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Questions tagged [numerics]

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0
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0answers
13 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 ...
0
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0answers
10 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 ...
0
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0answers
11 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: ...
0
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0answers
23 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 ...
4
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0answers
30 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
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0answers
32 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
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1answer
87 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
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0answers
54 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
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4answers
129 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 ...
0
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1answer
60 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
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1answer
61 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}...
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0answers
61 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=\...
0
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0answers
40 views

Derive a relation between multiple parameters, given constraints

I have a mixture of 4 liquids (their percent volumes being: $a,b,c,d$). By varying their volumes I get different data for the boiling point of the mixture. I have a sufficient number of such data (12, ...
2
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1answer
121 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 ...
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0answers
16 views

Attempting SOR and conjugate gradient with 2D BVP, is there something wrong with the problem? Or will matrix be ill-conditioned?

The goal is to use a Laplace equation to solve: $$a(x,y)(u_{xx} + u_{yy}) = f(x,y)$$ with boundary condition $u=0$ on the boundary $x:[-1,1] , y:[-1,1]$. The problem is that we are supposed to work ...
0
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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
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1answer
75 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
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1answer
148 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
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2answers
76 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 ...
0
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1answer
49 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
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2answers
237 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
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1answer
94 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
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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
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2answers
120 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
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1answer
109 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
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1answer
97 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
86 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
142 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
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2answers
92 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
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3answers
200 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
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1answer
111 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
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0answers
62 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
125 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
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0answers
64 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
39 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 ...
4
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0answers
65 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
vote
2answers
41 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
55 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
77 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
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1answer
108 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
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1answer
56 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
142 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
80 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
176 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
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0answers
38 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
110 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
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0answers
31 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
109 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
67 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 ...