Lecture

Implicit Methods: Retrograde Euler

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In course

MATH-251(a): Numerical analysis

This course presents numerical methods to solve mathematical problems such as systems of linear and nonlinear equations, function approximation, integration and differentiation, and differential equations. ...

Description

This lecture covers the retrograde Euler method, an implicit numerical scheme used to solve equations. The method involves seeking solutions of equations at each time step, introducing partitions, and resolving equations. It also discusses the concept of convergence order and the importance of regularity in numerical schemes.

Instructors (4)

Fabio Nobile

Teaching & PhD PhD Students Veronica Montanaro, Matteo Raviola, Francesca Bettinelli, Eliott Van Dieren, Riccardo Saporiti, Giacomo Mossinelli, Noé Stauffer, Fabio Zoccolan Past EPFL PhD Students Diane Sylvie Guignard, Francesco Tesei, Eleonora Musharbash, Matthieu Claude Martin, Davide Pradovera, Sundar Subramaniam Ganesh, Eva Vidlicková, Juan Pablo Madrigal Cianci, Andrea Zanoni, Thomas Simon Spencer Trigo Trindade Past EPFL PhD Students as codirector Michele Pisaroni Courses Advanced analysis II - ve

Daniel Kressner

Pablo Antolin Sanchez

Teaching & PhD PhD Students Philipp Christoph Weder, Davor Dobrota Past EPFL PhD Students as codirector Ondine Chanon Courses Analysis III MATH-202(c) The course studies the fundamental concepts of vector analysis and Fourier-Laplace analysis with a view to their use in solving multidisciplinary problems in scientific engineering. Analysis III (for IC) MATH-203(d) The course studies the fundamental concepts of vector analysis and Fourier-Laplace analysis with a view to thei

Espen Sande

Official source

Ontological neighbourhood

Mathematics

Analysis: Numerical analysis

Related concepts (32)

Explicit and implicit methods

Explicit and implicit methods are approaches used in numerical analysis for obtaining numerical approximations to the solutions of time-dependent ordinary and partial differential equations, as is required in computer simulations of physical processes. Explicit methods calculate the state of a system at a later time from the state of the system at the current time, while implicit methods find a solution by solving an equation involving both the current state of the system and the later one.

Numerical stability

In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algorithms for solving ordinary and partial differential equations by discrete approximation. In numerical linear algebra, the principal concern is instabilities caused by proximity to singularities of various kinds, such as very small or nearly colliding eigenvalues.

Runge–Kutta methods

In numerical analysis, the Runge–Kutta methods (ˈrʊŋəˈkʊtɑː ) are a family of implicit and explicit iterative methods, which include the Euler method, used in temporal discretization for the approximate solutions of simultaneous nonlinear equations. These methods were developed around 1900 by the German mathematicians Carl Runge and Wilhelm Kutta. The most widely known member of the Runge–Kutta family is generally referred to as "RK4", the "classic Runge–Kutta method" or simply as "the Runge–Kutta method".

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.

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.

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