Questions tagged [numerics]
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862
questions
4
votes
2answers
143 views
Effect of mesh size on solution curves for a 1D problem
I'm interested in studying the effect of mesh size on the behavior of the solution curves of 1D convection-diffusion problem.
$$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2} - ...
2
votes
1answer
130 views
Slope limiting for discontinuous Galerkin (DG) method
I had a question regarding the implementation of the TVB limiter for the RKDG method by Cockburn. I have seen that some implementations of the DG method use normalized Legendre polynomials such that ...
1
vote
0answers
61 views
Conjugacy in Non-linear Conjugate Gradient Descent
In linear conjugate gradient method, our goal is to solve the system of linear equations $$Ax = b$$ where A is a symmetric positive definite matrix, and that is equivalent to finding the minimizer of ...
1
vote
1answer
38 views
How to compute the mean 2D slice of a 3D set of data in MPI
I have a 3D set of data v(i,j,k), and I want to compute the mean 2D slice vmean(i,j) summing up the ...
1
vote
0answers
83 views
How to minimize a integral function using a constant step gradient method in Python?
I am developing a practical work of the following system of ode
\begin{align}x'(t) &= k_1h(t) - (k_2+k_3)x(t)\\
y'(t) &= k_3x(t)\end{align}
and $z(t) = (1-k_4)(x(t)+y(t))+k_4h(t)$, where $h(...
7
votes
1answer
172 views
evaluating $\coth(x) - 1/x$ for real $x$, on 2 “pieces”
The function $\coth(x) - 1/x$ has a removable singularity at 0.
Its Taylor series is:
$$
\coth(x) - 1/x = \frac{x}{3} - \frac{x^3}{45} + \frac{2x^5}{945} + \ldots
$$
I would like to evaluate the ...
3
votes
1answer
238 views
log-sum-exp trick for signed/complex numbers
I need to evaluate a sum of values that are on many different orders of magnitude in scale but might be signed. I’ve had great luck with the “log-sum-exp” trick for an unsigned version of my problem, ...
0
votes
0answers
63 views
A name for a numerical phenomena when using numerical methods
I have a nonlinear solver for equation $g= c_1f(x_1,y_1)+c_2f(x_2,y_2)$. Note that $c_1$ is much bigger than $c_2$. So after using Levenberg–Marquardt algorithm, I could only get $x_1$, $y_1$ and $...
1
vote
1answer
76 views
Type of Rosenbrock method by its coefficients
A Fortran code that solves stiff PDE systems contains the following arrays of Rosenbrock-Wanner method coefficients:
...
1
vote
1answer
659 views
What is the difference between Abaqus and Calculix contact input?
I would like to say first that am new at using Calculix.
I'm using Abaqus/CAE to create a cup deep drawing simulation and everything worked perfectly but my objective is to run the same exact ...
2
votes
1answer
106 views
pdepe or Crank-Nicolson? How much is pdepe good?
I am beginner in MATLAB and similar. I sow and discussed with my professors doing simulations some times: they wrote down a lot of calculus, most of them using Crank-Nicolson Method and so implement ...
5
votes
1answer
147 views
Complex differentiation of linear solvers
I have a linear system $$Ax=b$$ which I'm solving approximately, and I need to take the frechet derivative of x with respect to z. Were I solving the problem exactly (either analytically or to machine ...
0
votes
0answers
70 views
Comparison of diffusion time - theoretical value vs computed
This is a follow up to my previous post
I've been trying to compare the diffusion time obtained from theoretical
derivation(answered in my previous post) and what is obtained computationally, for a ...
2
votes
1answer
166 views
mesh dependence of numerical adjoint solution
I am solving the steady, two-dimensional adjoint Euler equations, $$A_x^T \partial_x \Psi + A_y^T \partial_y \Psi = 0$$, where $A_x = \partial F_x/\partial U$ and $A_y= \partial F_y/\partial U$ are ...
0
votes
2answers
195 views
What are the most important theorems in computational science? [closed]
I was reading the book: The Finite Element Method: Theory, Implementation, and Applications by Mats Larson and Fredrik Bengzon, in page 140 of this book they say this:
"The Lax-Milgram Lemma is ...
2
votes
1answer
93 views
Defining Current Density in a FEM model (MATLAB)
I'm attempting to solve the Poisson equation in 3D for a magnetic vector potential in the presence of a current source. To validate my code, I'm initially looking to reproduce the model described in ...
2
votes
0answers
40 views
How to account for a corner node with zero-flux condition at an extrapolated distance
I am trying to implement a numerical solver and am having troubles dealing with boundary conditions, especially in the corners.
I have a 2D mesh, and on the left I have a Dirichlet condition, on the ...
2
votes
0answers
36 views
Computing convolution of two characteristic function over a 1D Cartesian mesh
I am trying to compute the convolution of two characteristic functions over a Cartesian mesh. First, I define my Cartesian mesh of the interval $[0,1]$ as follows
$$
x_{i} = i \Delta x, i = 0, 1, 2\...
0
votes
1answer
89 views
Solution of thermal analysis using finite element
I want to solve a thermal analysis using finite elements. The governing equation is $$C \frac{dT}{dt}+K T = Q$$.
When using backward differencing for time, the resulting equation is quite straight ...
1
vote
0answers
55 views
Stably solve transport equation with source term
I am trying to solve a transport equation of the form for the variable $\psi(t,r)$
\begin{equation}
\partial_t\psi-\alpha(r)\partial_r\psi-\beta(r)^2\psi-f(t,r)=0
,
\end{equation}
where I am solving ...
0
votes
2answers
391 views
Boundary conditions for streamlines in enclosed flow
I am trying to solve Lid driven square cavity flow problem of Stokes equation using finite element method. I have boundary conditions for velocity as zeros on every boundary but u=1 on top boundary. ...
4
votes
1answer
105 views
Modelling flow through pipe networks
I'm trying to educate myself on modelling solute flows through pipe networks.
This is a follow up of my previous post here
$$\frac{\partial C}{\partial t} = - v\frac{\partial C}{\partial x}$$
While ...
2
votes
1answer
124 views
Does mass balance hold in convective diffusion
I'm trying to understand how convection-diffusion equations are solved in pipe flow modules available in CFD solvers.
$$
\frac{\partial C}{\partial t} + \nabla \cdot (\mathbf{v} C) = \nabla \cdot (D \...
0
votes
1answer
76 views
How do I get power from gaussian beam numerically?
I would like to get the power from a Gaussian beam given a set of points at which electric field is evaluated. Please follow my reasoning and tell me what assumption maybe are wrong
Power definition ...
0
votes
1answer
205 views
Solving differential equation in Python with discretized variable coefficients
I am trying to solve a differential equation with discretized variable coefficients which are calculated from a time serie.
In this case the Runge-Kutta step size is fixed by the frequency in the time ...
5
votes
2answers
190 views
Is the diffusion equation with Neumann and Dirichlet BCs well-posed?
I am considering the following diffusion equation:
$$\frac{\partial f}{\partial t} = \frac{\partial}{\partial x}[D(x,t)\frac{\partial f}{\partial x}]$$
over a grid ...
3
votes
0answers
73 views
Numerical integration with singularity term
In https://www.johndcook.com/blog/2012/02/21/care-and-treatment-of-singularities, the author explains the subtraction method to get rid of singularities when performing numerical integration.
The ...
4
votes
2answers
292 views
Chebyshev and Legendre expansions
I am looking at approximating my function $f(x)$ using a Chebyshev and Legendre series and I ran into this question.
Is interpolation using $n+1$ Chebyshev nodes the same as representing the function ...
6
votes
2answers
1k views
Runge-Kutta in the presence of an attractor
Suppose you are solving a system of equations numerically that possesses an attractor (no matter the initial conditions set, all the different solutions will approach a specific set of values that ...
7
votes
0answers
194 views
How to check if my stiffness matrix is correct
I built the stiffness matrix for the Poisson equation on a 2-dimensional domain with the shape of "almost" an octagon, using pyramid basis functions. I used almost to intend the fact that I have an "...
2
votes
1answer
55 views
Numerical integration of the dataset of a function
The energy equation for a spherically symmetric system is given by
$$\mathscr{E}=\frac{v^2(r)}{2}+\frac{c_s^2(r)}{\gamma-1}+\phi(r)$$
where $\mathscr{E}$ is the total energy, $v$ is the velocity of ...
4
votes
0answers
110 views
Integrators for Nonlinear/Stiff PDE
It was suggested I ask this question in this section. Anyway:
I have a particular nonlinear PDE of the form
$$
u_t(x,t)=iu_{xx}(x,t)+f(x,u(x,t))
\tag{1}
$$
Where f is some nonlinear function. With ...
8
votes
1answer
167 views
Radial integration of expensive function with Bessel weights
I need to calculate the integral
$$I = \int_0^R f(r)J_n\left(\frac{z_{nm}r}{R}\right)rdr$$
where $J_n$ is the $n^{\mathrm{th}}$ order Bessel functions of the first kind, $z_{nm}$ is its $m^{\mathrm{...
1
vote
1answer
146 views
Solve linear system with Newton-Raphson method
Is it possible to solve a linear matrix system $A x = b$ using the Newton-Raphson method? If yes, how can this be done? More special, how is the derivative build?
-1
votes
1answer
49 views
Integrate using Composite Simpson's rule
In a question, we have been given the speed of a car at time t= 0,2,4,6,.......,20 minutes.But it asks us to approximate the distance travelled by the car in 30 minutes using Composite
Simpson's rule. ...
7
votes
1answer
138 views
Numerically estimating expected value of f(x) when x is normally distributed
I need to estimate
$$
\mathbb{E}_x[f_i(x)] = \int_{\mathbb{R}^n} f_i(x) p(x) dx
$$
for many functions $f_i(x)$, where $p(x)$ is the density of a normal distribution. The evaluation of all the ...
-1
votes
1answer
246 views
Recursive Algorithm to Calculate Determinant via Expansion of Minors in C#
I have been recently trying to attempt to write an algorithm in C# that would calculate the determinant of a matrix via recursion using the expansion of minors method. I understand that there are ...
2
votes
0answers
158 views
What exactly is the cause(s) of blow-up for too-large step size in a method like RK4?
I have been working on creating a few home-made numerical methods, and I am using them to visualize text-book problems from my Strogatz dynamics textbook. It feels like a good way to learn numerical ...
1
vote
1answer
305 views
Implementing Robin Boundary condition (finite difference)
I'm interested in applying Robin boundary condition to a convection-diffusion problem in 1D.
In the following system,
$$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2} - v\frac{\...
1
vote
1answer
246 views
Runge-Kutta fourth order method. Integrating backwards
I am using a Runge-Kutta fourth order method to solve numerically the usual equation of motion of a background scalar field in curved spacetime with a quartic potential:
$\phi^{''}=-3\left(1+\frac{H^{...
1
vote
1answer
109 views
How to Break Coupled ODEs down to first order for Runge-Kutta
My question might seem a bit simple. I am trying to solve a system of ODEs using Runge-Kutta method. I am having difficulty breaking down the equations into a system of first order ones required ...
2
votes
1answer
70 views
How to robustly and numerically expand a $k$-order polynomial in two variables defined on a polygon domain?
Given a $k$-order polynomial in two variable $p(x, y)$ defined on a polygon domain $K$. And I want to numerically expand it to the following form
$$
p(x, y) = c_0 + c_1 x + c_2 y + c_3 x^2 + c_4 xy + ...
1
vote
0answers
51 views
Numerical Method for Equation System of two depending Equation Systems
I am searching a solution method for the following equation system of equation systems:
Let $A \in \mathbb{R}^{n \times n}$ be an invertible Matrix, $f, b_1, b_2 \in\mathbb{R}^n$ given vectors and $ ...
1
vote
2answers
55 views
Numerical integral with symbolic integral in exponent
Many times in fourier approximation we come across integrals such as
$$\int_0^1 e^{-\gamma\int_0^xu_0(\eta)d\eta}dx$$ where $\gamma$ is a constant and the data for $u_0$ is provided as a discretely ...
6
votes
1answer
163 views
Computing square root of diag(u)-uu'?
I need an efficient way to take square root of a matrix which is a sum of diagonal matrix and rank-1 matrix.
More specifically it's the following matrix
$$A=D-uu'=\text{diag}(u)-uu'$$
Where entries ...
3
votes
1answer
58 views
Bounding error of float32 matrix multiplication
Some numerical debugging led me to the minimal example below. I'm observing relative error of 0.75 on individual elements.
Is there a way to estimate/bound this error without resorting to higher ...
2
votes
1answer
69 views
Passing data as arguments in ODE45
I need to import data from file in order to describe the structure of a network.
I used the following:
weights = readtable('weights192.txt');
W = weights{:,:};
...
2
votes
0answers
71 views
Numerical integration of SDE: choice of $dt$ and algorithm
I am working on the following Stochastic Differential Equation (SDE) in the Quantum Mechanics context:
$$dX_{t} = a X_{t} dt + b X_{t} dW$$
where $X_{t}$ is my stochastic varible, $dt$ is my ...
1
vote
1answer
279 views
Weighted moving variance
i have a time-series and, in analogy with exponentially weighted moving average, i would like to compute the exponentially weighted moving standard deviation or variance in an efficient, numerically ...
Ertxiem - reinstate Monica
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2
votes
0answers
41 views
Inverses of many standard subspaces of one large matrix
i have a large rectangular invertible matrix M (about 5000x5000) and i have a loop in which i do the following for each iteration i (there are about 6000 iterations):
i am given a subspace S_i (which ...
