Questions tagged [numerics]

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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 ...
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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 ...
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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
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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
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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
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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
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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{...
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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
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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+\...
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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
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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}...
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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 Lentz’s algorithm. This statement is now ~15 years old, so I wonder has this gap in the ...
3
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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
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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
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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 ...
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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 ...
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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 ...
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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
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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}^...
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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
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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
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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
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0answers
73 views

Plot of ratio of two integrals:

Consider the following integrals $$ I_1(x) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy) − F(x −\mathrm iy)}{\mathrm e^{2πy}-1}, $$ And $$I_2(x) =\int_1^x F(t)dt$$ Where, $ F(z) = \sin^2[π\Gamma(z)/...
1
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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
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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
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3answers
241 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
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 } \...
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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}...
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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
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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
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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
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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 ...
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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
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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 ...
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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 ...
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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
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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
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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 ...
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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
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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
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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
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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?
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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 ...
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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 ...
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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
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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
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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
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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
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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
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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, ...

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