Questions tagged [finite-differences]

A method in numerical analysis which consists of approximating the derivatives of a solution of an ordinary or a partial differential equation. This leads to the solution of a linear system.

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Writing the implicit scheme using matrices

\A= \begin{bmatrix} 2 & -1 & 0 & . & . & . & 0 \\ -1 & 2 & -1 & . & . & . & . \\ 0 & . & . & . & . & . &. \\ . & . & . & . & . & . & 0 \\ . & . & . & . & -2 & 2 & -1 \\ 0 & . & . & . & 0 & -1 & 2 \end{bmatrix} So the…
David
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Closed form solution to $\frac{1}{a-1}= \log a$

I want to find a function that satisfies $$\Delta [f(x)]=f'[x]$$ Obviously the solution is the exponential function $f(x)=a^x$ with $a$ in between $2$ and $e$ because $\Delta[2^x]=2^x$ and $(e^x)'=e^x$. Thus the base should satisfy the…
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Implementing discrete Poisson equation wtih Neumann boundary condition

I understand how to implement a discrete 2D poisson solution with Dirchlet boundary conditions. Using http://en.wikipedia.org/wiki/Discrete_Poisson_equation#On_a_two-dimensional_rectangular_grid , you just replace any of the $u_{ij}$ that are on the…
David Doria
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Solution to the heat equation using a finite difference scheme

I have used a difference scheme and Fourier Analysis to find an expression for the solution to the heat Equation. My problem is plotting my solution. $w_{k,j,m}=(1-\Delta t\mu_k)^msin(k\pi x_j)$ $\mu_k=\frac{4}{(\Delta x)^2}sin^2(k\pi \Delta…
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In 1D FDTD, Do we expect power oscillations at PEC reclecting boundary?

Basically I am trying to check if my 1D FDTD code works fine and how to write quantity that is conserved all the way. In 1D FDTD should we expect that the power is conserved when the pulse is being reflected off a perfect electric conductor boundary…
Anonymous
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Question about Convergence Definition for Finite Difference Scheme

I have a question about the Convergence Definition for Finite Difference Scheme. The definition is given by Convergence: for one-step schemes approximating a IBVP to be convergent we compare $U(x,t)$ (true solution) and $U^n_m$ (numerical…
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In which cases iterative equations can be reduced to finite-difference equations?

In which cases iterative equations can be reduced to finite-difference equations and when they can't?
Anixx
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How to choose a proper numerical optimisation method

Given a problem in numerical analysis in finance/econometrics, how to decide whther to choose Monte Carlo, Newton Raphson , Finite Difference , Gradient descent? I had this silly misconception that all these methods pretty much do the same thing by…
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Fastest numerical way to solve steady-state reaction-diffusion equation

I have a reaction-diffusion equation in 2 dimensions of the typical form: $\frac{\partial u}{\partial t} = D\nabla^2u - \Phi(u(x))$ I want to stress that $\Phi(u(x))$, is not a constant, but depends on the location, where it can either be a source…
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Finite difference numerical differentiation

I needed to find an O(h2) method to find f'''(x). Using Taylor expansions, I found: $$f'''(x)=\frac{f(x+2h)-2f(x+h)-2f(x-h)+f(x-2h))}{2h^3} + O(h^2)$$. However, I have also found that: $$f'''(x)=\frac{f(x-3h)-6f(x-2h)+12f(x-h)-10f(x)+3f(x+h)}{2h^3}…
AccioHogwarts
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ODE and recurrence relation

I am trying to understand the following claim (I came across it while reading a paper): Consider the map (Standard/Arnold map) $T_{k}:(x,y)\mapsto(x+y+kf(x), y+kf(x))$, with $x\in\mathbb{R}/2\pi\mathbb{Z}$, $y\in\mathbb{R}$, $f$ is a periodic, real…
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Square pulse test of Upwind Finite Differences

I`m analyzing the numerical methods for the 1D convection equation for stability, consistency, and accuracy. I want to implement the methods to test on a square pulse. I know that I have to compare the numerical solution with the exact one using…
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Derivative of $\sqrt{x}$ Using Symmetric Derivative Formula

How do you find the derivative of $\sqrt{x}$ using the symmetric derivative formula? $$ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x-h)}{2h}. $$ I got stuck on trying to remove the h from the denominator.
Nicolas
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Finite differences Wrong number

The list is 1 2 4 8 16 26 42 64 93. A number in this list is wrong. Finite differences propagation error: \begin{array}{|c|c|c|c|} \hline f(x)& \Delta^1{f}& \Delta^2{f}& \Delta^3{f} \\ \hline f_{0}& \Delta^1{f_{0}}& \Delta^2{f_{0}}&…
valer
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Differential equation $\Delta^k f(x)$.

Let $f(x)$ be a function of real variable and let $\Delta f$ be the function $\Delta f=f(x+1)-f(x)$. For $k>1$, put $\Delta^k f=\Delta(\Delta^{k-1}f)$. Then $\Delta^k f(x)$ equals: $$\text{A) }\sum_{j=0}^{k} (-1)^{j} \binom{k}{j}f(x+j)$$ $$\text{B)…
Idkwoman
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