All Questions
848
questions
10
votes
2answers
435 views
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 ...
7
votes
1answer
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 ...
27
votes
11answers
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 ...
13
votes
1answer
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. ...
6
votes
2answers
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 ...
25
votes
5answers
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 ...
15
votes
4answers
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 ...
13
votes
2answers
964 views
Impose the compatibility conditions for mixed finite elements method in Stokes equation
$\newcommand{\v}[1]{\boldsymbol{#1}}$
Suppose we have following Stokes flow model equation:
$$
\tag{1}
\left\{
\begin{aligned}
-\mathrm{div}(\nu \nabla \v{u}) + \nabla p &= \v{f}
\\
\mathrm{div} \...
11
votes
6answers
6k views
Finite differences on domains with irregular boundaries
Can anybody help me to find the books on numerical solutions(finite difference and Crank–Nicolson methods) of Poisson and diffusion equations including examples on irregular geometry, such as a domain ...
4
votes
4answers
7k views
Efficient assembly of finite element matrix in MATLAB
Question
What is the most efficient algorithm for finding a row of a matrix which matches a given row? This is the same as a table lookup based on multiple criteria.
Context
Finite Element Matrices ...
15
votes
2answers
2k views
FeniCS: Visualizing high order elements
I've just started messing around with FEniCS. I am solving Poisson with 3rd order elements and would like to visualize the results. However, when I use plot(u), the visualization is just a linear ...
13
votes
5answers
2k views
Global maximization of expensive objective function
I am interested in globally maximizing a function of many ($\approx 30$) real parameters (a result of a complex simulation). However, the function in question is relatively expensive to evaluate, ...
11
votes
3answers
2k views
How should non-constant coefficients be treated with finite-volume first order upwind scheme?
Starting with the advection equation in conservation form.
$$
u_t = (a(x)u)_x
$$
where $a(x)$ is a velocity which depend on space, and $u$ is a concentration of a species which is conserved.
...
10
votes
2answers
703 views
How much regularization to add to make SVD stable?
I've been using Intel MKL's SVD (dgesvd through SciPy) and noticed that results are are significantly different when I change precision between ...
9
votes
5answers
1k views
How can I automate the process of optimizing the design of a physical object?
I'm trying to optimize a flow distributor in a tank such that the velocity and temperature distribution across any cross-section is relatively uniform. There are many parameters I can adjust to the ...
6
votes
2answers
4k views
How can I prove numerical diffusion in upwind scheme for transport equation
I was just implementing the upwind scheme for a linear transport equation $u_t + cu_x = 0$ where $c=0.5$ and I saw that the solution was indeed advected but over time it starts to diffuse. Can anyone ...
6
votes
5answers
10k views
How to solve block tridiagonal matrix using Thomas algorithm
Thomas algorithm can be used to solve a tridiagonal matrix:
$$
\begin{bmatrix}
{b_ 1} & {c_ 1} & { } & { } & { 0 } \\
{a_ 2} & {b_ 2} & {c_ 2} & { } & { }...
5
votes
2answers
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 ...
23
votes
6answers
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?
16
votes
3answers
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 ...
11
votes
4answers
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-...
11
votes
1answer
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 ...
10
votes
2answers
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 ...
8
votes
2answers
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) ...
8
votes
1answer
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 ...
6
votes
2answers
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 ...
4
votes
2answers
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 ...
4
votes
3answers
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, ...
21
votes
1answer
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 ...
15
votes
3answers
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 ...
14
votes
1answer
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 ...
12
votes
3answers
2k views
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 ...
11
votes
3answers
1k views
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 ...
10
votes
1answer
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 $B_0$...
9
votes
0answers
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 ...
9
votes
3answers
919 views
Iterative methods for indefinite systems without block structure
Indefinite systems of matrices appear for example in the discretization of saddle point problems by mixed finite elements. The system matrix can then be put in the form
$$\begin{pmatrix} A & B^t \...
7
votes
2answers
555 views
Matrix free finite elements method for visualization in process tomography
I am Computer Scientist and now I am interested in matrix multiplication on GPUs. My research are focused on matrix free finite elements method where I multiply sparse matrix. Sparse matrix could ...
7
votes
3answers
393 views
Strong coupling of a non-linear multiphysic problem: failure with Newton Raphson method
I am trying to solve a multiphysic problem using finite elements and a Newton Raphson solution scheme. I have two non-linear subsystems that are coupled bi-directionally.
The first subsystem includes ...
6
votes
4answers
2k views
Numerical integration of non-uniform acceleration samples
I'm given a stream of acceleration data with timestamps. The sampling is non-uniform.
Apart from Euler, is there a way to integrate the acceleration into velocity? Something more accurate or of ...
5
votes
1answer
323 views
Working with multi-dimensional functions
How would you represent functions of type $[-1, 1]^n \to \mathbb R \;$ for moderate $n$? How would you integrate them?
For small $n$ (1-2) such functions can be represented as histograms, vectors in ...
5
votes
1answer
2k views
Finite Difference Method Neumann Boundary Condition with Variable Coefficients
Disclaimer
In the process of typing up this question, I determine its solution. Since I went through the trouble of typing up the question in its entirety, I will post its answer as well. It may ...
3
votes
1answer
295 views
Random placement of euclidean points with constrained inter-point distances in a fixed area
I'd like to place as many random points as possible in a 2D square $S=[0,1]x[0,1]$ such that the euclidean distance $d$ between any two points $d$ is greater than a given value $b$ (b is small). I'm ...
2
votes
1answer
2k views
3D Solid 8 Node FEM Matlab Code
So this semester, I'm taking a Finite Element Method course at my graduate school. We started out making codes for 1D bars and came all the way to 8 node solid elements. However, I seem to have run ...
2
votes
1answer
3k views
Angular Velocity by Vector - 2D
This is originally a problem in programming, but since almost no one on Stackoverflow know how to solve this I went here instead; https://stackoverflow.com/questions/23003612/javascript-angular-...
0
votes
1answer
405 views
Find cfl condition
We have the advection equation $u_t+a u_x=0, a>0, 0<t<T_f, x \in \mathbb{R}$ with initial condition $u(0,x)=u_0(x)$.
Suppose that we have the following sheme:
I want to find the CFL ...
-1
votes
1answer
572 views
Pde problem with robin boundary condition
I have my pde 2D problem with robin condition (form: du/dn +ku=g) to solve with matlab. i have the exact function u and I want to find the function g in robin condition. How can i do it?
thanks for ...
15
votes
3answers
2k views
Why isn't my Matrix-Vector Multiplication Scaling?
Sorry for the long post but I wanted to include everything that I thought was relevant in the first go.
What I want
I am implementing a parallel version of Krylov Subspace Methods for Dense Matrices. ...
9
votes
3answers
2k views
How to use polylogarithm function in c++?
Is there any preprocessor directives that could be used to use the polylog function? Or is it included in cmath? If so, do you call it by Li or by polylog?
EDIT:
What I really am trying to do is give ...
9
votes
2answers
698 views
How to model a fishing rod (or a rope)?
I wish to model a fishing rod (or a rope) by joining short segments. (The segments may have equal (short) length but each segment should be assigned its own individual mass.) One segment will ...
8
votes
1answer
803 views
Newton iteration applied to nonlinear PDE
I'm having difficulty understanding how to apply Newton iteration to nonlinear PDEs and then use a fully implicit scheme to time step. For example, I want to solve Burgers equation
$$u_{t} + u u_{x} -...