What is Discretization: Definition and 30 Discussions

In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers. Dichotomization is the special case of discretization in which the number of discrete classes is 2, which can approximate a continuous variable as a binary variable (creating a dichotomy for modeling purposes, as in binary classification).
Discretization is also related to discrete mathematics, and is an important component of granular computing. In this context, discretization may also refer to modification of variable or category granularity, as when multiple discrete variables are aggregated or multiple discrete categories fused.
Whenever continuous data is discretized, there is always some amount of discretization error. The goal is to reduce the amount to a level considered negligible for the modeling purposes at hand.
The terms discretization and quantization often have the same denotation but not always identical connotations. (Specifically, the two terms share a semantic field.) The same is true of discretization error and quantization error.
Mathematical methods relating to discretization include the Euler–Maruyama method and the zero-order hold.

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1. I Numerically solving a non-local PDE

I have a PDE to solve numerically on the region ##x \in [0,1]## and ##t \in (0, \infty)##. It is of the form:$$\frac{\partial f(x,t)}{\partial t} = g(x,t) + \int_0^1 h(x, x') f(x', t) dx'$$The second term is the tricky part. The change in ##f(x,t)## at ##x## depends on the value ##f(x',t)## of...
2. A Discretization of diffusion equation of a fluid in movement

Hello, I want to model the thermal behaviour of a moving heat transfer fluid in 1D, with convective exchanges with the walls. I have obtained the following equation (1 on the figure). I have performed a second order spatial discretization with decentred schemes at the extremities (y = 0 and H)...
3. A Graph or lattice topology discretization

Mathematicians, I summon thee to help me identify which field deals with this stuff. I come here not as a physicist but as a sunday programmer trying to solve some numerical problems. I set out to model a lattice version of a smooth space. A discretization procedure not uncommon in physics, but...
4. A Discretization for a fourth-order PDE (and solution)

Hi. I have this PDE that governs an L x L plate (similar to the Poisson equation, it seems) with boundary conditions F = 0 and F" = 0 along the edges. I have successfully solved the problem by setting up an equality W = ∇2F then I solved the two PDEs simultaneously: W = ∇2F (boundary...
5. A PDE discretization for semi-infinite boundary?

Hi. Been a while since I logged in here, I missed this place. Anyway, I have a question (title). Is that even possible? Say for example I have the standard heat equation (PDE) subject to the boundary conditions: T(0,t) = To T(∞,t) = Ti And the initial condition: T(0,t) = Ti I am aware of how...
6. Engineering How Is Alpha Calculated in the Discretization of an RC Circuit?

Homework Statement [/B] Derive a discrete formula for an RC circuit for Vab[k] using the forward difference approximation. It should be of the form Vab[k + 1] = α Vab[k], and depend only on T, R, and C. For T = 0.076 s, R = 3 kΩ, and C = 10 mF, what is α? T is the period. The current is going...
7. What does the Navier-Stokes equation look like after time discretization?

Hi, I know the general form of the Navier Stokes Equation as follows. I am following a software paper of "Gerris flow solver written by Prof. S.Popinet" [Link:http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.374.5979&rep=rep1&type=pdf] and he mentions after time discretization he ends...
8. Discretize using a forward-Euler scheme

Homework Statement Consider the differential equation $$y'''-y''=u$$ Discretize (1) using a forward-Euler scheme with sampling period $$\Delta=1$$ and find the transfer function between u(k) and y(k) Homework Equations The Euler method is...
9. Investigating a Parabolic PDE algorithm

Homework Statement Hi - I'm on the last chapter of this book and am a bit stuck. I am given a very basic fortran program (code attached in the zip file) and asked to 'investigate its accuracy and stability, for various values of Δt and lattice spacings'. The program is an implementation of the...
10. MATLAB Transforming Complex Exponential to Discrete Vector Form

Hi, I want to transform a complex exponential with quadratic phase to discrete form, in other words to a vector form. can anyone help me with that? Thanks
11. Is the Discretization of this Differential Equation Accurate?

Hello, I am trying to solve the following differential equation in a tri-dimensional grid with grid spacing of hx , hy , and hz along the x, y, and z coordinates. $$\nabla\cdot(\epsilon\nabla\phi)=-4\pi\rho$$ Here, ε is a scalar quantity that is a function of x, y...
12. Finite difference discretization for systems of higher ODEs

How can I use finite difference to discretize a system of fourth order differential equations? for example: y(4)+5y(3)-2y''+3y'-y=0
13. Finite Differences-Semi discretization method on Heat Equation

Hi!, I'm working on a personal project: Solve the heat equation with the semi discretization method, using my own Mathematica's code, (W. Mathematica 9). The code: I'm having problems with the variable M (the number of steps). It works with M=1-5, but no further, I do not know what's going...
14. Discretization in cylindrical coordinates, unit thickness for azimuth?

I am setting up a numerical simulation from a 2D discretization of the heat equation in cylindrical coordinates. my spatial variables are radius (r), height (z), and azimuth (ø). The assumption is that there is no gradient along the azimuth direction (if temperature is T then dT/dø = 0)...

Hi, I would like to solve the steady-state incompressible Navier-Stokes equations by a spectral method. When I saw the classic primitive-variable finite element discretization of the time-dependent incompressible N-S, it turned out that the coefficient matrix of the derivatives of the unknowns...
16. Partial differential equation discretization. HELP D:

So figuratively, I'm trying to win a nuclear war with a stick. :smile: I did not take any course in PDEs, I just self-studied some of them, and now I'm toast. :smile: First, please feel free to hurl rocks at me if my simplification is incorrect...
17. Is Motion Possible in a Continuous Space?

I have quite a naive question, which doesn't really go deep into physics/mathematics.. :) Let's take seriously the idea that space is continuous. The questions is, how are we able to move in such a space? We know that in a continuous space (real numbers), between two points there are...
18. Derivative discretization with fixed boundary condition

Hi all, It may be a trivial question. But, if I have a PDE of variable u(x,t) -------------------------------- \dot{u} = f(u,\partial_x{u},..) with boundary condition : u(0,t) = u(L,t) =0. -------------------------------- Now I need to calculate \partial_x{u} for that can I define the...
19. Smoothed Particle Hydrodynamics discretization

Hi all, I want to use SPH discretization for my cooling rate but i don't know if its valid. So, this is the equation i want to discretize. L = (1/dT/dt) . ∫v(T) dT So, dT/dt = (1/L) . ∫v(T) dT dT/dt = (1/L). Ʃv(T) dT (with dT →0) The limits of the above integral are...
20. Optimal discretization and expansion order of arbitrary data

Hi all, I am trying to figure out 1) What to call my problem so I can better research the literature, and 2) see if anyone here knows of a solution. Essentially, I have a large set of f(x) vs x points (~20,000) which I need to split into subdomains in x, and within each subdomain...
21. Discretization of the divergence operator

I work with a grid-based code, this means that all of my quantities are defined on a mesh. I need to compute, for every point of the mesh the divergence of the velocity field. All I have is, for every cell of my mesh, the values of the 3-d velocity in his 26 neighbors. I call neighbors the...
22. CFD Discretization: Building a Higher Order Code Using Polynomial Fitting

Hi guys I was wondering if anyone on here could help me out. Essentially I am trying to build a higher order cfd code and struggling to work out where to start, I’m trying to build it using polynomial fitting so here’s my problem. Taking my domain as...
23. Kalman, White Noise, Sensor Specification, Discretization?

Hi. I have a few questions about sensor specifications and its implementation in a Kalman Filter and simulation of gyroscope/accelerometer output. Abbreviation used: d - discrete c - continuous Q1: From book: Aided Navigation - Farrell (you don't need the book to understand the...
24. Analysis of spatial discretization of a PDE

Hi everybody, I hope I am asking in the right forum. Let describe the problem as follows: I have a 1D heat equation. To solve it, I use finite-difference method to discretize the PDE and obtain a set of N ODEs. The larger N gives the better solution, i.e., the closer the solution to the...
25. Exploring Space Structure: Evidence & Implications of Discretization?

My QM is undergrad level and I don't keep up on recent breakthroughs. I'm curious what the status is on space structure. This article (Mecklenburg, 2011) uses the phrase "hidden substructure" http://prl.aps.org/abstract/PRL/v106/i11/e116803 is it similar to Zurek, 2001? "structure on...
26. Finite Difference Discretization of a Fourth Order Partial Differential Term

What is a finite-difference discretization for the partial differential term: \frac{\partial^4\phi}{\partial x^2\partial y^2} Thanks in advance.
27. MATLAB discretization of a 2D circular surface

1. Hi, I am trying to model the magnetic force between two cylindrical identical sized permanent magnets. I am using the Charge/Coulombian model to do this. This assumes that all of the "magnetic charge" is on the pole ends of the magnet, i.e. the flat surfaces at the top and bottom of each...
28. Discretization of Lz: Does m Have to Be Integer?

I just learned about L and Lz, I can accept the fact that L is discreted in quantum world, but it does not make sense at all for me that Lz too is bound in term of m multiple. I mean, if we choose an axis that is tilted just a little bit, then our value for Lz changes immideately and m is not an...
29. Discretization of the Poisson Equation across Heterointerface

Homework Statement Consider a 1D sample, such that for x < xb the semiconductor has a dielectric constant \varepsilon_{1}, and for x > xb has a dielectric constant \varepsilon_{2}. At the interface between the two semiconductor matierials (x = xb) there are no interface charges. Starting...
30. What Is Discretization in Temperature Change Analysis?

If I have a logged temperature change over time which makes up a plottet graph. I denote this change {{dT} \over {dt}} People have told me this can be solved using discretization, but I have no idea what that is. Apparently it is something like this: {{T_1 - T_2 } \over {\Delta...