Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Finite Differences Method: Error Division and Schemes

Description

This lecture covers the finite differences method, focusing on error division and schemes for solving differential equations. The instructor explains how to divide the error and presents various schemes for different scenarios, emphasizing the importance of accuracy and efficiency in numerical methods.

In MOOC

Numerical Analysis for Engineers

Official source

Ontological neighbourhood

Mathematics

Analysis: Numerical analysis

Related concepts (25)

Ordinary differential equation

In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with partial differential equations which may be with respect to one independent variable. A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form where a_0(x), .

Differential equation

In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.

Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts.

Numerical methods for ordinary differential equations

Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals. Many differential equations cannot be solved exactly. For practical purposes, however – such as in engineering – a numeric approximation to the solution is often sufficient. The algorithms studied here can be used to compute such an approximation.

Linear differential equation

In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form where a0(x), ..., an(x) and b(x) are arbitrary differentiable functions that do not need to be linear, and y′, ..., y(n) are the successive derivatives of an unknown function y of the variable x. Such an equation is an ordinary differential equation (ODE).

Related lectures (41)

Numerical Methods: Boundary Value Problems

Explores boundary value problems, finite difference method, and Joule heating examples in 1D.

Numerical Differentiation: Part 1

Covers numerical differentiation, forward differences, Taylor's expansion, Big O notation, and error minimization.

Finite Difference Method: Approximating Derivatives and Equations

Introduces the finite difference method for approximating derivatives and solving differential equations in practical applications.

Numerical Methods: Euler Schemes

Focuses on Euler schemes for numerical approximation of forces and velocities.

Implicit Euler Scheme