Questions tagged [numerics]
The numerics tag has no usage guidance.
3
votes
2answers
65 views
Analytical convergent sequence and numerical divergent sequence
Is it possible to construct a sequence that converges in theory but when computed numerically with a computer program is diverging.
I feel that today our computer programs doesn't allow such ...
-2
votes
0answers
33 views
python Simpson integration [on hold]
Currently, I am trying to apply a Crank-Nicolson method on a function that I want to evolve. However, I am only facing one drawback and it is the step when I normalize the initial function using the ...
2
votes
2answers
104 views
How does a stiff equation solver work?
I am trying to understand how stiff differential equations are solved.
For instance the equation,
$$\frac{\partial y}{\partial t} = \alpha\frac{\partial ^2 y}{\partial z^2}$$
can be solved using ...
3
votes
1answer
99 views
Derivation of backward differentiation formulas(BDF)
I have been reading upon numerical techniques that are used to solve stiff ordinary differential equations.
From the description given here, I could follow the steps till equation (5).
I am finding ...
1
vote
0answers
49 views
Implementing boundary condition
I'm studying the transport of species A in the blood vessels,
$$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2}-v\frac{\partial C}{\partial x}$$
At x=0, I want to use the ...
3
votes
1answer
80 views
Golub-Kahan-Lanczos Bidiagonalization Procedure implementation doesn't produce bidiagonal matrix
I'm trying to implement the aforementioned procedure using this website as a reference. At the end of the page the algorithm is described as follows:
I think I've mapped the given algorithm to code ...
3
votes
0answers
60 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
$$\mathbf{u}_{t}+\hat{A}(\mathbf{x})\mathbf{u}_{\...
1
vote
1answer
35 views
Wrong results for $2$ stage multistep method $y_{n+2} - y_n = h\left[(1/3)f_{n+2} + (4/3)f_{n+1} + (1/3)f_n\right]$
I need to fix a code to utilise the $2$ stage multistep method :
$$y_{n+2} - y_n = h\left[(1/3)f_{n+2} + (4/3)f_{n+1} + (1/3)f_n\right]$$
Since this is an implicit method, I used a Newton-Raphson ...
-2
votes
0answers
25 views
Two exercises from Bertil Gustafsson's Time-Dependent Problems and Difference Methods
I would like some help on these two exercises from Gustafsson's textbook:
Ex. 2.3.2: Calculate the exact number of arithmetic operations required to advance by one step the implicit scheme $$(2.3.3) (...
-1
votes
0answers
69 views
Numerical analysis: least squares method
Basically I came up with a function where I need to approximate it using the least squares method. I had no trouble computing the constants on MATLAB and got to the following plot:
Now as you see the ...
4
votes
0answers
62 views
Solution of constrained system of ODEs
Can someone point me in a direction to solve this kind of integral constrained system of ODEs.
\begin{align}
&\int_0^{1/2}\dot{y}^2(t)=p\\
&2\lambda_1\ddot{y}(t)+\pi cos(\pi y(t))=0\\
&y(...
-1
votes
0answers
32 views
Question on implementing boundary conditions
Let's consider a 1D heat diffusion problem,
$\frac{dT}{dt} = \alpha\frac{\partial^2T}{\partial x^2}$
This is most often discretized in space.The discretization results in a tridiaginal matrix.
...
1
vote
2answers
37 views
Determine conditions on parameters (for consistency) on RK method $y_{n+1} = y_n + ha_1f(t_n,y_n) + ha_2f(t_n + b_1h, y_n + b_2hf(t_n,y_n))$
I'm asked to find the conditions on the coefficients $a_1,a_2,b_1,b_2$ in the RK method
$$y_{n+1} = y_n + ha_1f(t_n,y_n) + ha_2f(t_n + b_1h, y_n + b_2hf(t_n,y_n))$$
such that is consistent of (a) ...
0
votes
1answer
49 views
Numerical solution of non-linear first order partial differential equation (HJB)
I am trying to solve a simple optimal control problem using the Hamilton-Jacobi-Bellman equation, numerically in Python. This is proving to be rather difficult as I end up having to solve the ...
4
votes
0answers
73 views
Comparing sum of floating points
I am currently working on a numerical algorithm involving a lot of floating point arithmetic, involving some badly conditioned problem sets.
I am using the relation $|x - y| / (\max(|x|, |y|, 1)) \...
1
vote
1answer
78 views
Operation count for GMRES
One can use GMRES as it is, but there is also a version of GMRES called k-step restarted GMRES, which is used for large matrices, where $k$ is some fixed number of steps after which we take a new $x_0$...
5
votes
1answer
47 views
Mass matrix and BDF time integration
I have a system of nonlinear equations on the general form:
\begin{align}
\mathbf{M}(\bar{y})\dot{\bar{y}} =\bar{f}(\bar{y},t)
\end{align}
Where $\mathbf{M}(\bar{y})$ is a matrix and $\bar{f}$ is a ...
3
votes
2answers
134 views
Asymptotic error of forward Euler
I'm trying to understand the asymptotic error behaviour of forward Euler (finite difference method), as timesteps are decreased (refined), so I feel trust in the method of manufactured solutions (MMS)....
3
votes
0answers
77 views
Why the MIRACLE of Lanczos/CG-like?
Lanczos/Arnoldi/Rietz/CG-like algorithm share the same core strategy...
In each, a little miracle appears, most of the Gram-Schmidt inner products are zeroes! In others words, new direction need only ...
3
votes
1answer
153 views
Computational complexity of Newton's method
the classical Newton's method for non-linear systems of equations is $x_{k+1} =x_k-J_F(x_n)^{-1} F(x_n)$. In pratice, rather than compute the inverse of the Jacobian matrix, one solves the systems $...
1
vote
0answers
34 views
Numerically solving a system of parabolic PDEs and 1st order ODEs
I'm trying to solve the following system of differential equations numerically. What are the available finite difference approaches and matlab solvers to solve such a system? Other approaches to solve ...
6
votes
1answer
106 views
How do I integrate a function defined over an arbitrary area?
Let's say, I have a compact area $S$ (for example a circle, a square or some arbitrary polygon) and a function $f: S \rightarrow \mathbb{R}$. I want to numerically calculate the Integral
$$ \int_S f(\...
2
votes
0answers
27 views
Numerical stability while modeling wave equation in staggered grid
I have modeled a simple wave equation given by:
\begin{align}
& \begin{cases}
u_t = v_x \\
v_t = u_x
\end{cases}
\end{align}
Boundary conditions given on interval $M=[-1,1]$ by:
\begin{align}
u(...
5
votes
1answer
96 views
Numerical solution of two coupled nonlinear eigenvalue problems
I would like to numerically solve the following system of coupled nonlinear differential equations:
$$
-\frac{\hbar^2}{2m_a} \frac{\partial^2}{\partial x^2}\psi_a + V_{ext}\psi_a +
\left( g_a |...
-1
votes
1answer
66 views
Relation of Condition of a Matrix and Convergency
Can anybody explain me the relation between the condition of a Matrix and the convergency of a problem.
For example how is the relation between the condition of the stiffness Matrix occuring in FEM ...
1
vote
0answers
70 views
why I cannot find explicit finite difference for elliptic equation
Let us think on the Poisson equation $\nabla^2 u(\bf{x})=\rho(x)$ with Neumann boundary conditions, with $\bf{x}=\it (x,y)$ in 2D.
Here is a stencil with central differences in both $x$ and $y$ (...
1
vote
2answers
65 views
Guaranteed equality between binary results with increasing MPI processes
Testing on an MPI scientific code for compressible flow dynamics I noticed that the results may depend on the number of processors used for the calculation. In fact, comparing the binary files they ...
3
votes
1answer
71 views
Derivatives of Approximate Matrix inverses
I am cross posting this question to the mathermatics stack exchange. please find it either at this link, https://math.stackexchange.com/q/2952989/430980, or below:
I have a question concerning the ...
2
votes
0answers
63 views
Efects from the boundary in advection equation [duplicate]
I am implementing the advection equation $u_x+(1/c)u_t=0$ following a Crank-Nicholson finite difference scheme. The equation for this is
\begin{eqnarray*}
-\frac{\gamma}{4} w_{n-3 j+1} + w_{n-2 j+1} ...
1
vote
0answers
85 views
How to solve an implicit ODE with forward Euler?
Consider the implicit ODE
$$
M(y)\dot{y} = F(t,y)
$$
If $M$ is non-singular for all $y$
How to use the forward-Euler method to numerically solve for $y$ without inverting $M(y)$?
I only came out ...
1
vote
1answer
104 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
54 views
PML boundary conditions
I set up two one-way wave equations for constant velocity $c$ in one-dimension. When I implement them I get a highly unstable (divergent) solution. I wonder if someone could give me a suggestion about ...
2
votes
0answers
72 views
How to numerically optimize affine transformations?
I need to optimize affine transformations for of a set of triangles using energy function based on the connectivity.
The energy of an edge $e_j$ between triangles $T_a, T_b$ is given by
$$
E_j = \...
2
votes
1answer
87 views
Euler Method Instability. Why?
I am currently enjoying writing computational codes as a hobby. Right now I've worked out an Euler method and results are pretty good with up to $x=1$. Over $x=1$, instability starts to kick in. May I ...
-2
votes
1answer
44 views
Calculate accuracy order/rate
I was doing error analysis of numerical scheme and I get $L_1$ error for each grid size with $N$ element. I was searching reference to compute accuracy order/rate from that error data but doesn't find ...
0
votes
4answers
234 views
Which Linux OS for computer science and numerical analysis
I am switching from OSX to Linux, which I am familiar with but have no greater experience. There are several different alternatives: archlinux, gnome, ubuntu etc. Which would you recommend for a PhD ...
2
votes
0answers
30 views
Sensitivity Lagrangian solution general case
I have asked this question already on maths and mathoverflow.
Just a question about a literature reference. I am writing a paper for engineers.
Usually for the Lagrange multiplier problem
$$
\...
1
vote
0answers
31 views
Efficient numerical optimization of an “almost separable” function
I have come across an optimization problem with the following objective function:
$$f(x_0,y_0,z_0,x_1,y_1,z_1,...,x_N,y_N,z_N) = \sum_{i=0}^N f_i(x_i,y_i,z_i, \alpha(x_{i+1}-x_i) + \beta(y_{i+1}-y_i))...
9
votes
0answers
105 views
Are there shortcuts for numerically approximating systems of ordinary differential equations when autonomous?
Existing algorithms for solving ODEs handle functions $\frac{dy}{dt} = f(y, t)$, where $y \in \mathbb R^n$. But in many physical systems, the differential equation is autonomous, so $\frac{dy}{dt} = f(...
4
votes
2answers
108 views
Estimating error at mid timesteps for Runge-Kutta methods
When a timestep $h$ is rejected using a Runge-Kutta pair, such as Dormand–Prince, the algorithm resumes from the same initial point $t$ with a smaller timestep. A different idea is to resume at an ...
4
votes
0answers
103 views
Compile-time error control vs. interval arithmetic?
I use interval arithmetic for reliable computing. Now, a procedure coded in a good implementation of interval arithmetic takes perhaps about eight times as much as the same procedure carried out ...
8
votes
1answer
69 views
Benchmark problems for eigenvalue reordering algorithms sought
Every real matrix $A$ can be reduce to real Schur form $T = U^T A U$ using an orthogonal similiary transform $U$. Here the matrix $T$ is quasi-triangular form with 1 by 1 or 2 by 2 blocks on the main ...
1
vote
1answer
130 views
Does a generic method for solving a system of PDEs exist?
There are generic methods for solving systems of ODEs numerically. Are there generic methods for PDEs? If so, what are they? If not, why not? To elaborate...
Any set of ODEs can be written in ...
1
vote
0answers
46 views
What does the Jackson Kernel measure?
A certain filter I'm writing uses two different kernels. The Fejer kernel (which is common) and the Jackson kernel:
$$ \Delta_T(x) = T \,\left( \frac{\sin \pi T x}{\pi T x}\right)^2 \quad\text{and}...
3
votes
1answer
224 views
How does fmincon in MATLAB calculate gradients?
I am trying to solve numerically a constrained optimisation problem in MATLAB, and I am wondering how the fmincon function calculates gradients when one isn't ...
5
votes
0answers
104 views
Cressman interpolation and objective analysis
I have read this question and answer – Interpolation of scattered data to a regular grid in python and I am doing something similar as I have temperature values of the atmosphere at different heights ...
2
votes
1answer
68 views
How can a CG solver solve a non positive definite sparse matrix
I am using the CUSP CG solver and I ran it on couple of sparse matrices from the University of Florida sparse matrix collection. The solver was able to solve non positive definite sparse matrices. My ...
2
votes
1answer
153 views
Solve 3-D Heat equation with Neumann boundaries
I want to solve the Poisson PDE for heat flow in a 3-D solid cube with given dimensions $x$, $y$, and $z$:
$$\rho C\frac{\partial T}{\partial t} = k \Delta T$$
The cube is irradiated with a constant ...
1
vote
0answers
79 views
Basic approach for numerical solution of PDE
I'm looking for some guidance on how to write a program to numerically solve a PDE. As an example for comparison in 1D:
$$\frac{d^2u}{dx^2} = f\;\;\;\;u(0) = 0\;\;\;\;u(1) = 0$$
We could try 6 ...
2
votes
1answer
50 views
Oscillation term in a posteriori error estimator
Assume that in the a (residual type) posteriori error estimator of some PDE is a term of the form $h_T\|g\|_{L^2(\Omega)}$ involved where $h_T$ is the diameter of an element and $g$ is some known data ...
