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16
votes
1answer
12k views

How should boundary conditions be applied when using finite-volume method?

Following from my previous question I am trying to apply boundary conditions to this non-uniform finite volume mesh, I would like to apply a Robin type boundary condition to the l.h.s. of the domain (...
13
votes
2answers
9k views

Periodic boundary condition for the heat equation in ]0,1[

Let us consider a smooth initial condition and the heat equation in one dimension : $$ \partial_t u = \partial_{xx} u$$ in the open interval $]0,1[$, and let us assume that we want to solve it ...
22
votes
4answers
3k views

How to incorporate the boundary conditions with the Galerkin method?

I've been reading some resources on the web about Galerkin methods to solve PDEs, but I'm not clear about something. The following is my own account of what I have understood. Consider the following ...
10
votes
1answer
6k views

Are there any heuristics for optimizing the successive over-relaxation (SOR) method?

As I understand it, successive over relaxation works by choosing a parameter $0\leq\omega\leq2$ and using a linear combination of a (quasi) Gauss-Seidel iteration and the value at the previous ...
17
votes
2answers
8k views

Null-space of a rectangular dense matrix

Given a dense matrix $$A \in R^{m \times n}, m >> n; max(m) \approx 100000 $$ what is the best way to find its null-space basis within some tolerance $\epsilon$? Based on that basis can I then ...
5
votes
3answers
885 views

Computing accurate fluxes with FEM

I have solved Poisson equation on a 3d domain with neumann and dirichlet boundary condition. I get the potential, take the gradient for each element and integrate on a surface of an element, I do this ...
17
votes
3answers
67k views

How to determine the amount of FLOPs my computer is capable of

I would like to determine the theoretical number of FLOPs (Floating Point Operations) that my computer can do. Can someone please help me with this. (I would like to compare my computer to some ...
13
votes
3answers
8k views

Understanding how Numpy does SVD

I have been using different methods to calculate both the rank of a matrix and the solution of a matrix system of equations. I came across the function linalg.svd. Comparing this to my own effort of ...
10
votes
3answers
4k views

Applying Dirichlet boundary conditions to the Poisson equation with finite volume method

I would like to know how Dirichlet conditions are normally applied when using the finite volume method on a cell-centered non-uniform grid, My current implementation simply imposes the boundary ...
4
votes
2answers
7k views

Tikhonov regularization in the non-negative least square - NNLS (python:scipy)

I am working on a project that I need to add a regularization into the NNLS algorithm. Is there a way to add the Tikhonov regularization into the NNLS implementation of scipy [1]? [2] talks about it, ...
8
votes
1answer
2k views

PDE discretization with the method of rothe and the method of lines (Modular implementation)

The Heat equation is discretized in space with FV (or FEM), and a semi-discrete equation is obtained (system of ODEs). This approach, known as the method of lines, allows to easily switch from one ...
4
votes
2answers
2k views

The meaning of conservative discretization in Galerkin FEM and Discontinuous Galerkin

I do understand the meanning of "conservative discretization" within the FVM/FDM framework, indeed it is well explained in this post. Now, according to the table in this slide (pp.8), it concludes: ...
16
votes
7answers
1k views

Does Computational Science involve programming?

I read about computational science on Wikipedia, but my understanding is not very clear. Does computational science involve programming? How different is computational science from computational ...
15
votes
2answers
1k views

Estimation of condition numbers for very large matrices

Which approaches are used in practice for estimating the condition number of large sparse matrices?
1
vote
1answer
510 views

More Smearing with decreasing timestep in advection problems

I find it kind of counter intuitive, that the result of an advection gets more smeared out at the borders when decreasing the timestep (which should make it more accurate). Let there be a equally ...
16
votes
4answers
2k views

Why can't Householder reflections diagonalize a matrix?

When computing the QR factorization in practice, one uses Householder reflections to zero out the lower portion of a matrix. I know that for computing eigenvalues of symmetric matrices, the best you ...
10
votes
2answers
458 views

Eigenvectors of a small norm adjustment

I have a dataset that is slowly changing, and I need to keep track of eigenvectors/eigenvalues of its covariance matrix. I've been using scipy.linalg.eigh, but it'...
7
votes
1answer
426 views

How to directly compute the inverse of an ill-conditioned dense matrix

I know that it is generally a bad idea to compute the inverse matrix directly. However, if it is necessary to compute the inverse of an ill-conditioned invertible dense matrix, then what can I try? ...
6
votes
1answer
151 views

Roots of a function for eigensystem

I want to find the roots for $\kappa$ for the equation $$\sqrt{\alpha - 1} \cos{\left (\frac{\sqrt{2} \sqrt{\alpha - 1}}{2 \sqrt{\epsilon}} \right )} \cosh{\left (\frac{\sqrt{2} \sqrt{\alpha + 1}}{2 \...
2
votes
1answer
502 views

Finite difference discretization on a circle

I am trying to discretize the differential operator $\frac{d^2}{dx^2}$ acting on $S^1 = [0,1]$ using finitely many points around a circle at $0, \frac{1}{N}, \frac{2}{N}, \dots, \frac{N-1}{N}$. Here ...
2
votes
1answer
2k views

How can I implement the implicit Euler method for a small nonlinear system of ODEs?

I am trying to solve a system of coupled ODEs: $$ \begin{align} \frac{dn_A}{dt} & = e\left[j(t) - f\, θ_H\sinh\left(\frac{g\,n_A}{T}\right)\right] \\ \frac{dθ_H}{dt} & = a\left[bP\,(...
5
votes
2answers
1k views

First order finite volume spatial discretization of the heat equation on an unstructured triangle mesh

Consider a scalar field $u$ on an unstructured triangle mesh which is constant on each face. Let $A_i$ be the area of triangle $T_i$, $N(i)$ the set of triangles sharing an edge with $T_i$, and $L_{...
4
votes
1answer
108 views

Apply for a cluster for scientific computing from a developing country?

I don't have access to a computer cluster in my university. Is there website that accepts applications for free access to a computer cluster for scientific computing? Further information: I am in ...
3
votes
1answer
113 views

Solving a non-linear heat equation with the galerkin method gives negative values

I am trying to solve a non-linear time-dependent heat equation $$\partial_tT=\nabla \left(k_T(T)\nabla T\right) + f$$ using the galerkin method, with neumann boundary conditions. For linearization of ...
52
votes
3answers
33k views

What are the conceptual differences between the finite element and finite volume method?

There is an obvious difference between finite difference and the finite volume method (moving from point definition of the equations to integral averages over cells). But I find FEM and FVM to be very ...
37
votes
4answers
31k views

How does the MATLAB backslash operator solve $Ax=b$ for square matrices?

I was comparing a few of my codes to "stock" MATLAB codes. I am surprised at the results. I ran a sample code (Sparse Matrix) ...
31
votes
4answers
2k views

Why is local conservation important when solving PDEs?

Engineers often insist on using locally conservative methods such as finite volume, conservative finite difference, or discontinuous Galerkin methods for solving PDEs. What can go wrong when using a ...
37
votes
7answers
2k views

What programming paradigms should I be investing in if I want my code to run on petascale machines in the future?

It's pretty clear from a survey of the top500 that the industry is trending towards an exponential increase in processing cores. The largest supercomputers all use MPI for communication between nodes,...
54
votes
17answers
3k views

Is it common not to use libraries for standard numerical algorithms, and why?

A lot of numerical algorithms (integration, differentiation, interpolation, special functions, etc.) are available in scientific computation libraries like GSL. But I often see code with "hand-rolled" ...
17
votes
6answers
41k views

Python vs FORTRAN

Which one is better: FORTRAN or Python? And I guess that in both cases you need Gnuplot, am I right? I'm working on a Windows machine at the moment. I'd like to use it to get numerical solutions for ...
18
votes
1answer
7k views

What is the general idea of Nitsche's method in numerical analysis?

I know that the Nitsche's method is a very attractive methods since it allows to take into account Dirichlet type boundary conditions or contact with friction boundary conditions in a weak way without ...
28
votes
4answers
16k views

Dealing with the inverse of a positive definite symmetric (covariance) matrix?

In statistics and its various applications, we often calculate the covariance matrix, which is positive definite (in the cases considered) and symmetric, for various uses. Sometimes, we need the ...
22
votes
1answer
12k views

Why is Newton's method not converging?

I am using PETSc's nonlinear solver package SNES to solve a system of nonlinear equations obtained by discretizing a partial differential equation. How can I determine why the solver is not converging ...
27
votes
7answers
24k views

What is the fastest way to calculate the largest eigenvalue of a general matrix?

EDIT: I am testing if any eigenvalues have a magnitude of one or greater. I need to find the largest absolute eigenvalue of a large sparse, non-symmetric matrix. I have been using R's ...
27
votes
5answers
4k views

Are there any famous problems/algorithms in scientific computing that cannot be sped up by parallelisation

Are there any famous problems/algorithms in scientific computing that cannot be sped up by parallelisation? It seems to me whilst reading books on CUDA that most things can be.
18
votes
3answers
841 views

Strategies for unit testing and test-driven development

I'm a huge advocate of test-driven development in scientific computing. It's utility in practice is just staggering, and really alleviates the classic troubles that code developers know. However, ...
15
votes
3answers
2k views

multigrid method to solve PDE

I need simple explanation of the Multigrid Method or some literature about this. I am familiar with iterational methods including BiCGStab,CG,GS,Jacobi and preconditioning, but I am a beginner with ...
16
votes
2answers
4k views

Stopping criteria for iterative linear solvers applied to nearly singular systems

Consider $Ax=b$ with $A$ nearly singular which means there is an eigenvalue $\lambda_0$ of $A$ that is very small. The usual stop criterion of an iterative method is based on the residual $r_n:=b-Ax_n$...
8
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 ...
5
votes
3answers
1k views

Books on mathematical foundation of finite element methods

After reading three books about finite element method, with two of them covering also finite volume and grid generation, I found myself lost when I have to discuss these topics with library developers ...
17
votes
8answers
1k views

Is there any open-source or easy-to-access software that can simplify algebraic expressions like $x^{2}+2x+3, x=\sqrt{2}t-1$?

I always calculate things by hand, but now my comrades are getting nasty and making a lot of repetitive exercises involving just plugging things in like the expression above. I am particularly ...
16
votes
1answer
4k views

When is Newton-Krylov not an appropriate solver?

Recently I have been comparing different non-linear solvers from scipy and was particularly impressed with the Newton-Krylov example in the Scipy Cookbook in which they solve a second order ...
15
votes
1answer
5k views

How to derive the Weak Formulation of a Partial Differential Equation for Finite Element Method?

I have taken a basic introduction to Finite Element Method, which did not emphasize a sophisticated understanding of a 'weak formulation'. I understand that with the galerkin method, we multiply both ...
13
votes
1answer
1k views

Can the advection equation with variable velocity be conservative?

I am trying to understand the advection equation with variable velocity coefficient a bit better. In particular I don't understand how the equation can be conservative. The advection equation, $$ \...
11
votes
5answers
2k views

Numerical derivative and finite difference coefficients: any update of the Fornberg method?

When one want to compute numerical derivatives, the method presented by Bengt Fornberg here (and reported here) is very convenient (both precise and simple to implement). As the original paper date ...
10
votes
2answers
394 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 ...
8
votes
3answers
28k views

Difference between Nodes and CPUs when running software on a cluster?

I'm looking into moving some computations of mine to a data center to get more computation power. In the context of this process, I am getting confused by the differentiation of a computation node and ...
11
votes
1answer
16k 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
385 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
667 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 ...

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