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5
votes
1answer
2k views

PDE - Conservative form, conservative methods and discrete conservation

I cannot find a reference explaining clearly and rigorously the links between the notions of conservative form for a PDE, a conservative numerical method and discrete conservation. I would be very ...
5
votes
2answers
470 views

How to parallelize a banded direct solver?

I have a linear system whose matrix that is diagonally dominant, non-symmetric, but banded. Since the band-radius is 2 (producing only 5 variables per equation), a banded direct solver (gaussian ...
5
votes
1answer
1k views

method of frozen coefficients and its relation to von Neumann stability analysis

I am considering two equations $$u_t=a(x)u_{xx}$$ and $$v_t=b(x)v_x$$ as classical representatives of the parabolic and hyperbolic family of equations. If $a(x)=a$ and $b(x)=b$ were constants, to show ...
4
votes
1answer
2k views

Is there a mesh generator that will generate zero thickness elements for interfaces?

I have a geometry and a finite element method where there are a couple internal interfaces across which I want to enforce some fairly simple jump conditions. One recommendation I've gotten has been ...
4
votes
1answer
280 views

adjoint method for reaction-diffusion problem

I'm trying to code a parameter estimation for a reaction-diffusion problem. Namely, knowing the distribution of tumor density $u$ at time $0$ and $T_f$ ($u^0$ and $u^f$), what are the best ...
4
votes
3answers
518 views

Quality Measures for Various Pseudo-Random Number Generators

According to this paper, Ideally, a pseudorandom number generator would produce a stream of numbers that: are uniformly distributed, are uncorrelated, never repeats itself, ...
3
votes
1answer
313 views

Finite Elements Weak Formulation generalization

I am struggling with an equation that represents the Weak form of Galerkin method: $ \phi^{T}F(\textbf{u})\sim \int_{\Omega}^{ } \phi.f_{0}(\mathit{u},\nabla \mathit{u}) + \nabla\phi:f_{1}(\mathit{u},...
3
votes
1answer
964 views

Implementation of gradient zero boundary conditon in advection-diffusion equation

My question is about Finite Element Method. I want to know how to implement "gradient zero" conditions to advection-diffusion equations in conservative form like, $\frac{\partial \rho}{\partial t} + ...
2
votes
1answer
199 views

Imposing boundary conditions for PDE quadratic eigenvalue problem

I have a quadratic eigenvalue problem of the form: $$(A_2 s^2 + A_1 s + A_0)\hat{v} = 0$$ where $s$ is the eigenvalue. The matrices $A_i$ contain derivatives up to order six, and I have six boundary ...
2
votes
1answer
3k views

Introduction to Lattice Boltzmann methods [closed]

I am trying to learn the Lattice-Boltzmann method and was looking for some good beginner resources explaining the method. I have been looking at some codes online, but have been having trouble ...
2
votes
1answer
121 views

Calculation of the EFIE integral

I need help computing the following integral: $$ \int_{}\frac{(1+jk|\vec{r}-\vec{r}^\prime|)e^{-jk|\vec{r}-\vec{r}^\prime|}}{|\vec{r}-\vec{r}^\prime|}d\vec{r}^\prime $$ in this integral $\vec{r}$ ...
2
votes
1answer
148 views

Physical interpretation of divergence theorem

In a diverging pipe section like the following, the pipe of radius $r$ splits into two pipes of radius $r/2$. Consider a solute transported by convection from node 1. $$\frac{\partial C}{\partial t}...
2
votes
0answers
130 views

Heisenberg Model python : Specific heat capacity for spin 2

I have the correct plot for specific heat capacity when I am using the formula which is $C_V$ = differentiation of entropy with respect to temperature. However, When I try to calculate $C_V$, by using ...
1
vote
1answer
260 views

Calculate inverse of dense matrix with entries of very different magnitude

I need to calculate the inverse of a dense matrix, with some elements taking values as high as 1e9 and some around 1e2. What would be the best method to do it? Note: I am more concerned about the ...
1
vote
1answer
372 views

FEM on tet10 element: negetive determinant at the Gauss point

I am trying to implement a fem code on tet10 elements. I follow the lecture notes for tet10 implementation given in http://www.colorado.edu/engineering/CAS/courses.d/AFEM.d/AFEM.Ch10.d/AFEM.Ch10.pdf ...
0
votes
2answers
141 views

Adaptive numerical integration of a univariate vector integrand

Background & Problem formulation I'm trying to write a simple program in C++ that performs adaptive numerical integration of vector valued integrands (in one variable), i.e. $$\int_a^b \bar{f}(...
0
votes
0answers
1k views

Finite differences and Neumann boundary conditions

I am dealing with a highly nonlinear system of two PDEs. I already have a code to solve the system in case of Dirichlet boundary conditions. The explicit system is: $$ \begin{eqnarray*} \partial_{t}u ...
11
votes
1answer
313 views

For software submitted to ACM TOMS, how does the ACM software license agreement interact with other licenses?

The journal Association for Computing Machinery Transactions on Mathematical Software (ACM TOMS) publishes many articles on numerical algorithms that include software implementations. According to ...
8
votes
1answer
1k views

Shortley-Weller finite difference method

can you give me a link for a good and simple explanation of the Shortley-Weller finite-difference scheme? I tried to google it but all I get is (inaccessible) academic publications. I also tried ...
8
votes
2answers
219 views

How should I report profiling/timing information about my code?

I've seen a lot of publications in Computational Physics journals use different metrics for the performance of their code. Especially for GPGPU code, there seems to be a great variety of timing ...
7
votes
3answers
519 views

C - OpenMP, MPI, Serial Program

I'm part of a Computational Science course and come from a non-programming background, so please forgive me my ignorance. I'm working on a set of code in C to numerically solve the Navier Stokes ...
7
votes
1answer
264 views

How to avoid overflow error in program that computes product of two numbers, such that when one is big enough to cause overflow, other is $0$?

Let us say that I have a function like so: def f(x): return g(x)*h(x) Now, g(x) and ...
6
votes
3answers
4k views

How to find QR decomposition of a rectangular matrix in overdetermined linear system solution?

While trying to find cell-centered gradients in finite volume method computation of incompressible fluid flow I get over-determined linear system. This is a well known "cell based least-square" ...
5
votes
2answers
304 views

Approximating and visualizing basins of attraction

I am working on estimating the position and orientation (pose) of a model (rigid object) from its silhouette in an image. For this, I have constructed an error measure between the model in its pose ...
5
votes
3answers
3k views

Discrete Poisson Equation with Pure Neumann Boundary Conditions

I'm trying to implement the Helmholtz-Hodge Decomposition in 2D, which states that a vector field is composed by a rotational free component, a divergence free component and a harmonic component. ...
5
votes
2answers
551 views

Perturbation of Cholesky decomposition for matrix inversion

I am looking for a computationally cheap way to compute $x$ such that $$(L L^T + \mu^2 I)x = y$$ where $L \in \mathbb{R}^{n \times n}$ is a lower triangular definite positive matrix (with some very ...
5
votes
2answers
480 views

What does symmetrize mean? (imposing multifreedom constraints to stiffness matrix)

I have a small FEM implementation program. And I want to add imposing multifreedom constraints (MFC) feature to it. The theory of master-slave method is given here (page 10 for general case). ...
4
votes
1answer
316 views

Interpolating a mathematical function using a Hermite Cubic Finite Element Space

I have a Hermite Cubic Finite Element Space on a computer in the form of Matlab m-files. More specifically, I can evaluate four "shape functions" $N_1, N_2, N_3,$ and $N_4$, for which the following ...
4
votes
3answers
220 views

Is there a minimum angle requirement for cells in the finite volume method?

In his talk "What is a good linear finite element?", Shewchuk states that small dihedral angles in linear tetrahedra elements cause ill-conditioning of the stiffness matrix. Do small dihedral angles ...
4
votes
1answer
516 views

LU Decomposition of PSD Matrix + Diagonal Matrix

If I have a psd, symmetric matrix $\mathbf{A}$ and I need to do LU decomps on $\mathbf{B_i}= \mathbf{A} + \mathbf{D_i}$ (where $\mathbf{D_i}$ is a diagonal psd matrix, where $\mathbf{D_i}$ changes ...
4
votes
1answer
91 views

Linear programming boundedness

Assume the optimal value of a primal problem is bounded. Is the following statement true? If the primal problem is bounded, then its dual problem is bounded as well.
4
votes
1answer
179 views

Fast algorithm for computing cofactor matrix

I wonder if there is a fast algorithm, say ($\mathcal O(n^3)$) for computing the cofactor matrix (or conjugate matrix) of an $N\times N$ square matrix. And yes, one could first compute its determinant ...
4
votes
1answer
2k views

Partial trace algorithm (original)

In general, is there a partial trace algorithm (ideally for systems of any size) that can be coded using basic matrix operations found in software like Mathematica or Maple? All of the methods I'm ...
4
votes
1answer
138 views

Fast and free server for computing

I have to calculate a huge differential equation. With my laptop, it's going to be computed for several days. Is there a free (I need just for 3 days) fast server for scientific calculations? My ...
4
votes
1answer
174 views

Compute hypergeometric function ratio: $\frac{_{2}F_{1}(a+1,b;c;x)}{_{2}F_{1}(a,b;c;x)}$?

I need a numerically stable way to compute the following ratio: $$\frac{_{2}F_{1}(a+1,b;c;x)}{_{2}F_{1}(a,b;c;x)}$$ All the parameters are real numbers, with $a< 0$,$\ $ $b,c > 0$ and $0<x&...
3
votes
2answers
728 views

NP-Completeness

Consider an instance of non-convexoptimization problem: It seems that this problem is NP-complete. How can I find a suitable reduction for this?
3
votes
2answers
386 views

Get equation for a curve which intersects x at seemingly randomly distributed points?

Is there any type of function that when graphed would show a curve which intersects the x axis multiple times, with each point being an arbitrary distance from the last? I mean, not like a trig ...
3
votes
1answer
251 views

non-smooth convex c++ solver

I happened to know that there are advanced established techniques for non-smooth convex optimization in research. For example, these two papers: Nesterov, "Smooth minimization of non-smooth functions"...
3
votes
1answer
212 views

Finding zeroes of an infinitely differentiable function of ~100 to ~1000 variables

I have a function that is not only infinitely differentiable, but it is also very cheap to calculate any of those derivatives. It looks like: $f(\boldsymbol{C}, \boldsymbol{x})=\sum_{i} C_{i} \prod_{...
3
votes
1answer
168 views

How many operations are needed for LAPACK's zgesv to solve a linear system?

I have a linear system of complex numbers. I am using LAPACK' zgesv (actually I am using intel MKL LAPACKE, but I am assuming the algorithm is the same). No assumption can be made about the system. I ...
3
votes
0answers
206 views

logsumexp with one very large term and many very small terms

I want to compute an expression of the form: $$L = \ln\sum_i e^{x_i}$$ Suppose that there are many small terms, say $e^{x_i} \approx \epsilon$. If there are $N_\epsilon$ such terms, their ...
3
votes
2answers
503 views

DST using FFT routine

Please can you help me with my problem? On Wikipedia, in article Discrete sine transform, this is written (chapter Computation): "Although the direct application of these formulas would require O(N2) ...
3
votes
1answer
5k views

Why a finite difference scheme would give second order of accuracy in norm L2 but 1.5 with L1 (while 1 with Linf)?

My finite difference scheme for the 2D Euler equations is second order accurate in theory, since all the terms are second order accurate, with the advective terms being third order. So I expect a rate ...
2
votes
1answer
573 views

Crank-Nicolson algorithm for coupled PDEs

Assumed I have the following two coupled equations $$\begin{split} \partial_tA&=a_0AB\\ \partial_tB&=b_0AB \end{split} $$ but I am not sure how to calculate them. One approach is a crank-...
2
votes
1answer
89 views

MATLAB's ode45 not dealing with initial conditions well [RESOLVED]

*Concern highlighted in yellow *Solution at bottom I have a differential equation to solve for the motion of an electron: $$ \frac{d^2v}{dt^2} = \frac{1}{\gamma^6}\left( \frac{eE}{\tau m} - \left( \...
2
votes
2answers
348 views

In practice, what are the most useful ways to visualize 2d fluid flow, to tell what is happening in the simulation? Esp for verification and debugging

My simulation creates a 2d grid of vectors and scalars (EDIT of velocity, depth etc), at 60 frames per second. Is it correct? It looks sort of right... but who knows? How can I tell what's happening - ...
2
votes
1answer
2k views

4th order Runge-Kutta Method for Driven Damped Pendulum

Although I've been looking everywhere, I have been unable to find an answer to my question so here it is. For a driven damped pendulum the equation of motion in dimensionless units is, $$\alpha(\...
1
vote
1answer
92 views

Integrating a function and then plotting it's graph in Matlab

I've been working on this code for some hours and it seems I'm doing something wrong which I just can't figure out. I have a function which I integrate and then plot it's values. But the graph I get ...
1
vote
1answer
1k views

Paraview: Multiple csv files to time series

I have multiple CSV files with point-coordinates and, possibly, multiple data values attached to them, i.e., the rows look like x y z data1 data2 ... Is there ...
1
vote
1answer
891 views

How can I call the Boost C++ odeint Runge-Kutta integrator for a system of ODEs?

I would like to use Boost C++ odeint Runge-Kutta integrator on a system that looks like this : $$\ddot x = - \frac A{||x||^3} * x $$ $ x $ is a vector in 3D space, so basicaly $ x(i, j, k) $ $ \...

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