# All Questions

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### Where can I find a good reference for the stability properties of several methods of solving parabolic PDEs?

Right now I have a code that uses the Crank-Nicholson algorithm, but I think that I would like to move to a higher-order algorithm for timestepping. I know that the Crank-Nicholson algorithm is stable ...
3k views

### computing turbulent energy spectrum from isotropic turbulence flow field in a box

I have my 3 dimensional velocity flow-field u, v and w at a given instant of time from DNS using pseudo-spectral method. I need to calculate the energy spectrum ( in Fourier space ) as a function of ...
8k views

### Robust algorithm for $2 \times 2$ SVD

What is a simple algorithm for computing the SVD of $2 \times 2$ matrices? Ideally, I'd like a numerically robust algorithm, but I'll like to see both simple and not-so-simple implementations. C code ...
17k views

### How to formulate lumped mass matrix in FEM

When solving time dependent PDE's using the finite element method, for example say the heat equation, if we use explicit time stepping then we have to solve a linear system because of the mass matrix. ...
403 views

### Introduction to computational science?

I'm a high school student interested in computational science, and I would like to learn more about it. This year I took AP Computer Science for that reason, but except for some very basic gambling ...
684 views

### Is there software that can autogenerate numerically-accurate floating point C routines from symbolic formulae?

Given a real function of real variables, is there software available that can automatically generate numerically-accurate code to calculate the function over all inputs on a machine equipped with IEEE ...
2k views

### Testing numerical optimization methods: Rosenbrock vs. real test functions

There seem to be two main kinds of test function for no-derivative optimizers: one-liners like the Rosenbrock function ff., with start points sets of real data points, with an interpolator Is it ...
964 views

2k views

### Does the finite element method impose any restrictions on the Peclet number for numerical stability?

Background on finite volume method When discretising the flux with a central difference stencil of the the advection-diffusion equation restriction $\frac{ah}{d} < 2$ must be observed for the ...
4k views

### Future of OpenCL?

The OpenCL programming paradigm promises to be a royalty free opens standard for heterogenous computing. Should we invest our time in developing software based on OpenCL? Pros/cons?
5k views

### Finding which triangles points are in

Suppose I have a 2D mesh consisting of nonoverlapping triangles $\{T_k\}_{k=1}^N$, and a set of points $\{p_i\}_{i=1}^M \subset \cup_{k=1}^N T_K$. What is the best way to determine which triangle each ...
2k views

### Matrix exponential of a skew-Hermitian matrix with fortran 95 and LAPACK

I'm just getting tucked into fortran 95 for some quantum mechanics simulations. Honestly, I've been spoiled by Octave so I've taken matrix exponentiation for granted. Given a (small, $n\leq 36$) skew-...
874 views

### Smallest eigenvalue without inverse

Suppose $A\in\mathbb{R}^{n\times n}$ is a symmetric, positive definite matrix. $A$ is big enough that it's expensive to solve $Ax=b$ directly. Is there an iterative algorithm for finding the ...
544 views

### Which novel data structures are used in adaptive FEM?

A lot of adaptive FEM libraries use more advanced mesh data structures to handle adding/removing nodes, edges, triangles, tetrahedra, etc. For example, the p4est library uses octree data structures ...
267 views

### calculating eigenvector components of a given vector

I have some vector $V$ which can be decomposed into the eigenspace of the hermitian sparse operator $M$: $V = \sum_i v_i \hat{m}_i$ Is there a way to find the $\hat{m}_i$ (the eigenvector itself) ...
490 views

### Fourth order IMEX Runge-Kutta method

I am looking for the Butcher tableau of a fourth order accurate Runge-Kutta method with IMEX splitting. I have been reading the ''classical'' paper on the subject by Ascher, Ruuth and Spiteri as well ...
356 views

### What guidelines should I use when choosing a scalable linear solver?

There are many different linear solvers, some which work best for diagonally dominant matrices, some for symmetric, some for positive definite ones, some for banded matrices, etc... There are direct ...
8k views

### Visualising Maxwell's equations using MATLAB

I asked this question to help me understand what is going on in one of Maxwell's equations. I am happy with following through the maths on paper now, but would like to use MATLAB to take it one step ...
2k views

### A good, simple book/resource on Parallel Programming in C++ for scientific computing

I am a Mechanical Engineering grad student, currently working on a project which will be scaled up in the new future to require quite some processing power. I am using C++ for the code that I have, ...
3k views

### Diagonal update of a symmetric positive definite matrix

$A$ is a $n \times n$ symmetric positive definite (SPD) sparse matrix. $G$ is a sparse diagonal matrix. $n$ is large ($n$ >10000) and the number of nonzeros in the $G$ is usually 100 ~ 1000. $A$ has ...
708 views

### PDEs in Many Dimensions

I know that most methods of finding approximate solutions to PDEs scale poorly with the number of dimensions, and that Monte Carlo is used for situations that call for ~100 dimensions. What are good ...
579 views

### Is there a multigrid algorithm that solves Neumann problems and has a convergence rate independent of the number of levels?

Multigrid methods usually solve Dirichlet problems on levels (e.g. point Jacobi or Gauss-Seidel). When using continuous finite element methods, it is much less expensive to assemble small Neumann ...
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### What's the current state of the art regarding algorithms for the singular value decomposition?

I'm working on a header-only matrix library to provide some reasonable degree of linear algebra capability in as simple a package as possible, and I'm trying to survey what the current state of the ...
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### Numerically stable explicit solution of small linear system

I have an inhomogeneous linear system $$Ax=b$$ where $A$ is a real $n\times n$ matrix with $n\leq 4$. The nullspace of $A$ is guaranteed to be of zero dimension so the equation has a unique ...
2k views

### Sensitivity of BFGS to initial Hessian approximations

I'm trying to implement the Broyden-Fletcher-Goldfarb-Shanno method to find the minimum of a function. I need two initial guesses $x_{-1}$ & $x_0$ and an initial Hessian Matrix approximation $B_0$...
577 views

### Numerical implementation of the Dirichlet-to-Neumann map

I am solving the Dirichlet problem $$\begin{cases} \Delta u = 0, \\ u|_{\partial D} = f, \end{cases}$$ in a $2d$ domain $D$ using the finite element method. What I want to get is the ...