Lecture

Numerical Computing: Stability and Error Analysis

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In course

MATH-451: Numerical approximation of PDEs

The course is about the derivation, theoretical analysis and implementation of the finite element method for the numerical approximation of partial differential equations in one and two space dimensions. ...

Description

This lecture covers numerical computing, focusing on stability and error analysis. The instructor explains the concepts of stability, consistency, and convergence of numerical schemes, emphasizing the importance of error analysis. The lecture also delves into the truncation error and its impact on the accuracy of computations.

Instructor

Annalisa Buffa

Teaching & PhD PhD Students Gonzalo Bonilla Moreno, Philipp Christoph Weder, Yannis Dirk Voet, Fabio Marcinno, Ferhat Sindy, Francesco Sala, Markus Renoldner, Mohamed Ben Abdelouahab, Davor Dobrota Past EPFL PhD Students Luca Coradello, Ondine Chanon, Riccardo Puppi Courses Numerical approximation of PDEs MATH-451 The course is about the derivation, theoretical analysis and implementation of the finite element method for the numerical approximation of partial differential equations in on

Official source

Ontological neighbourhood

Mathematics

Analysis: Numerical analysis

Related concepts (22)

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.

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.

Round-off error

In computing, a roundoff error, also called rounding error, is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. Rounding errors are due to inexactness in the representation of real numbers and the arithmetic operations done with them. This is a form of quantization error.

Numerical integration

In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of definite integrals. The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for numerical integration, especially as applied to one-dimensional integrals.

Error analysis (mathematics)

In mathematics, error analysis is the study of kind and quantity of error, or uncertainty, that may be present in the solution to a problem. This issue is particularly prominent in applied areas such as numerical analysis and statistics. In numerical simulation or modeling of real systems, error analysis is concerned with the changes in the output of the model as the parameters to the model vary about a mean. For instance, in a system modeled as a function of two variables Error analysis deals with the propagation of the numerical errors in and (around mean values and ) to error in (around a mean ).

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