What is Finite difference: Definition and 144 Discussions

A finite difference is a mathematical expression of the form f (x + b) − f (x + a). If a finite difference is divided by b − a, one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, especially boundary value problems.
Certain recurrence relations can be written as difference equations by replacing iteration notation with finite differences.
Today, the term "finite difference" is often taken as synonymous with finite difference approximations of derivatives, especially in the context of numerical methods. Finite difference approximations are finite difference quotients in the terminology employed above.
Finite differences were introduced by Brook Taylor in 1715 and have also been studied as abstract self-standing mathematical objects in works by George Boole (1860), L. M. Milne-Thomson (1933), and Károly Jordan (1939). Finite differences trace their origins back to one of Jost Bürgi's algorithms (c. 1592) and work by others including Isaac Newton. The formal calculus of finite differences can be viewed as an alternative to the calculus of infinitesimals.

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  1. Tertius

    Correct Setup for Finite Difference to Calculate Quantum Tunneling

    I thought I solved the problem in answering my own post a few days ago, but the tunneling probability vs. energy trend is clearly wrong. I've remade the post because I have totally changed my approach and need a better understanding of the boundary setup. Overall description: a plane wave...
  2. person123

    My Python Library For Finite Difference Method

    I recently made a Python library for modelling (very basic) finite difference problems. The Github readme goes into details of what it does and how it works, and I put together a Google Colab with some examples (diffusion, advection, water wave refraction) with interactive visuals. I'd love to...
  3. yashabyadav

    A Need help implementing finite difference method for BPM

    I am trying to apply finite difference scheme for Beam propagation method by following this paper. I was wondering if anyone can share their code if they have implemented this method. I can share my code which is not working as expected and can get some insights if possible.
  4. F

    A Finite difference scheme for y'(t)=a*y(t)

    It seems no finite difference scheme is stable for a>0, dt>0, correct?
  5. F

    I Finite difference scheme for u'=u^2, u0=1, 0<t<2

    u'(t)=u(t)^2, u(0)=1, 0<t<2 What finite difference scheme can overcome the difficulty when t->1+ and help the solution jump to the negative branch?
  6. F

    I Conservation of finite difference for vibration equations

    Let's discuss whether the energy under a finite difference (FD) scheme is conserved. Take the simplest vibration eq mx''+kx=0, which one will use a FD scheme to solve. The energy is mx'^2/2+kx^2/2. Whether the energy is conserved doesn't depend on the FD scheme for the ODE but upon the FD scheme...
  7. T

    What are the limitations of the Explicit Finite Difference Model?

    Homework Statement:: Discuss the limitation of the Explicit Finite Difference Model. Relevant Equations:: no formula Hello there, I have to discuss the limitations of using the Explicit Finite Difference model to calculate a 2D Heat Diffusion through an aluminium place, however, I really...
  8. D

    A Numerical Hartree Fock with Finite Difference Matrices for Helium

    Here is the paper again: https://www.mdpi.com/2218-2004/6/2/22?type=check_update&version=2#related_content For a class project I need to calculate the energy levels of atoms using the Hartree Fock method as presented in this paper which essentially brute forces the calculation using finite...
  9. L

    Finite difference method for the diffusion-advection equation

    So for my scheme I obtained ##\frac{\mu}{h^2} U_{p}+(\frac{v_{1}}{2 h}-\frac{\mu}{h^2})U_{E}+(\frac{v_{2}}{2 h} - \frac{\mu}{h^2})U_{N} - (\frac{v_{1}}{2 h}+\frac{\mu}{h^2})U_{W} - (\frac{v_{2}}{2 h} + \frac{\mu}{h^2})U_{N} + \tau = f## however I am not sure this is correct. I am quite new to...
  10. patrykh18

    I Numerical solution to the Schrodinger eqn. using Finite Difference Method

    As part of my project I was asked to use the finite difference method to solve Schrodinger equation. I see how you can turn it into a matrix equation, but I don't know how to solve it if the energy eigenvalues are unknown. Are there any recommended methods I can use to determine those...
  11. T

    A Lecture notes on Finite Difference Methods

    I have lately been working with Numerical Analysis and I am using Finite Difference Methods for Ordinary and Partial Differential Equations by Randall J. LeVeque. It was recommended to me by a friend of mine (physicist) https://epubs.siam.org/doi/book/10.1137/1.9780898717839?mobileUi=0&...
  12. yecko

    Conduction - finite difference method

    Homework Statement Homework Equations Finite difference method The Attempt at a Solution I have tried two different approaches, but still i am wrong in the question. Can anyone guide me how to attempt this question? Thank you
  13. HumanistEngineer

    A Finite Difference Scheme - Plume Entrainment

    I am working on modelling of a heat storage tank. More specifically, I need to find out the transient temperature variation through the tank height. My question is about the plume entrainment considering the case that hot water is injected from the bottom part of a heat storage tank. This...
  14. P

    Finite difference conjugate heat xfer: internal flow

    Hi. I have been trying to solve this problem that has been keeping me up at night for a coupe weeks at least. If anyone can help me, I would be greatly appreciated. Hot air enters a cylindrical duct. The duct has some R-value and radiation and convection is being accounted for on the outside...
  15. mertcan

    A Runge Kutta finite difference of differential equations

    Hi PF, initially I would like you to focus on that link https://books.google.com.tr/books?id=Dkp6CwAAQBAJ&pg=PA389&lpg=PA389&dq=runge+kutta+method++is+tvd+proof&source=bl&ots=47ULQDVwcC&sig=e2zjdnXENJ7WxBbrf6hXkSouvLI&hl=tr&sa=X&ved=0ahUKEwjU5Z2XsbXZAhUMCMAKHWpnATQ4ChDoAQhKMAQ#v=onepage&q=runge...
  16. T

    Finite difference method; finding the 'Stencil'

    Homework Statement Determine the Finite Difference Method stencil for approximating a second derivative u''(x) at a discrete set of nodes with maximum accuracy for stencil of sizes (0,4) (off-centered). My questions: I think I am able to answer the question I am just not sure about what is...
  17. L

    A Convergence order of central finite difference scheme

    For example, when we solve simple 1D Poisson equation by finite difference method, why three point central difference scheme on uniform grid (attached image) is second order method for solution convergence? I understand why approximation of first derivative is second order (and that second...
  18. M

    A Is the discretization for the PDE correct?

    Hi PF! I'm trying to finite difference a FTCS PDE $$h_t+\partial_x\left[h^3(h_{xxx}+(Kf'(h)-G)h_x\right] = 0$$ where ##K## and ##G## are constants. ##f'(h) = -(n(h^*/h)^n-m(h^*/h)^m)/h##. Boundary conditions are ##h_x=h_{xxx}=0## at both ends of the ##x## domain (however long you want to make...
  19. M

    A Better way to find Finite Difference

    Given a function ##\psi## having ##N## components, how would you (as fast as possible) construct ##\psi'## also having ##N## components? I thought about taking a forward finite difference approach on the first ##N-1## components of ##\psi## to generate ##\psi'## and then a backward finite...
  20. H

    A Finite difference of fourth order partial differential

    What is a finite difference discretization for the fourth-order partial differential terms \frac{\partial u}{\partial x}k\frac{\partial u}{\partial x}\frac{\partial u}{\partial x}k(x,y)\frac{\partial u}{\partial x} and \frac{\partial u}{\partial x}k(x,y) \frac{\partial u}{\partial y}...
  21. maistral

    A Finite difference formulation ideas (journal verification)

    Hi. I am trying to simulate this paper since apparently I have a lot of time. Scrolling down to the last page, he simulated a transient 2D heat conduction plate with composite slabs on it. Darkest one is copper, lighter one is steel, lightest one is glass. If you look closely, the authors said...
  22. maistral

    A 2D Finite Difference formulation in polar coordinates.

    So I have this PDE: d2T/dr2 + 1/r dT/dr + d2T/dθ2 = 0. How do I implement dT/dr || [r = 0] = 0? Also, what should I do about 1/r? This is actually the first time I am going to attack FDF in polar/cylindrical coordinates. I can finite-difference the base equation fairly decently; I am just...
  23. A

    A 2D Finite Difference Convergence Rate Issues

    I have completed a 2D finite difference code in MATLAB that has a domain of (0,1)x(0,1) and has Dirichlet Boundary Conditions of value zero along the boundary. I get convergence rates of 2 for second order and 4 for fourth order. My issue now is that I'm now wanting to change the domain to a...
  24. Riverbirdy

    Finite difference Method of Wave Equation

    Hi, Physics forum! Just a little push of my doubts I hope somebody could help me with my confusion of one of our home works. I know that all boundary conditions are zero. My doubt is how do I interpret (x,y,0)=0.01 source in the figure? Where is it located in the grid. I am hoping someone...
  25. M

    Heat Transfer, Finite difference, Curved geometry

    Homework Statement Homework Equations I could really use a push on how to approach this problem. My primary problem is it asks for the heat flux into the page, which makes no sense to me as that is the z direction and this is in the x/y plane. If anyone could explain this problem and maybe...
  26. A

    I Solving Poisson's Equation Using Finite Difference

    I am using the Finite Difference Method to solve Poisson's equation \frac{\partial \phi}{\partial z^2} = \frac{\rho}{\epsilon} To do it is discretized according to the Finite Difference Approximation of the second order derivative yielding the following set of equations for each grid point...
  27. F

    Finite difference method derivation PDE

    Homework Statement Which algebraic expressions must be solved when you use finite difference approximation to solve the following Possion equation inside of the square : $$U_{xx} + U_{yy}=F(x,y)$$[/B] $$0<x<1$$ $$0<y<1$$ Boundary condition $$U(x,y)=G(x,y)$$ Homework Equations Central...
  28. apgt512

    Time dependent heat equation

    Homework Statement Solve the time dependent 1D heat equation using the Crank-Nicolson method. The conditions are a interval of length L=1, initial distribution of temperature is u(x,0) = 2-1.5x+sin(pi*x) and the temperature in the ends of the interval are u(0,t) = 2; u(1,t) = 0.5. Homework...
  29. J

    I Solving the 3D Poisson Equation Using Finite Difference/Volume

    Hi, I'm attempting to solve the 3D poisson equation ∇ ⋅ [ ε(r) ∇u ] = -ρ(r) Using a finite difference scheme. The scheme is simple to implement in 3D when ε(r) is constant, and I have found an algorithm that solves for a non-constant ε(r) in 2D. But I am having trouble finding an algorithm...
  30. A

    I Finite difference method for Schrödinger equation

    Suppose I want to solve the time-independent Schrödinger equation (ħ2/2m ∂2/∂x2 + V)ψ = Eψ using a numerical approach. I then discretize the equation on a lattice of N points such that x=(x1,x2,...,xN) etc. Finally I approximate the second order derivative with the well known central difference...
  31. M

    I Difference Equation Boundary Conditions0.

    This question is inspired by Gilbert Strang's Course on Computational Science and Engineering, MIT 18.085. Consider the three matrices Fixed-Fixed $$K=\begin{bmatrix} 2 &-1 & 0 &0 \\ -1&2 & -1 &0 \\ 0 & -1 &2 & -1 \\ 0 & 0 & -1 & 2 \\ \end{bmatrix} $$ Free-Fixed $$T=\begin{bmatrix} 1 &-1 & 0 &0...
  32. T

    Python Using backward vs central finite difference approximation

    I am solving the simple 2nd-order wave equation: $$ \frac {\partial ^2 E}{\partial t^2} = c^2 \frac {\partial ^2 E}{\partial z^2} $$ Over a domain of (in SI units): ## z = [0,L=10]##m, ##t = [0,t_{max} = 10]##s and boundary/initial conditions: $$ E(z=0) = E(z=L) = 0 $$ $$ E(t=0) =...
  33. A

    I Finite difference Hamiltonian

    Suppose I am given some 1D Hamiltonian: H = ħ2/2m d2/dx2 + V(x) (1) Which I want to solve on the interval [0,L]. I think most of you are familiar with the standard approach of discretizing the interval [0,L] in N pieces and using the finite difference formulas for V and the...
  34. jdawg

    Using finite difference to solve DE

    Homework Statement d2T/dx2 = 5*(dT/dx) - 0.1*x = 0 T(0) = 50 T(10) = 400 (Δx) = 2I've figured out how to do these problems when Δx = 1, but when it equals any other number it goes wrong. I know you start by plugging in the algebraic approximations for the differential elements, I think maybe my...
  35. F

    MATLAB Finite difference numerical integration or ode45?

    I'm trying to numerically solve the time dependent Schrödinger equation and I've been told that the best approach is to numerically integrate using a finite difference method, however I don't understand why I couldn't just use ode45 to solve it?! Is the finite difference (interpolation) method...
  36. A

    B Finite difference problem

    I found this part of forum the most relevant to this theme so excuse me if I missed. This year I'm doing a high school summer project related to quantum mechanics. Anyway I'm using finite difference method to solve Schrodinger equation. Before starting to work on a project I decided to get some...
  37. S

    A Finite Difference solver for 2D Elasticity equations

    Hi I've been trying to get a simple solution to the 2D Navier-Lame equations using finite difference on a rectangular grid. I want to see the displacements, u and v, when a simple deformation is imposed - e.g. top boundary is displaced by 10%. The equations are as follows: \begin{eqnarray*}...
  38. evinda

    MHB How is the Finite Difference Method Applied in Wave Approximation?

    Hello! (Wave) We consider the finite difference method for the approximation $\left\{\begin{matrix} -u''(x)+q(x)u(x)=f(x)\\ u'(a)=u'(b)=0 \end{matrix}\right.$ and let $K$ be the $(N+2) \times (N+2)$ matrix of the method. Let $v \in \mathbb{R}^{N+2}, v=\begin{pmatrix} v_0\\ v_1\\ \dots\\...
  39. I

    Finite difference method to solve first-order, multivariable

    I'm trying to replicate the model presented in this [paper](http://www.sciencedirect.com/science/article/pii/S1359431103000474), which is basically to model heat and mass transfer along a one-dimensional duct. There are four characteristic equations for this problem : Momentum conservation...
  40. evinda

    MHB How to Solve a Boundary Value Problem Using the Finite Difference Method?

    Hello! (Wave) I want to solve numerically the following boundary value problem: $\left\{\begin{matrix} -u''+qu=f & , x \in [a,b]\\ -u'(a)+d_1 u(a)=0 & \\ u'(b)+d_2 u(b)=0 & \end{matrix}\right.$ where $q(x) \geq 0 \forall x \in [a,b], d_1, d_2 \geq 0$. We consider the uniform partition of...
  41. A

    Finite difference Schrodinger equation

    I am simulating electrons inside a cylindrical well like the one shown on the first figure. My current work has been on solving the Schrodinger equation numerically for the above potential and then finding corrections to the solution such that it is consistent with Poissons equation. To do so...
  42. N

    Finite Difference Method for non-square grid

    Hi, I have written some codes for the finite difference solution of diffusion equation (\frac{\partial c}{\partial t}= D {\nabla^2 c}, where c is the species concentration and D is the diffusion coefficient) as follows: DO k= 1, tsteps+1 DO i = 2, zsteps DO j = 2, rsteps...
  43. A

    Finite difference Poisson's equation

    I am trying to solve the following eigenvalue differential equation numerically: ∇2ψ = Eψ , where the coordinate system is polar coordinates and the boundary condition is ψ(R,Φ)=0, where R is the radius of the disk i am working on. To solve it I am using a finite difference scheme, but there...
  44. M

    Finite Difference Approach for a Moving Boundary Problem

    Hi PF! I was wondering if anyone could help me with a finite difference question? The problem I am doing is a 1-D space and time problem, so ##z## (space variable, from left to right) and ##t## (time) are my independent variables and my dependent variable is ##h##, the height, governed by a PDE...
  45. P

    Heat equation order of accuracy (Crank-Nicolson)

    Hi, Let's consider the heat equation as \frac{\partial T}{\partial t}=\alpha \frac{{{\partial }^{2}}T}{\partial {{x}^{2}}} In order to have a second accuracy system, one can use the Crank-Nicolson method as \frac{{{\partial }^{2}}T}{\partial {{x}^{2}}}\approx \frac{1}{2}\left(...
  46. Feldman Sia

    Backward Finite Difference Heat Equation error

    I had these code in this forum but comes out error as below, any suggestion? Error 1 error C4430: missing type specifier - int assumed. Note: C++ does not support default-int c:\users\username\documents\visual studio 2010\projects\fdm 001\fdm 001\explicit 001.cpp 27 Error 2...
  47. S

    MATLAB Can You Solve This Finite Difference Equation Using Matlab?

    hi dear i have a question. i have equation (1/α ) dT/dt =d2T/dr2 +1/r dT/dr +d2T/dz T=T(r,z) T(Ri,z)=Ti T(Ro,z)=To T(r,0)=To dT(r,L)/dz =0 by finite difference method O(h^3) and this question's MATLAB program. is there anyone who can do it ? it is very important for me tnx
  48. B

    Unsteady vorticity transport equation: codes available?

    I would like to reproduce results from a much older code to test a new one. I only have the old code's results in the form of plots, not data, but I need data. The older code solves the unsteady vorticity transport equation in 2D with a constant kinematic viscosity term. I'm interested in 2-D...
  49. K

    Poisson PDE in polar coordinates with FDM

    I want to solve a Laplace PDE in a polar coordinate system with finite difference method. and the boundary conditions: Here that I found in the internet: and the analytical result is: The question is how its works? Can I give an example or itd?Thanks
  50. B

    RHS of Laplace's Equation is f(u(x,y))

    Homework Statement I need to (computationally) solve the following linear elliptic problem for the function u(x,y): \Delta u(x,y) = u_{x,x} + u_{y,y} = k u(x,y) on the domain \Omega = [0,1]\times[0,1] with u(x,y) = 1 at all points on the boundary.Homework Equations [/B] I know that I...