All Questions

Suppose we have an element in the polar coordinate as $0\leq r\leq 1$, $0\leq\theta\leq \alpha$. What are the correct basis functions in this element? For this kind of mesh, there are a lot of ...

Suppose we use a preconditioned iterative solver for a linear system. If the initial state for the solver can be chosen very close to the exact solution - does this reduce requirements for the ...

Let $p$ be a polynomial with floating-point coefficients and let $a$ be a floating-point value. Is there a method for accurately evaluating the sign of $p(a)$ in floating-point arithmetic? I don't ...

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 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 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'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 ...

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 ...

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 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 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 $\vec{r}$ ...

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 ...

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 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 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 ...

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 ...

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 ...

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 ...

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'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" ...

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 ...

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'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 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 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 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 ...

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 ...

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 ...

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&...

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 ...

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 ...

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 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 ...

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"...

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 ...

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) ...

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

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 ...

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 ...

*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( \...

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-...

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 am trying to obtain the weak form of the Navier-Cauchy equation, which is $$- \rho \omega ^2 \textbf{U} - \mu \nabla ^2 \textbf{U} - (\mu + \lambda) \nabla (\nabla \cdot \textbf{U}) = \textbf{F}$$ ...