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# Tag Info

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 ...
• 12.4k

### 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 ...
• 2,805
Accepted

• 56.1k

### 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

### 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 ...
• 8,701
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
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: ...
• 12.4k

### 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 ...
• 12.4k

### 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 ...
• 3,513
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 ...
• 56.1k
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....
• 56.1k
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 ...
• 56.1k
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, ...
• 796
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 ...
• 832

### Any FEM book recommendations that focus on stability and proofs on error bounds?

I would suggest the books below: Susanne C. Brenner , L. Ridgway Scott - The Mathematical Theory of Finite Element Methods Daniele Boffi , Franco Brezzi , Michel Fortin - Mixed Finite Element Methods ...
• 2,821
Accepted

### 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 ...
• 901

### 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 ...
• 8,582
Accepted

### 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 ...

### 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 ...
• 16.6k
Accepted

### Truncated power series algebra implementation

Julia TPSA.jl being developed at https://github.com/bmad-sim/TPSA.jl. Should be operational soon.
• 166
I can write my experiences here because I do not have any book references at hand. Consider a triangular element with the corner points $\boldsymbol{x}_i \in \mathbb{R}^2$, $i=1,2,3$. The degrees of ...