All Questions

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 ...

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 ...

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 ...

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 ...

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 ...

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

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},...

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} + ...

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 ...

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 ...

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 $\vec{r}$ ...

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}...

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 ...

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 ...

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 ...

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}(...

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 ...

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

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 ...

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 ...

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 ...

Let us say that I have a function like so: def f(x): return g(x)*h(x) Now, g(x) and ...

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" ...

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 ...

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. ...

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 ...

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). ...

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 ...

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 ...

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 ...

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.

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 ...

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 ...

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 ...

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&...

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

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 ...

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"...

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_{...

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 ...

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 ...

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) ...

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 ...

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-...

*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( \...

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 - ...

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(\...

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 ...

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 ...

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) $ $ \...