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Python - calculation time derivative and laplacien by finite differences

I would like to determine a temporal derivative and a Laplacian by the finite differences method to solve the heat equation in a 1-dimensional case. My aim is to get the sources term that is why I ...
5
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0answers
162 views

Elliptical problem with Robin BC

Working in finite differences, I am using a transformation on the temperature variable $\Theta = \int_{T0}^T \kappa(T)dT$ to linearize the steady-state heat equation into a Poisson equation $-\...
5
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0answers
87 views

Interface Formulation at Finite Volume Boundaries when using the Dual Mesh

When using the dual mesh (vertex-centered) for finite volume methods, you end up with a cell center at the boundaries between materials. It is possible that the equations being solved in each ...
5
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0answers
150 views

Methods for integrating black box functions on a non-uniform grid

If i have some function expressed as points on a non-uniform grid (I'm specifically interested in logarithmic grids, but general results are also interesting), and I want to integrate it, I believe ...
5
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0answers
465 views

Finite volume method

I have question connected with finite volume method. Consider equation $$\frac{\partial u}{\partial t}=\operatorname{div}A\nabla u +f . \quad x\in \overline{B}_{1} (0)\subset \mathbb{R}^3 -\text{unit ...
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0answers
205 views

Conjugate residual/gradient convergence checking in practice

Let's say we want to solve $Ax=b$ ($A$ symmetric positive /semi/definite) with the conjugate residual/gradient method. $A$ comes from FEM where the mesh is being refined. The exact solution is $x_*$ ...
5
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0answers
87 views

How do I perform chebyshev interpolation from a to b with custom angle range?

Typically Chebyshev interpolation from $-1$ to $1$ with angle from $0$ to $\pi$: $\xi_j=\cos \left ({\pi j \over N}\right )$ $x_j=(1+\xi_j) * {L \over 2}$ $w$: $w_0=\pi/(2N)$ $w_{1,...,N-1}=\pi/(N)$...
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0answers
84 views

Computing linear combinations of sines and cosines (phasors)

I have a finite series that looks like this: $f(t) = \sum^n_{i=0} A_i cos(\Theta_i + \omega_i t) + B_i sin(\Theta_i + \omega_i t)$ That is, a finite series of pairs of phasors. What's the state of ...
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0answers
242 views

Sound Waves Simulation in 3D Environment

I want to do a simulation of sound waves including wave propagation, absorption, and reflection in 3D space. I did some research and I found this question in stackoverflow but it talks about ...
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701 views

Finding roots of systems of equations with a Jacobian that is singular everywhere

Given $a\in\mathbb{R}^{mn\times n}$, find a $C\in\mathbb{R}^{n}$, $x\in\mathbb{R}^{m\times n}$ such that $$ 0 = f_{k}(\boldsymbol{C}, \boldsymbol{x}):=\sum_{i=1}^{m} C_{i} \left(\prod_{j=1}^{n} a_{kj}^...
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239 views

Negative viscosity stabilized by fourth order terms

I am trying to solve a "Navier-Stokes"-type problem where the viscosity is negative. Of course this renders the equation unstable and thus I add a fourth order term, so the entire equation becomes: $$...
5
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0answers
115 views

What are the governing equations solved in coupled atmosphere-ocean models?

In (hydrostatic) atmospheric general circulation models, for example the so-calle Primitive Equations, consisting of the horizontal momentum equation, the hydrostatic balance, the continuity equation, ...
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188 views

Time-stepping for coupled nonlinear PDEs

What are good references for time-stepping of the coupled incompressible Navier-Stokes-heat equation (Boussinesq flow), $$ \begin{cases} \rho\left(\dot{\mathbf{u}} + \mathbf{u}\cdot\nabla \mathbf{u}\...
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347 views

Understanding the meaning of Computational Order of Convergence

I am a postgraduate student with interest in numerical methods for solving nonlinear systems of equations. I have read some papers that discussed about 'computational order of convergence' for some ...
5
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0answers
1k views

Using MINPACK for curve fitting: implementation?

I need to implement a non-linear fitting algorithm in Fortran and chose to use MINPACK's flavor of the Levenberg-Marquardt algorithm as a basis for the least-squares stuff. However, I seem to ...
5
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0answers
79 views

Order of convergence of Scrodinger eq. with CN scheme

I'm trying to solve numerically the 1-dim time dependent Schrodinger equation using the Crank Nicolson scheme and the Thomas algorithm to solve the tridiagonal matrix. The physical system consists of ...
5
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0answers
694 views

Generating pseudo-random orthonormal bases for random projection

I am performing series of random projections i.e. projecting the input matrix onto randomly generated orthonormal bases (of much lower dimensionality). The projection is just a matrix multiplication ...
5
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0answers
84 views

Non-convergance when calculating temperature/heat flows through a section of rock

I am attempting to calculate temperature of section of rock in the earth as a function of vertical position in the rock and time. Along with it I am calculating the heat flow through the rock as a ...
5
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0answers
602 views

frozen coefficient vs. constant coefficient

This is a follow up to the question about the method of frozen coefficients. I think it deserves to be a separate question. The frozen coefficient problems are obtained by fixing the coefficients of ...
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195 views

Finding quadrature weights for a given set of points? How to select points such that all weights are positive?

Currently, I fit a Finite Element solution of a PDE on a spectral basis. The matrices ($R^{25000\times 2000}$) of the corresponding system of linear equations are highly ill-conditioned ($\kappa \...
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0answers
1k views

How does GAMG in OpenFOAM really work?

I use OpenFOAM for CFD simulations. A very popular preconditioner is GAMG which needs a low number of iterations per a time step in SIMPLE or PISO solvers that are used to simulate the fluid flow. I ...
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0answers
202 views

Discontinuous Galerkin for flow through porous media

I am struggling with DG methods for 2 phase flow through porous media. I managed to get the global pressure, total flux equations to work with an unconditionally stable mixed FE DG formulation as ...
5
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0answers
224 views

Adaptive mesh data structure for Fast Marching Method to overcome RAM limit

On an uniform mesh of positions in space $\ (x_i,y_j,z_k)$: $$\ x_i = x_0 + i\Delta x,\quad i=0,\ldots,n_x$$ $$\ y_j = y_0 + j\Delta y,\quad j=0,\ldots,n_y$$ $$\ z_k = z_0 + k\Delta z,\quad k=0,\...
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172 views

Solving diffusion PDE using finite differences

I need some hints on how to solve this diffusion equation ($\alpha, k_1,k_2$ and $k_3$ are constants): $$ {\partial P \over \partial y} + k_1 {\partial P \over \partial t} + \alpha P = {1 \over k_2} ...
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0answers
118 views

How to choose a stable PML for pseudo-spectral method with strongly varying velocity

My friend was working on this, and he asked me about the stability of PML while applying on pseudo-spectral method, I believe his concern was how to choose the difference(if the difference should be ...
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0answers
700 views

Perron-Frobenius theorem on general real symmetric matrices

From the Perron-Frobenius theorem, it might be concluded that the spectral radius is the largest eigenvalue for positive matrices, ie, matrices with strictly positive entries. In other words, the ...
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101 views

Time-stable spectral decomposition algorithm

Consider an $n \times n$ real, time-dependent matrix $A(t)$ such that $A(t) = A(t)^{T} > 0$ and $A(t)$ is continuous on $[a,b]$. Then it is posible to specify a matrix $S(t) \in SO(n)$ such that $S(...
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432 views

Why is my lower convex hull extraction algorithm not working?

Recently, I wrote an algorithm to obtain a delaunay triangulation of a random point set in $I=[-10,10]$x$[-10,10] \subset R^2$ by projecting these points onto the 3 dimensional paraboloid $z=x^2+y^2$, ...
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0answers
118 views

Wavelets frame for $L^2[0,\infty)$

I need a wavelet frame for $L^2[0,\infty)$. Moreover, the wavelet should be twice differentiable and with continuous second order derivatives. Hopefully, the wavelet should have compact support (...
5
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0answers
2k views

Two-chordless cycle extraction from a failed comparability graph recognition

I have implemented a comparability graph recognition algorithm from M.C. Golumbic's "Algorithmic graph theory and perfect graphs" book. It is hinted in Fekete, Schepers, and van der Veen's "Exact ...
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135 views

Probabilistic algorithms for matrix approximation

Considering regular matrix approximation inequality || $A - QQ^TA $|| < e where we try to approximate matrix $A$ by a lower rank orthonormal matrix $Q$. I've read an article on probabilistic ...
5
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0answers
259 views

Convergence rate of Monte-Carlo variance estimate

What is the convergence rate for Monte-Carlo variance estimate for a random variable $X \in {L^q}(\Omega ,R),2 < q < 4$?
5
votes
1answer
556 views

Finite volume a posteriori error estimation

I'm wondering what alternatives there are to a grid convergence study to judge solution accuracy for a given grid resolution when doing steady-state RANS simulations on an automatically generated ...
5
votes
1answer
243 views

Robust smoothers for geometric multigrid

I'm searching for robust smoothers for geometric multigrids. By robust I mean: Effective for high order approximations (say spectral element, spectral Discontinuous Galerkin), Parallel (suitable for ...
5
votes
1answer
144 views

complexity constants in median computations same as that of general quantiles?

I would like to know whether the constant in the time complexity of computing the median is different from that of computing general quantiles. In R for example: ...
4
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0answers
119 views

Do computational scientists think about the statistical aspects of using least squares?

Correct me if I'm wrong, but when least squares is used in the computational science community, it's typically not in the context of regression. It could be used to solve for gradients in discretized ...
4
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0answers
68 views

Stability of Crank-Nicholson for advection diffusion equation for spatial discretization other than finite differences second-order centered

Crank Nicholson is a time discretization method (see 4th equation here). From what I see around, you can use different space discretization, such as Finite elements. But for the linear advection-...
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0answers
52 views

Continuous limit and nonlinear functional analysis

I have a kind of general question about approximation schemes in nonlinear functional analysis. Given a nonlinear map $\Phi$ from an open set (in an infinite dimensional Banach space) of functions to ...
4
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0answers
56 views

Efficient computation of marginalized multivariate normal likelihood

In general,if we know that the marginal Gaussian distribution for some variable $\textbf{x}$ and a conditional Gaussian distribution for some $\textbf{y}|\textbf{x}$ of the forms: $$p(\textbf{x}) = \...
4
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0answers
64 views

Optimize linear equation using inner products and subject to L1 norm

I have a linear system of the form $A x = b$ where $A$ and $b$ are known, $A$ is "square", and $\lvert b \rvert_1 = \lvert x \rvert_1 = 1$. Unfortunately, I am working in a framework that ...
4
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0answers
228 views

Fastest matrix library for Android (with GPU is possible)

I was working on an Android app that requires some linear algebra with matrices. The matrices will be somewhat medium-sized as they are not too small or too big. I was originally using jBlas because ...
4
votes
1answer
185 views

Scipy Spline Interpolation Parameter

Documentation in scipy.interpolate (found at https://docs.scipy.org/doc/scipy/reference/tutorial/interpolate.html) states: "The parameter variable is given with the keyword argument, u, which ...
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0answers
110 views

Integrators for Nonlinear/Stiff PDE

It was suggested I ask this question in this section. Anyway: I have a particular nonlinear PDE of the form $$ u_t(x,t)=iu_{xx}(x,t)+f(x,u(x,t)) \tag{1} $$ Where f is some nonlinear function. With ...
4
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0answers
53 views

Is it Grid/Cluster/Cloud Computing? How are those terms defined?

There are three very connected and widely used terms: Grid and grid computing Cluster and cluster computing Cloud and cloud computing In many situations, it is not obvious which term to use, as I ...
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0answers
72 views

Fast approximate solver for vehicle routing problem

I need to solve capacitated asymmetrical vehicle routing problem with time windows on ~30k points. Time limits for calculations are 2 hours. I've tried using Clarke and Wright savings algorithm, it is ...
4
votes
0answers
171 views

MATLAB: Compute the Schwarz-Christoffel transformation symbolically

Suppose we have a conformal mapping from the unit disk in the $\omega$ plane onto the exterior of a polygon in the $z$ plane. The Schwarz-Christoffel mapping in this case is defined as: $$f(u) = A - ...
4
votes
0answers
83 views

Block matrix and DSYRK

I want to compute the matrix $$ A = \sum_{i=1}^N v_i v_i^T $$ where each $v_i$ is a given vector of length $2500$, so that $A$ is $2500 \times 2500$, and my $N$ is about 2 million. Rather than call ...
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0answers
213 views

Does quantum espresso and VASP use same same self-consistent-field procedure?

The codes Quantum Espresso (QE) and Vienna Ab initio Simulation Package (VASP) both use plane wave basis sets and psuedopotentials. Most of the codes in both implementation of DFT uses Fortran code. ...
4
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0answers
73 views

Library for solving multidimensional (n > 3) hyperbolic PDE systems

Does there exist a library (in any programming language) for solving (numerically) systems of multidimensional first-order linear PDEs in the form $$\mathbf{u}_{t}+\hat{A}(\mathbf{x})\mathbf{u}_{\...
4
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0answers
66 views

Given a list of intervals, find region that is contained by the largest number of those intervals

Start with 1d case. Say I have lots of 1d intervals $[s_i, e_i]$ and I want to find an interval $[s^*, e^*]$ to maximise the count of interval $i$ such that $[s_i, e_i]\supseteq [s^*, e^*]$. 1d case ...

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