Questions tagged [numerics]
The numerics tag has no usage guidance.
802
questions
2
votes
1answer
67 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 ...
0
votes
0answers
29 views
Numerical simulation for a bounded process. Is slight deviation a “normal” fact?
Suppose I have to numerically simulate a process $\{y_t\}$ such that $y_t\geq0$ $\forall t\in\mathbb{N}$, with $t$ denoting time-step.
Let's suppose I use MonteCarlo with $\mathscr{N}$ simulation ...
1
vote
0answers
29 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
votes
1answer
60 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
votes
1answer
82 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
86 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
49 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
vote
0answers
33 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
1answer
55 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
$$
\frac{1}{1 + \frac{1^2}{1 + \frac{2^2}{1 + \frac{3^2}{1+\...
1
vote
0answers
78 views
Is Romberg integration method implemented as weighted function values numerically correct?
I have to integrate expression f(x) * g(x) for many different functions f but just one g.
I ...
5
votes
1answer
138 views
Accurate computation of Gauss-Kuzmin entropy
The Gauss-Kuzmin distribution gives the probability of an integer appearing as a partial denominator in the continued fraction of a real number $x$ as
$$
P(a_k = k) = -\log_2\left(1 - \frac{1}{(k+1)^2}...
0
votes
0answers
22 views
Error analysis of Modified Lentz's method
In Numerical Recipes, the authors state:
There is at present no rigorous analysis of error propagation in Lentzās algorithm.
This statement is now ~15 years old, so I wonder has this gap in the ...
3
votes
0answers
36 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
88 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
votes
1answer
127 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
votes
0answers
23 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
votes
0answers
74 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 ...
0
votes
0answers
60 views
Change one inlet boundary condition
This is a problem in modeling in hydraulic fracturing field. It's quite long so hopefully someone can patiently read and help me.
The equation numbers are match those of the reference paper by ...
3
votes
0answers
55 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}^...
0
votes
0answers
26 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:
$$ \dot{x_1} = f_1(\bar{x_1})+g_1(\bar{x_1})x_2, \ x_1(0) = 0.2 $$
$$ \dot{x_2} = f_2(\bar{x_2})+g_2(\bar{x_2})x_3, \...
-1
votes
1answer
95 views
Numerical solution for gradient(slope)
Abstract
I have the next equation to find a force, for my problem:
$$U=-\int \vec{m}\small{(x)}\times \vec{B}(x)dV$$
$$\vec{F}=-\nabla U$$
Considering 3-dimensional space with x,y,z coordinates, ...
19
votes
4answers
3k views
Is half precision supported by modern architecture?
I am new to computer science and I was wondering whether half precision is supported by modern architecture in the same way as single or double precision is. I thought the 2008 revision of IEEE-754 ...
1
vote
0answers
73 views
Plot of ratio of two integrals:
Consider the following integrals
$$
I_1(x) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy) ā F(x ā\mathrm iy)}{\mathrm e^{2Ļy}-1},
$$
And
$$I_2(x) =\int_1^x F(t)dt$$
Where, $ F(z) = \sin^2[Ļ\Gamma(z)/...
1
vote
3answers
125 views
Runge-Kutta method for an ODE with initial value which is root of denominator
I wrote a code in Fortran to solve this differential equation using RK4 method:
$$
\frac{dy}{dx}=A\sqrt{\frac{B}{y}+\frac{C}{y^2}}
$$
$A$, $B$, and $C$ are some known constants. The problem is that ...
5
votes
0answers
99 views
How to numerically evaluate this double Integral?
I want to evaluate the following integral:
$$\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
241 views
Comparison of integrals with a function:
Consider the following integral:
$$S(q)=\int_{x=2}^q\sin^2\left(\frac{Ļ\Gamma(x)}{2x}\right)dx$$
And consider the functions :
$$R(q)=\frac{q}{\log(q)}$$
$$T(q)=\int_2^q\frac{1}{\log(x)}dx$$
I ...
0
votes
0answers
56 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 } \...
0
votes
0answers
29 views
Implementation of nonlinear optimization for Generalized Nash-Equilibrium
I have to find a solver for $\begin{equation}
\min_{x^{\nu}} \Theta_{\nu}(x^{\nu},x^{-\nu})
\end{equation}$ with $x^{\nu} \in X_{\nu}$ which is a convex set.
$x^{*}$ needs to satisfy $$\nabla_{x^{\nu}...
0
votes
0answers
28 views
Markov chain Monte Carlo with stopping time
I asked the same question two days ago on MSE, but received no answer. So I post it here in hope to get any suggestion. As long as I have answer, I will close the other one.
Let $(X_t)$ be a ...
1
vote
1answer
96 views
Why is my Cahn-Hilliard simulation separating out so finely?
I am trying to simulate the Cahn-Hilliard equation using Python, but the 2 fluids aren't separating into big blobs, as desired, under any conditions. I'm setting up (what I think is) an orthogonal ...
0
votes
0answers
15 views
Shallow water equations: boundary conditions for sub- and super-critical flow
This question is (sort of) a continuation of this previously asked question. I am wondering about, in general, how we construct well-posed boundary conditions (both continous and numerical) for flow ...
1
vote
0answers
38 views
Advice for a topic in a seminar
I am a master student in Mathematics, and I have to prepare a seminar for a course in mathematical methods for applied sciences. I have a good background in numerical analysis for ODEs, PDEs and hence ...
0
votes
3answers
74 views
Changing variables in integral to avoid infinity
I want to write a code in Fortran to solve this integral numerically: $$\int_{1095}^\infty \frac{dx}{x\sqrt{(x+644.153)(4.17 \cdot 10^{-5} x+0.145)}}$$ What is the best method for it?
I tried Monte-...
6
votes
1answer
253 views
What kind of a researcher am I?
So far, I've worked a bit in modeling, simulations and simple lab experiments, and I've really enjoyed all three research methods to approach a single research question. I can write tricky (in terms ...
1
vote
0answers
25 views
Unable to achieve semi-linear running time in computation of continuant
I am trying to compute the continuant of a list of numbers $a_0, a_1,...,a_n$, defined by the recursion relation: $K_{n+1} = a_{n+1} K_n + K_{n-1}$ and $K_0 = 1$ (see Wikipedia).
I am trying to use ...
1
vote
0answers
20 views
Convection equation expanded in Legendre polynomials
In some areas of physics, for example radiation transport in plasmas, it is beneficial to re-express the convection equation in terms of a Legendre polynomial expansion. This results in a set of ...
1
vote
1answer
73 views
Galerkin method for heat equation
I'm working out the Galerkin method for the heat equation $$\frac{\partial u}{\partial t} - \frac{\partial^2 u}{\partial x^2} = 0$$ subject to $u(0,t)=0,u_x(1,t)=v(t)$.
I want to use a Fourier basis ...
0
votes
1answer
36 views
Coupled pdes of the first order
May question is about possible approaches to solve the following system
$$
\begin{array}{rcl}
\nabla{n}&=&n\,\mathbf{E},\\
\nabla\cdot\mathbf{E}&=&1-n,
\end{array}
$$
in general with ...
1
vote
0answers
59 views
Bifurcation points on homotopy path by numerical continuation?
I am trying to implement an algorithm that finds (possibly) all solutions of a system of nonlinear polynomial equations $$F(X) = 0$$ I thought about using the (convex) homotopy $$H(X,T) = TF(X) + (1-T)...
2
votes
0answers
71 views
Numerical method for harmonic oscillator with jumping constant
Let $k_1 \neq k_2$ be positive reals, $t_0 > 0$ and consider the following Cauchy problem in $[0,+\infty)$:
\begin{cases}
y(t) + k(t)y''(t) = 0 \newline
y(0) = 1/\sqrt{k_1} \newline
y'(0) = 0,
\end{...
2
votes
1answer
129 views
Solving numerically a linear ODE
I start by saying that I do not have a strong background in numerical analysis, so I may miss some basic things or make trivial mistakes.
Motivated by some problems in digital signal processing, I ...
3
votes
1answer
85 views
Can you compare integer part of two fractions without division?
Suppose we need to compare
$\left \lfloor{a / b}\right \rfloor $
and
$\left \lfloor{c / d}\right \rfloor $
.
One way would of course be to calculate $a/b$ and $c/d$ by division. Is their a faster way?
1
vote
0answers
28 views
Tackling multiscale problem in numerical simulation
In a dusty plasma system there are more than one component with different masses, i.e, electrons, ions,neutrals and dust grains. Accordingly, there are more than one temporal and spatial scales ...
0
votes
1answer
108 views
Time complexity of numerical finite differences
I have a function $f:\mathbb R^N\to \mathbb R$ and I would like to compute all the partial derivatives of $f$ w.r.t. the $N$ input. What is the computational complexity using the (ones-sided) finite ...
1
vote
2answers
52 views
What space-time points should a known coefficient function be evaluated at when using the Lax-Friedrichs scheme to solve the transport equation?
For a scalar quantity $u = u(x, t)$, I'm considering the transport equation
\begin{align}
u_t + au_x &= 0, \qquad{x\in[0, L], \ t\in[0, T]}
\\
u(0, t) &= u_{\text{in}},
\\
u(x, 0) &= f(x).
...
0
votes
1answer
93 views
Normalized legendre and quadrature basis for discontinous Galerkin method
I've successfully implemented a 1D DG code with non-normalized Legendre basis and I've now moved onto developing a 2D code using tensor products. For my 2D code I've chosen to have normalized ...
0
votes
1answer
45 views
Reinch's modification to the Clenshaw recurrence gives no improvement
The classical Clenshaw recurrence (see Algorithm 3.1 here) is less accurate as $x\to \pm 1$. So Reich proposed a modification to it, which is discussed as Algorithm 3.2 as well as by Oliver.
While ...
6
votes
2answers
216 views
Numerical flux and source term in FVM (Burger's like equation)
I'm trying to solve the following equation with FVM
$$u_t + f(u)_x = g(u)$$
where $g$ is some smooth function of $u$ and $f(u) = \frac{u^2}{2}$. This is really similar to Burger's equation, except ...
1
vote
0answers
68 views
Compute mass matrix in vibrations problem by using finite element method
I have to compute the mass matrix of a Hexahedral mesh.
There are 3 methods to compute mass matrix. I'm interested in one method which consists of dividing the mass of the element by the number of ...
0
votes
1answer
41 views
Showing stability of numerical scheme: Decaying norm implies stability?
I have a little trouble formulating my question since I am not really sure what conclusions I am supposed to draw from the results I have obtained. I am sorry for the long problem formulation below, ...
