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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Proper writing of IBVP PDE & Finite Difference Implementation

I've seen a few examples (see slide 4 ) of the 2D heat equation described as $$ \begin{cases} u_t(t,x,y) = \nabla^2 u(t,x,y), \quad t > 0, \quad (x,y) \in (0,L) \times (0,H), \\ u(t,0,y) = ...
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26 views

eigenvalues inequality finite differences

I have $x,y\in[0,1]^2$, $a\in[0,A]$ $t\in[0,T]$ and the mesh points $x_j = j \, \Delta x, j=0,\ldots,J$; $y_l = l \, \Delta y, l=0,\ldots,L$; $a_k = k\Delta a, k=0,...,K$ and $t_n = nh, n=0,\ldots,N$ ...
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1answer
66 views

What is this type of function called? How can I translate it to a different origin?

A factory produces 1 widget per week. A builder builds 1 factory each week. A construction firm trains 1 new builder each week. Partially-produced things do not produce anything. Starting with 1 firm, ...
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1answer
23 views

Having trouble with discretization and boundry value problems

I have the following homework question: Consider the boundary value problem $y''(x) + 5y'(x) − (2 + x)y(x) = e^x$ on $x ∈ (0, 2)$ with boundary conditions $3y(0) + y'(0) = 5$ and $y'(2) = 7$. ...
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1answer
26 views

How do I write the generic finite difference approx of f'(x) using Lagrange interpolating polynomial approximation?

I have the following homework problem: (10 points) Differentiation Formulas by Lagrange Interpolating Polynomials. (a) Write the generic finite difference approximation to f'(x) using the Lagrange ...
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5 views

What is the parameter of the Streamline-Diffusion Method?

Recently, I want to apply streamline-diffusion(SD) method to solve Burgers Equation. \begin{align} u_t + uu_x - \varepsilon u_{xx} &= 0,\\ u(0,t) = u(X,t) &= 0,\\ u(x,0) &= u_0(x),\ \ \ ...
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1answer
39 views

Expressing a function's value using finite differences

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ and let $x = (x_0, x_1, x_2, \dots)$ be a sequence of pairwise distinct real numbers. For every $n \in \{1, 2, \dots\}$ and every ordered $(n+1)$-tuple ...
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16 views

How to make a 9-point two-dimensional stencil for a elliptic operator?

I want use a finite difference schem to discretizate the elliptic operator: $$ \nabla \cdot \left( k(x, y) \nabla p\right), $$ where $k(x, y)$ a positive scalar function and $p$ is the unknow. We can ...
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19 views

Stability of a scheme for one way wave equation.

Multiplying the scheme \begin{equation*} \dfrac{v_m^{n+1} - v_m^n}{k} + \dfrac{a}{2}\left(\dfrac{v_{m+1}^{n+1} - v_m^{n+1}}{h} + \dfrac{v_m^n - v_{m-1}^n}{h}\right) = 0 \end{equation*} by $v_m^{n+1} + ...
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19 views

Show that $\Delta^{-1}\sin\alpha x = - \cos(\alpha x - \frac{α}{2})/(2\sin \frac{\alpha}{2}) + c(x)$ for fixed $\alpha\in \Bbb{R}$.

Show that $\Delta^{-1}\sin\alpha x = - \cos(\alpha x - \frac{α}{2})/\left(2\sin \frac{\alpha}{2}\right) + c(x)$ for fixed $\alpha \in \Bbb{R}$. I can find a few tables for anti-difference tables ...
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24 views

Finite Difference Method for Heat Equation with Neumann Boundary

I have read the book of Morton and Mayers. In chapter6, it said that the explicit finite difference scheme of a heat equation, $\frac{U^{n+1}_j-U^{n}_{j}}{\Delta ...
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20 views

Laplace equation in polar coordinates [closed]

Hy, I'm new in here but i have a question please help me. I want to solve a Laplace PDE in a polar coordinate system with finite difference method, but I have a problem with boundary conditions at r = ...
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2answers
36 views

Given this operator what is inverse operator?

Given operator $$\Delta_{sym}[f(x)]=\frac{f(x+\varepsilon)-f(x-\varepsilon)}{2\varepsilon}$$ what is inverse operator in terms of summations? For instance, given operator ...
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493 views

Has anybody ever considered “full derivative”?

When differentiating we usually take a limit and drop the infinitesimal terms. But what if not to drop anything? First, we extend the real numbers with an infinitesimal element $\varepsilon$ which ...
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24 views

How to transform $\frac{\partial^2 u}{\partial t^2}=\frac{\partial^3 u}{\partial x^3}$ into a system of first order PDE's and finite difference matrix

So I have this equation: $\frac{\partial^2 u}{\partial t^2}=\frac{\partial^3 u}{\partial x^3}$ and I need to transform it into a system of first order PDE's. I was thinking like this: ...
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1answer
22 views

In the finite difference formulas, how can we pick h to give a certain tolerance?

There's lots of questions on here about finite differences. In particular, picking the 'best' h value. But what if I want to find the biggest 'h' which bounds to a given tolerance? On first glance, I ...
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11 views

Finite Differences Scheme for Poisson Nernst Planck Equations

I want to solve numerically a simple version of Poisson Nernst Planck equations in 1-D on the domain $[0,\ell]$: $C_t=-j^c_x=(C_x +\rho\phi_x)_x$ $\rho_t=-j^\rho_x=(\rho_x +C\phi_x)_x$ ...
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19 views

Riccati Finite Difference Equation:

Question: Consider the finite difference equation in Zeilberger notation: $$D_{1,x} [y] = a_0(x) + a_1 (x) y + a_2 (x)y^2 $$ Which in functional form is: $$ y(x+1) - y(x) = a_0(x) + a_1 (x) y + ...
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1answer
18 views

Proof that strictly tri-diagonally dominant matrix has an inverse

We are given the following theorem of which we need only know the result. Theorem Suppose an $n\times n$ matrix $A= (a_{ij})$ is tri-diagonal with $a_{i,i-1}a_{i,i+1} \neq 0$, for each ...
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16 views

Finite Difference method for nested derivatives

I'm moderately experienced with finite difference methods, but I'm hoping that somebody has better intuition than I do. Suppose I have the following expression which I want to evaluate via finite ...
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1answer
42 views

Construct a symmetric Poisson matrix from $\nabla \cdot (\rho\nabla p)$

Let $\rho$ and $p$ be 2D scalar fields. How do I construct a symmetric Matrix $A$ so that is satisfies \begin{equation} Ap = \nabla \cdot (\rho\nabla p) \end{equation} in finite differences? I am ...
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18 views

Computing the CFL number for non-linear advection

Consider $u_t -(1-2t)u_x=\phi(t,x) $ with $u(x,0)=u_0(x)$ for integrable $\phi$. The solution for this PDE is given by $ u(x,t)= u_0(x+t+t^2) + \int_0^t \phi(x(z),z) dx$. For a consistent scheme ...
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1answer
19 views

Writing the implicit scheme using matrices

\A= \begin{bmatrix} 2 & -1 & 0 & . & . & . & 0 \\ -1 & 2 & -1 & . & . & . & . \\ 0 & . & . & . & . & . &. \\ ...
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1answer
62 views

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 ...
2
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1answer
77 views

Infinite sum involving ascending powers

I was working on a problem involving finite differences and came across the following sum as a general solution to a problem I was working with $$ 1 + x + \frac{1}{1 + a}x^2 + \frac{1}{1 + a} ...
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16 views

Finite differences

I don't know finite differences but I have reached a step in a proof where I need them. In particular, I need to know the $3k$th differences of $4\cdot 1^{3k}, 4\cdot 2^{3k},...,4\cdot (3k)^{3k}$. I ...
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54 views

Finite difference solution of steady-state diffusion equation with variable material properties

I'm trying to use a finite difference method to solve the steady-state neutron diffusion equation in a nuclear reactor: $$ D(x) \nabla^2 \phi(x) + \left( \frac{\nu(x)}{k} \Sigma_f(x) - \Sigma_a(x) ...
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44 views

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. $w_{k,j,m}=(1-\Delta t\mu_k)^msin(k\pi x_j)$ ...
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37 views

Laplace problem with elliptic BC - FINITE DIFFERENCES

I am experiencing some trouble in solving the problem $u-\Delta u = f$ with periodic boundary conditions. First question: Is the problem always solvable? If I am not mistaken $-\Delta u = f$ ...
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18 views

Order of accuracy of an scheme

How does one find order of accuracy for the following scheme (see picture below). The picture shows two equations instead of the typical one equation which confuses me a lot. You can read about the ...
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24 views

Irregular grid for finite differences PDE solution

I got a project in a class I'm taking. In this project, I need to solve fluid dynamics and heat equations. Up to here, the problem is not so complicated, however the chamber shape is the problem: as ...
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1answer
41 views

Calculus over integers for derivative/integral of factorial?

One usually introduces the Gamma function to define a derivative of the factorial. However couldn't one define a derivative over integers like $$ f'(n) = \frac{f(n+1) - f(n)}{1} ? $$ Such a discrete ...
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10 views

Discretization of Variable Coefficients

I'm trying to discretize something of the form $$ a(x)\frac{df}{dx}. $$ Now, with $x_{i}\equiv i \Delta x$ and $a_{i}\equiv a(x_{i})$ (similarly for f), this is what I think it should be: $$ ...
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48 views

Approximate Laplace Operator with Central Difference in Polar Coordinates

I'm trying to approximate the Laplace operator in polar coordinates with the central difference quotient and I know how to do this in cartesian coordinates, but with polar coordinates I just feel ...
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3answers
50 views

Finite differencing of the diffusion (heat) equation

I am attempting to code a problem for a meteorology class. Our initial equation was as follows: $$\tfrac{\partial u}{\partial t} = \nu \tfrac{\partial^2 u}{\partial x^2} (*)$$ We were then assigned ...
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1answer
33 views

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 ...
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15 views

Deriving solution of Poisson equation by considering special integral & finite differences

I know how to solve the Poisson Equation on the unit disc $\{ (x,y) \in \mathbb R^2 : x^2 + y^2 < 1 \}$ by separation of variables, and then rearranging to get the Poisson Kernel. But I am asked to ...
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43 views

Diffusion of a chemical species inside a Y-shaped tube

I'm trying to model diffusion of a chemical species X inside a Y-shaped tube, whose diameter (thickness) is constant everywhere. The diffusion constant of X is $D$ ($\mu$m$^2$/s), so the concentration ...
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1answer
16 views

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 ...
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36 views

How to improve stability of numerical solutions to partial differential equations

This is a quite general question, but I am working with a system of partial differential equations in two variables. There is one time direction $t$ and one spatial direction $z$ and the numerical ...
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68 views

Stability of Lax-Wendroff Approach for Advection Equation

The Problem: I am attempting to solve the following problem in 1D over a periodic region: "In one dimension, the mass density $\rho$ is advected with velocity $v$, so that it follows the equation: ...
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20 views

Discretisation of a product of two functions

Suppose I have two functions, $f(x,t)$ and $g(x,t)$, and for an upwind scheme I want to use the quantity $\partial_x (fg)$ to solve the advection equation $$ \frac{\partial f}{\partial t} + ...
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36 views

Finite difference scheme for the continuity equation

I am currently trying to solve a system of PDE's numerically, one being the equation; $$ (1)\quad \partial\rho/\partial t + \partial(\rho v)/\partial x = 0 $$ I have been reading up on ...
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340 views

Generalisation of $(n+3)^2-(n+2)^2-(n+1)^2+n^2=4$

After seeing the neat little identity $(n+3)^2-(n+2)^2-(n+1)^2+n^2=4$ somewhere on MSE, I tried generalising this to higher consecutive powers in the form $\sum_{k=0}^a\epsilon_k(n+k)^p=C$, where $C$ ...
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17 views

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 ...
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126 views

Numerical scheme to 1D advection equation

I am trying to numerically solve a system of equations which model the early universe in 1D. The equations I am stuck on are; $$ (1)\quad \partial\rho/\partial t + \partial(\rho v)/\partial x = 0 ...
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39 views

First order and second order derivative backwards approximation on non uniform grids

Assume we have a function f(x) and we have samples coming in, except they are not at uniform intervals. (e.g. you have f(x) at x = 0.1, 0.2, 0.7, 1.9, 2.8... etc). What is the correct way to compute ...
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29 views

Determine the order of dissipation for a finite difference scheme

Consider the PDE $u_t + au_x = f$ and the finite difference scheme: $$ \frac{3v_m^{n+1} - 4v_m^n + v_m^{n-1}}{2k} + a\frac{v_{m+1}^{n+1} - v_{m-1}^{n+1}}{2h} = f_m^{n+1} $$ I need to determine the ...
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286 views

Applying Central Difference (Finite Difference Method) in MATLAB

I was given a rather complicated few problems to solve in MATLAB using the central difference method, and I'd like some help figuring out how to translate this into code. The goal is to discretize ...
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1answer
43 views

Matlab solution for non-homogenous heat equation using finite differences

Given the following PDE (non-homogenous heat equation): $$ u_t(x,t) = c^2u_{xx}(x,t) + f(x,t) $$ $$ u(0,t) = u(l,t) = 0 $$ $$ u(x,0) = g(x) $$ $$ 0 < x < l ; t > 0 ; c > 0 $$ I wrote the ...