Questions tagged [numerics]
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4
votes
1answer
63 views
Going back in time in an initial value problem
Consider an initial value problem (IVP) $y'=f(t,y)$ with the initial value given by $y(t=0) = 0$.
If I need to find $y(t^*)$, hence finding the path for $y$ in $t \in [0,t^*]$ and $t^*<0$; is the ...
0
votes
0answers
38 views
Numerical integration(principal value) [on hold]
I need to solve this integral numerically, as you can see, the first term of this integral reduces to a principal value two dimensional integral and a one dimensional integral (due to dirac delta). In ...
2
votes
1answer
56 views
How to show the stability of $L^2$ projection?
If $\mathcal{T}_h$ is a regular and quasi-uniform triangulation of $\Omega$, and $V_h$ is the $H^1$-conforming linear finite element space. Moreover, let $P_h$ be the $L^2$ projection to $V_h\subset H^...
2
votes
1answer
71 views
In iterative methods, are matrix decompositions considered useful for implementation?
When we study an iterative method from textbooks, for example, see the Gauss-Seidel Method, the given matrix is decomposed with suitable splittings. In the example, $A = L+U$. So we can proceed with ...
1
vote
0answers
30 views
How to use Wolfe-Powell step-size control in quasi-Newton method?
I'm trying to find the minimum of a function using the quasi-Newton method with the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm.
But I want to change the following implementation, so that:
1) ...
2
votes
1answer
38 views
Computing face fluxes in FVM
In FVM, we have to compute fluxes at some face of a cell. There are many ways to compute this face flux value, but the most common and easiest way involves some simple averaging at the face. However, ...
0
votes
0answers
25 views
Fit exponential convergence
I'm working with a numerical algorithm whose output $y$ asymptotically approaches a certain unknown value $a$.
I expect an exponential convergence, i.e. the data $y$ given by my algorithm should be ...
4
votes
1answer
146 views
Accurate way of getting the square root inverse of a positive definite symmetric matrix
What is the most accurate algorithm to get the square root inverse of a positive definite symmetric matrix? I am not looking as much for efficiency, though using quadruple precision computation is out ...
0
votes
0answers
29 views
How to show equivalence between two programs?
Consider the following space $A = \{(x_1,x_2,x_3)\in \mathbb{R}^3|x_1+x_2+x_3 = 1\}.$ Then say that we want to minimize a function $J(y):\mathbb{R}^{3}\to \mathbb{R}$ subjected to the constraint that $...
-2
votes
1answer
77 views
How to : numerical integration by quadrature in C language / remove NaN
What I wanna solve it the problem following
( by quadrature method )
I want to get two arrays of data ( z & tau )
from z[0], tau[0] to z[2249], tau[2249].
Since the integrand diverges at z=0.9, ...
2
votes
0answers
33 views
How to determine the order of convergence of the Euler-Maruyama method?
This question is originally posted in Quant.StackExchange but has been unanswered for some time so I ask in here.
To make this simple let us consider the Geometric Brownian Motions (GBM).
My ...
1
vote
1answer
79 views
Dirichlet to Neumann Operator
EDIT: I am trying to specify my Question. Also I am not going to clearify which spaces I use, because I am only interested in the basic idea.
I am looking at a standard elliptic second order PDE:
\...
0
votes
1answer
141 views
Solving $n$ coupled equations numerically in Matlab
I would like to solve the following equations simultaneously and numerically for all $X, Y, Z, W$ where i = 1:Nw, j = 1:Nl, k = 1:K.
$W_\text{net1}$, $W_\text{net2}...
0
votes
0answers
40 views
Singular Spectrum Analysis Explanation
I need you to help me understand the Singular Spectrum Analysis algorithm. I already read a lot of articles about the subject but they never answered my questions like what is the mathematical reason ...
0
votes
1answer
57 views
Fast Poisson solver (with Dirichlet BC zero) on a *truncated* Cartesian 3D grid
I find myself in the position of having to solve
$-\Delta u = f$ on a subset of Cartesian grid points that don't necessarily form a cuboid domain subject to a homogenious Dirichlet boundary condition ...
2
votes
2answers
66 views
Stability of Crank-Nicolson for $u_t = iu_{xx}+2iu$
I want to use the Crank-Nicolson scheme to solve the equation
$$u_t = iu_{xx}+2iu$$
Here's the analysis: Suppose we make a grid, with $k = dt$ and $h = dx$, the usual notation, and also $u_j^n = u(...
0
votes
0answers
55 views
The final Boundary Condition is Unknown, Is Backward Euler is still valid to be implemented?
I am working on conductive polymer modeling and supposed to do one-dimensional diffusion model in the thickness of the polymer, however, due to the small value of thickness in micro, when I use the ...
1
vote
0answers
29 views
Why does the correlation function of this stochastic differential equation starts at different points?
I am working with the following differential equation:
The equation is $$x=\beta +\sqrt{2D} \xi(t)$$ where $\xi(t)$ is a white noise term, with a reflecting wall boundary conditions.
After solving ...
2
votes
0answers
51 views
How to implement adaptive step size Runge-Kutta Cash-Karp?
Trying to implement an adaptive step size Runge-Kutta Cash-Karp but failing with this error:
...
3
votes
1answer
45 views
Stability region of explicit midpoint method
Consider the explicit midpoint method, i.e
$$y_{n+1}-y_{n-1} = 2hf(y_n).$$
I'm asked to apply this method to the linear test equation, $f(y_n) = \lambda y_n,$ in order to find the method's stability ...
5
votes
0answers
77 views
Numerical methods for the continuity equation with Sobolev vector field
Consider the continuity equation
$$
\partial_t \rho(x,t) + \operatorname{div} (b(x,t) \rho(x,t)) = 0, \qquad t \in [0,T], \quad x \in \mathbb R^N,
$$
with $b \in L^1((0,T), W^{1,p}(\mathbb R^N))$.
...
0
votes
0answers
38 views
Time sampling changes solution
I'm currently trying to solve a problem using numerical methods. The set-up is rather long, so I apologize in advance...
TL;DR: My solutions change depending on how big my steps are and I don't know ...
2
votes
1answer
105 views
Why the numerical solution of advection-dominant problem is challenging
In many CFD text books, usually there is a dedicated chapter for advection term discretization.
Why discretization of such term in advection-dominated problems and near the discontinuities is ...
0
votes
0answers
57 views
Need help applying Implicit Eulers Method together with Newtons Method on Burgers' Equation
From the inviscid Burgers' equation: $\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x} = 0$, I get the discretization $\frac{u_i^{n+1}-u_i^n}{\Delta t}+\frac{(u_i^{n+1})^2-(u_{i-1}^n)^2}{\...
3
votes
4answers
152 views
Numerical integration in Python with unknown constant
I’d like to solve the below equation for the unknown $T$:
$$\int_0^\infty \frac{x^2}{\exp\left(\frac{x}{T}\right)-1}\kappa_x \mathrm{d}x = C,$$
where $C$ is a known constant and $\kappa_x$ is some ...
1
vote
1answer
61 views
Is there a better way to do run time analysis than this?
I currently have 2 different functions with options to vectorise them: acc_rej_sine(max_iter, algorithm=None) and ...
1
vote
1answer
65 views
Approximation Error in a Finite Difference Approximation of the Square of Derivative
First Part: (First-order derivative)
Assuming $f$ is an infinitely differential function everywhere, the Taylor series of $f(x + h)$ at $x$ is
\begin{align}\tag{1}
f(x + h) = f(x) + hf'(x) + \frac{1}...
0
votes
0answers
69 views
Analytic vs discrete understanding of PDE
The PDE I am working with:
$$\partial_tu = \nabla \cdot (a(x)\nabla u)-\beta(x)u\\
\partial_nu=0, x \in \Omega \subset \mathbb{R}^2\\
\beta(x)>0$$
Integrate the PDE:
$$\int_\Omega \partial_t u=\...
3
votes
1answer
124 views
What does the exponential function mean in numerical ODE solving formulas?
I'm trying to read papers on numerical ODE algorithms and I always seem to stumble upon huge amounts of exponentials multiplied by each other. For example in New families of symplectic splitting ...
0
votes
1answer
43 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
79 views
Actual global error vs theoretical global error: How to combine theory with practice
I have implemented an Adams Bashforth 4 method to solve an Initial Value Problem for an ODE and I am testing it against the test equation:
$y'=\lambda y$ with $y(0)=1$ with the exact solution: $y(t)=...
2
votes
1answer
153 views
Mass Matrix and how to handle it (ODEs) - References
I'm interested in a good reference/paper about how to handle numerically a mass-matrix system as
\begin{align}
\mathbf{M}(t,y)\dot{y} =F(y,t)
\end{align}
I know that such a problem can be solved by ...
7
votes
2answers
81 views
Algebraic multigrid for complex valued matrices
Assume one uses the classical AMG with Ruge-Stuben coarsening and direct interpolation for solving real valued problems. How can this approach be recycled to also solve complex valued problems like ...
1
vote
1answer
64 views
Adam Bashforth 4 method: how to determine starting values and stil keep the the order of accuracy
I am using an Adam Bashforth 4 method to solve an IVP problem so I need other numerical method to estimate the first 3 values. I am very much interested in finding a way to estimate the first 3 values ...
6
votes
2answers
240 views
Is Highams' computation of mean worth the price?
In Accuracy and Stability of Numerical Algorithms, equation 1.6a, Higham gives the following update formula for the mean:
$$
M_{1} := x_1, \quad
M_{k+1} := M_{k} + \frac{x_k - M_k}{k}
$$
Ok, one ...
0
votes
1answer
97 views
calculating integral for an ODE system
I have an ODE system defining a mathematical model of a biological system, say
$$
\frac{da}{dt}=f_1(a,b,\ldots,z,p)\\
\frac{db}{dt}=f_2(a,b,\ldots,z,p)\\
\cdots\\
\frac{dz}{dt}=f_n(a,b,\ldots,z,p)
$$
...
4
votes
2answers
98 views
SLATEC Routine Computes Givens Rotation in Unexpected Way
Some Background
I am working on a C++ translation of a SLATEC routine, R1UPDT, which performs a Givens rotation:
$$r = \frac{1.0}{\sqrt{a^2 + b^2}}$$
Usually, ...
1
vote
2answers
134 views
How I could calculate L2 norm of an unstructured grid?
I want to calculate L2 norm of a 3D unstructured grid to compare my simulation results in two different mesh sizes as coarse and fine. I read this answer and it seems in three-dimensional space, I ...
2
votes
1answer
110 views
Poorly conditioned, easily evaluated sum for unit testing
I am looking for examples of poorly conditioned sums which can rapidly be evaluated, for the purposes of unit testing.
I'm currently using the series representation for $\ln(2)$:
$$
\sum_{n=1}^{\...
1
vote
1answer
99 views
Relation between conjugate gradient method and finite elements method
What is difference beetwen this two method? Are these methods far from each other or are these methods complement each other? Could you take an example?
4
votes
1answer
91 views
Numerically find the nearest positive semi definite matrix to a symmetric matrix
I have a symmetric matrix $M$ which I want to numerically project onto the positive semi definite cone.
To do so, I decompose it into $M = QDQ^T$ and transform all negative eigenvalues to zero. (...
0
votes
1answer
62 views
Approximation of ODE solution using Taylor series methods
This is my first post on here, so please excuse mistakes if any.
I am trying to plot out the difference between two ODE solvers based on Taylor series:
1st order acccurate: $x(t_0 + h) = x(t_0) + ...
2
votes
2answers
152 views
Generate high n quantum harmonic oscillator states numerically
How can I generate the higher $n$ quantum harmonic oscillator wavefunction (in position space) numerically? Here, higher means around $n=500$, or say $n=2000$, where $n$ is the $n$th oscillator ...
4
votes
2answers
95 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
3answers
206 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
120 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
63 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
176 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
65 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
$$\mathbf{u}_{t}+\hat{A}(\mathbf{x})\mathbf{u}_{\...
1
vote
1answer
41 views
Wrong results for $2$ stage multistep method $y_{n+2} - y_n = h\left[(1/3)f_{n+2} + (4/3)f_{n+1} + (1/3)f_n\right]$
I need to fix a code to utilise the $2$ stage multistep method :
$$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 ...
