21
votes
Accepted
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 ...
- 12.3k
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 ...
- 2,054
14
votes
Accepted
$L^2$-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\...
- 12.2k
13
votes
Accepted
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-...
- 8,602
12
votes
Accepted
Benchmarks for Gröbner bases and polynomial system solution
I posted some benchmarks here:
http://www.cecm.sfu.ca/~rpearcea/mgb.html (archived copy)
These are for total degree orders. To solve systems you typically need to do more work. Timings are for a ...
- 236
11
votes
Accepted
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 ...
- 351
10
votes
Accepted
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'...
- 12k
10
votes
Accepted
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 ...
- 8,370
9
votes
Benchmarks for Gröbner bases and polynomial system solution
Googling benchmark polynomial systems leads to a few hits, including the University of Mannheim's Computer Algebra Benchmark Initiative. Sadly, most of these are ...
- 12.2k
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$ ...
- 91
9
votes
Accepted
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 ...
- 12.2k
9
votes
Accepted
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 ...
- 18.5k
9
votes
Accepted
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, ...
- 1,305
9
votes
Accepted
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 ...
- 747
8
votes
Accepted
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 \...
- 54.2k
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 ...
- 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 ...
- 8,602
7
votes
Accepted
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-...
- 1,738
7
votes
Accepted
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:
...
- 12.3k
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 ...
- 12.3k
6
votes
Benchmarks for Gröbner bases and polynomial system solution
Always keep in mind that the results of any benchmark will depend, in addition to the problem's size, on the base field over which the polynomial ring is defined (rational numbers or integers modulo ...
- 3,994
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 ...
- 3,523
6
votes
Accepted
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 ...
- 54.2k
6
votes
Accepted
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....
- 54.2k
6
votes
Accepted
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 ...
- 54.2k
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, ...
- 361
5
votes
Accepted
Fourth order IMEX Runge-Kutta method
I think the work of Kennedy and Carpenter (mentioned already by @GoHokies) is still the definitive study on this topic. The journal paper can be found here; for some reason Google Scholar only ...
- 16.4k
5
votes
Accepted
Resources for solving mixed left and right matrix equations
This is a Sylvester equation, although normally the matrices would all be square. Even for rectangular matrices, the Bartels-Stewart method applies (also A Hessenberg-Schur Method for the Problem $AX+...
- 11.4k
5
votes
Accepted
2d Euler manufactured solutions
This 2004 paper by Roy et. al in Int. J. Numer. Meth. Fluids 2004; 44:599–620 (DOI: 10.1002/d.660) should contain exactly what you're looking for:
ftp://ftp.demec.ufpr.br/CFD/bibliografia/...
- 2,176
5
votes
Introduction to Lattice Boltzmann methods
Back in August 2011, I followed a beginner's course (LBM workshop) on LBM in Canada.
The resources for that course are still up and contain a nice tutorial covering a wide range of topics: theory, ...
- 277
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