers are cool, but did you try to solve your problem with some kind of sparse direct solver? If not, try it first. State–of–the–art preconditioning techniques for ...
56th's user avatar
  • 901
5 votes

Leveraging scipy for matrix free finite elements

I would say that the implementation + verification + unit testing would take you more than just 3 weeks. Although, if you are planning to invest that time, you might add that capabilities to ...
nicoguaro's user avatar
  • 8,525
5 votes
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Resources on mesh generation for finite element methods

Starting a community answer, in line with your model question Mesh Generation Mesh Generation: Application to Finite Elements (P.-L. George and P. Frey) Hermes, Lyon, 2000. A clear primer on the core ...
5 votes

Time integration of wave equation

It's not really meaningful to talk about integrating the equation in form A or B, since one way to integrate A is to first transform to B and then discretize. You can only really compare the actual ...
David Ketcheson's user avatar
5 votes
Accepted

Truncated power series algebra implementation

Julia TPSA.jl being developed at https://github.com/bmad-sim/TPSA.jl. Should be operational soon.
DavidS's user avatar
  • 166

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