Questions tagged [numerics]

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99 views

Remez algorithm convergence

I have implemented the Remez algorithm in Python where all calculations were done with the Python mpmath library. I have noticed that sometimes the $|E_{max}|$ and $|E_{min}|$ do not monotonically ...
2
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1answer
175 views

Finite element (1D) for steady state non-linear problem

I need to solve with linear finite elements the equation $$\frac{\partial }{\partial x}\Bigl(\text{sgn}(x) u \Big) +\frac{\partial}{\partial x} \Bigl[ \sqrt{u} \frac{\partial u}{\partial x} \Bigr] =0$...
1
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1answer
123 views

Non-Linear advection diffusion with nondifferetiable advection term

I'm looking at Murray's book: Mathematical biology: an introduction , first volume, pag. 404 In particular, I'm interested to solve the following PDE: $$\partial_t u = \partial_x (\text{sign}(x) u) + \...
2
votes
1answer
168 views

Ill-condioned Linear System and Gaussian Elimination

Suppose that I have a linear system $Ax=b$ such that $A$ is ill-conditioned. Can I say that it is dangerous to find a solution with Gaussian Elimination for this system, or does there exist some class ...
1
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0answers
81 views

How can I practice multivariable root-finding?

Recently, I've been reading up on various root-finding / optimization algorithms such as the Levenberg-Marquardt method, Gauss-Newton, Conjugate Gradient, trust-region and trust-region-dogleg. I've ...
1
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2answers
122 views

Imposing pressure variation instead of Dirichlet boundary conditions on Finite Element Method

I always see Finite Element codes solving PDE with Dirichlet or Neumann boundary conditions. But, I have a problem now consisting of a straight cylinder with a circular base (a simple 3D tube), with ...
0
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0answers
113 views

Solution of Cahn-Hilliard equation

I need to solve the Cahn-Hilliard equation $$\frac{\partial u}{\partial t} = \Delta(f(u) - \epsilon^2\Delta u), \hspace{.5cm}(x, t)\in \Omega\times(0, T],$$ using mixed formulation \begin{equation}\...
0
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0answers
47 views

Norm estimates if adjoints can't be computed

Assume that you have two linear maps $A$ and $V$. For a given $x$ (of appropriate dimension) you can compute $Ax$ numerically, and for any $y$ (of appropriate dimension) you can calculate $V^Ty$ ...
5
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1answer
121 views

Accurate and efficient computation of the logarithm of the ratio of two sines

For exploratory work related to special function implementations, I need to compute $\log \frac{\sin y}{\sin x} $, where $0 \le x \le y \le 2x < \frac{\pi}{2}$. Cases with $x \approx y$ in ...
2
votes
1answer
66 views

Parity for artificial dissipation term in a finite-difference solution

I have a doubt regarding the signal of the dissipative term in a finite difference solution for an equation of the form $$ \frac{\partial u}{\partial x}+f(x)=0, u(0)=0 $$ In which $f$ is an odd ...
0
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0answers
58 views

numerical solution to pde on an ellipse

Looking for advice on discretization (preferably finite difference) schemes for pdes on curves in general, but in this case it is an ellipse (so given by $(a\cos(r), b\sin(r)$). The problem is the ...
2
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1answer
81 views

Find quadrature points and weights

I'm struggling with the following problem: What is the maximum degree of exactness that we can obtain with the following quadrature >formula $$\int_0^1 f(x)\frac{1}{\sqrt{x}}dx \approx w_0 f(x_0) +...
0
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1answer
120 views

Interpreting multivariable root-finding results from Matlab's fsolve algorithm

Edit: So I was able to get the same value of r that's given, when coding up the sum of squares of function values directly in the script file, rather than on the Command Window. So, maybe there's a ...
0
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2answers
364 views

No flux Neumann boundary condition for non-stationary PDE equivalent to Dirichlet boundary?

When using no flux Neumann boundary conditions (i.e. zero derivative to the normal on the boundary) in a non-stationary PDE, I don't seem to recognize the difference to using Dirichlet boundary ...
3
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1answer
257 views

How to use numerical integration to calculate the surface area of a superellipsoid?

I am working in an application in which I need to calculate the surface area of a superellipsoid. I have read that there is no closed form solution (see here), so I am trying to compute it using ...
0
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2answers
68 views

Grid Independence Study

Is the change in time step necessary for the grid independent study? As the CFL is based on the relation between dt and dx. In mesh independent study, only change should be mesh i.e, dx isn't it so?
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0answers
28 views

Set of integrators do not consistently solve an equation in Python

I must solve the following second order differential equation: $\delta \phi^{''}_{\mathbf{k}}+(3-\epsilon)\delta \phi^{'}_{\mathbf{k}}+\left(\frac{k^2}{a^2 H^2}+\frac{V_{,\phi\phi}}{H^2}-6\epsilon +4\...
0
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1answer
106 views

Extracting data from VTK simulations using C++

I have been given a few numerical simulations regarding fluid mixing and have been asked to extract a few parameters from them using C++. Altogether there are about 1000 VTK files per simulation, and ...
4
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1answer
224 views

Imposing no flux boundary conditions on variants of the Cahn-Hilliard equation using Finite Differences in Python

I have been looking into simulations of phase separation in variants of the Cahn-Hilliard system and have been running into issues with implementing no flux boundary conditions on certain variants. ...
0
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0answers
62 views

The error in SOR algorithm suddenly falls to zero when it reaches 1e-7 range

I am solving the Poisson equation for heterojunction using Fortran90. I use the SOR algorithm to arrive at the potential profile. I see the weird behavior where the error (the difference between the $...
0
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0answers
32 views

Abnormalities when using SOR to solve the Poisson Equation

I am trying to solve the Poisson equation for Heterostructures using SOR. The equation to solve looks lik I have discretized the Poisson equation using finite difference and my code is written in ...
2
votes
1answer
94 views

Computing Series of $ke^{-(x - h)^2}$

I asked this question on the Computer Science stack exchange (https://cs.stackexchange.com/questions/128710/faster-computation-of-ke-x-h2), but it appears that it is more appropriate in Computational ...
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0answers
39 views

Discretization formula for a system of two differential equations. "Solution to one of these is the initial condition of the other". In which sense?

Consider the following stochastic differential equation \begin{equation} dy=\left(A-\left(A+B\right)y\right)dt+C\sqrt{y\left(1-y\right)}dW\tag{1} \end{equation} where $A$, $B$ and $C$ are parameters ...
0
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1answer
107 views

interface value on the error equation

https://www.jstor.org/stable/pdf/2157482.pdf, here I have a problem in last equation of (2.6) in section (2.1). When they are considering error equation on the interface $\Gamma$ they get $e_v^{(n)} = ...
0
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1answer
115 views

Which scheme for inhomogeneous convection-diffusion equation with highly variable coefficients?

I have a 1D convection-diffusion equation $\sigma_t = a(x,t) \sigma_{xx}+b(x,t)\sigma_x+f(x,t)$ defined on the unit interval, with nonzero Neumann boundary conditions at both ends. It should be noted ...
2
votes
1answer
116 views

Calculate stable time step DG method for advection-diffusion

For stable time steps for the RKDG method for transport equations we require that $$ \Delta t \le \frac{\Delta x CFL}{(2k + 1)|\lambda|}, $$ where $\lambda$ is the eigenvalue of our conservation law ...
1
vote
1answer
108 views

Linear system with an l1-norm constraint

I have a saddle-point system of the form \begin{equation} \begin{bmatrix} A & B \\ B^T & O \end{bmatrix}\begin{bmatrix} x\\ y \end{bmatrix} = \begin{bmatrix} f \\ \vec{0} \end{bmatrix}, \end{...
1
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0answers
43 views

Euler Explict Method for solving the PDE for option prices under the Schwartz mean reverting model. Numerical Finance

I have to solve the following PDE for a Call option : $\partial_tV + \{ \alpha - (\mu - \lambda/ \alpha -log(S))\}S\partial_SV + 1/2 \sigma^{2}S^{2}\partial_{S}^{2}V - rV = 0$ Where V(S,t) is the ...
2
votes
2answers
145 views

Accelerating convergence of a generalized continued fraction

I wish to compute $$ \frac{1}{1 + \frac{1^3}{1 + \frac{2^3}{1 + \frac{3^3}{1+\cdots} } } } $$ to high accuracy. To start, I tried computing $$ \frac{1}{1 + \frac{1^2}{1 + \frac{2^2}{1 + \frac{3^2}{1+\...
1
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0answers
110 views

Is Romberg integration method implemented as weighted function values numerically correct?

I have to integrate expression f(x) * g(x) for many different functions f but just one g. I ...
5
votes
1answer
153 views

Accurate computation of Gauss-Kuzmin entropy

The Gauss-Kuzmin distribution gives the probability of an integer appearing as a partial denominator in the continued fraction of a real number $x$ as $$ P(a_k = k) = -\log_2\left(1 - \frac{1}{(k+1)^2}...
3
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0answers
64 views

Numerical calculation of the Berry connection

I'm doing some numerical calculations involving Hermitian matrices, and derivatives of the eigenvectors. Essentially, I have an n x n, Hermitian matrix H(x), which is dependent on some continuous ...
5
votes
1answer
113 views

Accurately Computing a Positive Vector in the Nullspace of a Matrix

I'm sure this question has been asked before yet after many hours of searching I am unable to find a definitive answer. The problem at hand is solving the linear system: $$A \mathbf{x} = \mathbf{0}$$ ...
0
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1answer
146 views

Red flags for numerical computing?

I've learnt the hard way that you should avoid: computing small numbers as the difference of two large numbers evaluating chaotic functions with imprecise inputs. Are there any other red flags a ...
0
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0answers
30 views

At what l/d ratio will a frame element start to behave as a shell element?

I'm working in ETABS. There are few columns of dimension 300mmx1400mm. The height of building is 36.6 meter above ground level and the dimension of building is 26mx68m. I'm getting the time period of ...
0
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0answers
145 views

A parallelized GMRES solver?

My application calls for solving a dense, 40,000 x 40,000, ill-conditioned linear system. The native SciPy GMRES solver with preconditioning has worked well for my application and solving a single ...
3
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0answers
81 views

How to construct a Fortin Operator for Crouzeix-Raviart Element?

I want to prove the LBB condition for the Stokes Equations discretised by the Crouzeix-Raviart element. The continuous Stokes Equation in the weak formulation is Find $u \in H_0^1(\Omega, \mathbb{R}^...
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0answers
36 views

Simulating the response of nonlinear system with stiff differential equations

I want to simulate the response of a nonlinear system given in the following form: $$ \dot{x_1} = f_1(\bar{x_1})+g_1(\bar{x_1})x_2, \ x_1(0) = 0.2 $$ $$ \dot{x_2} = f_2(\bar{x_2})+g_2(\bar{x_2})x_3, \...
-1
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1answer
102 views

Numerical solution for gradient(slope)

Abstract I have the next equation to find a force, for my problem: $$U=-\int \vec{m}\small{(x)}\times \vec{B}(x)dV$$ $$\vec{F}=-\nabla U$$ Considering 3-dimensional space with x,y,z coordinates, ...
20
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4answers
4k views

Is half precision supported by modern architecture?

I am new to computer science and I was wondering whether half precision is supported by modern architecture in the same way as single or double precision is. I thought the 2008 revision of IEEE-754 ...
1
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0answers
77 views

Plot of ratio of two integrals:

Consider the following integrals $$ I_1(x) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy) − F(x −\mathrm iy)}{\mathrm e^{2πy}-1}, $$ And $$I_2(x) =\int_1^x F(t)dt$$ Where, $ F(z) = \sin^2[π\Gamma(z)/...
1
vote
3answers
156 views

Runge-Kutta method for an ODE with initial value which is root of denominator

I wrote a code in Fortran to solve this differential equation using RK4 method: $$ \frac{dy}{dx}=A\sqrt{\frac{B}{y}+\frac{C}{y^2}} $$ $A$, $B$, and $C$ are some known constants. The problem is that ...
5
votes
0answers
129 views

How to numerically evaluate this double Integral?

I want to evaluate the following integral: $$\int_{0}^{60} \ \left(\int_{0}^{2z} 0.5\cdot t \left(\mathrm{erf}(t-a) -1 \right)J_{0}(qt)\mathrm{d}t \right)^2 \mathrm{exp}\left(-\frac{(z-a)^2}{2s^2}\...
2
votes
3answers
268 views

Comparison of integrals with a function:

Consider the following integral: $$S(q)=\int_{x=2}^q\sin^2\left(\frac{π\Gamma(x)}{2x}\right)dx$$ And consider the functions : $$R(q)=\frac{q}{\log(q)}$$ $$T(q)=\int_2^q\frac{1}{\log(x)}dx$$ I ...
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0answers
61 views

Difficulty to prove the coercivity of $a(u, v)=\int_{\Omega} \nabla u \cdot \nabla v$

I'm stuck at proving the coercivity of the bilinear functional in the variational formulation of the problem: \begin{array}{c} -\nabla^{2} u=f \quad \text { in } \Omega \\ u=g_{D} \text { on } \...
1
vote
1answer
114 views

Why is my Cahn-Hilliard simulation separating out so finely?

I am trying to simulate the Cahn-Hilliard equation using Python, but the 2 fluids aren't separating into big blobs, as desired, under any conditions. I'm setting up (what I think is) an orthogonal ...
1
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0answers
41 views

Advice for a topic in a seminar

I am a master student in Mathematics, and I have to prepare a seminar for a course in mathematical methods for applied sciences. I have a good background in numerical analysis for ODEs, PDEs and hence ...
0
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3answers
89 views

Changing variables in integral to avoid infinity

I want to write a code in Fortran to solve this integral numerically: $$\int_{1095}^\infty \frac{dx}{x\sqrt{(x+644.153)(4.17 \cdot 10^{-5} x+0.145)}}$$ What is the best method for it? I tried Monte-...
5
votes
1answer
275 views

What kind of a researcher am I?

So far, I've worked a bit in modeling, simulations and simple lab experiments, and I've really enjoyed all three research methods to approach a single research question. I can write tricky (in terms ...
1
vote
0answers
26 views

Unable to achieve semi-linear running time in computation of continuant

I am trying to compute the continuant of a list of numbers $a_0, a_1,...,a_n$, defined by the recursion relation: $K_{n+1} = a_{n+1} K_n + K_{n-1}$ and $K_0 = 1$ (see Wikipedia). I am trying to use ...

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