Concept

Numerical methods for partial differential equations

Numerical methods for partial differential equations is the branch of numerical analysis that studies the numerical solution of partial differential equations (PDEs). In principle, specialized methods for hyperbolic, parabolic or elliptic partial differential equations exist. Finite difference method In this method, functions are represented by their values at certain grid points and derivatives are approximated through differences in these values. Method of lines The method of lines (MOL, NMOL, NUMOL) is a technique for solving partial differential equations (PDEs) in which all dimensions except one are discretized. MOL allows standard, general-purpose methods and software, developed for the numerical integration of ordinary differential equations (ODEs) and differential algebraic equations (DAEs), to be used. A large number of integration routines have been developed over the years in many different programming languages, and some have been published as open source resources. The method of lines most often refers to the construction or analysis of numerical methods for partial differential equations that proceeds by first discretizing the spatial derivatives only and leaving the time variable continuous. This leads to a system of ordinary differential equations to which a numerical method for initial value ordinary equations can be applied. The method of lines in this context dates back to at least the early 1960s. Finite element method The finite element method (FEM) is a numerical technique for finding approximate solutions to boundary value problems for differential equations. It uses variational methods (the calculus of variations) to minimize an error function and produce a stable solution. Analogous to the idea that connecting many tiny straight lines can approximate a larger circle, FEM encompasses all the methods for connecting many simple element equations over many small subdomains, named finite elements, to approximate a more complex equation over a larger domain.

Official source

https://en.wikipedia.org/wiki/Numerical_methods_for_partial_differential_equations

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related courses (32)

PHYS-443: Physics of nuclear reactors

In this course, one acquires an understanding of the basic neutronics interactions occurring in a nuclear fission reactor as well as the conditions for establishing and controlling a nuclear chain reaction. ...

MATH-251(c): Numerical analysis

Le cours présente des méthodes numériques pour la résolution de problèmes mathématiques comme des systèmes d'équations linéaires ou non linéaires, l'approximation de fonctions, l'intégration et la dérivation, et les équations différentielles. ...

MATH-507: Scientific machine learning

This course covers topics of scientific machine learning. Part I focuses on sequential-in-time training of nonlinear parametrizations for numerically solving partial differential equations. Part II treats generative modeling of physical processes. This is a project-based course.

Related lectures (32)

Finite Element Method: Basics and Applications

Introduces the Finite Element Method for solving PDEs and demonstrates its application through examples and Comsol Multiphysics.

Numerical Methods for Boundary Value Problems

Covers numerical methods for solving boundary value problems using finite difference, FFT, and finite element methods.

Numerical Methods: Boundary Value Problems

Covers numerical methods for solving boundary value problems, including applications with the Fast Fourier transform (FFT) and de-noising data.

Related publications (32)

The semi-explicit nonsmooth Newmark-β time integrator for robust unilateral contact in dynamic fragmentation simulations

Numerical simulations of solids undergoing dynamic fragmentation, a problem characterized by dynamic fracture and dense contacts, require accurately capturing the transition from a solid continuum to a collection of interacting fragments. We use the finite-element method with the extrinsic cohesive zone model for fracture. For contact, conventional penalty-based methods often exhibit numerical instabilities in dynamic collision-rich settings. To address this, we adapt and validate a novel semi-explicit time

2027

High-performance coupling for monolithic microreactor analysis with preCICE

The simulation of monolithic microreactors poses unique challenges, which are typically addressed by the use of specialised codes and solvers that are tightly coupled with one another. Various numerical methods and discretisation schemes may be employed in resolving the involved physics. This demands a flexible and high-performance coupling framework that can integrate diverse solvers. Here, we employ the preCICE library for external coupling, verifying our implementation for microreactor analysis involving

Elsevier BV2026

A new kinetic-MHD code based on the Case-Van Kampen approach for studying magnetic fusion relevant instabilities

Kinetic effects of bulk and fast particles in tokamaks can significantly impact the stability predictions of Magnetohydrodynamic (MHD) modes. Hybrid kinetic-MHD models are typically used to estimate these effects. Many such existing codes compute the kinetic effects by solving a reduced drift-kinetic or gyrokinetic equation semi-analytically, which often involves approximations limiting the type of orbits that can be accurately reproduced. Others solve particle orbits as a time evolution problem, requiring

EPFL2026

Graph Chat

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.

Official source

https://en.wikipedia.org/wiki/Numerical_methods_for_partial_differential_equations

About this result

Related courses (32)

PHYS-443: Physics of nuclear reactors

MATH-251(c): Numerical analysis

MATH-507: Scientific machine learning

Related lectures (32)

Finite Element Method: Basics and Applications

Introduces the Finite Element Method for solving PDEs and demonstrates its application through examples and Comsol Multiphysics.

Numerical Methods for Boundary Value Problems

Covers numerical methods for solving boundary value problems using finite difference, FFT, and finite element methods.

Numerical Methods: Boundary Value Problems

Covers numerical methods for solving boundary value problems, including applications with the Fast Fourier transform (FFT) and de-noising data.

Related publications (32)

The semi-explicit nonsmooth Newmark-β time integrator for robust unilateral contact in dynamic fragmentation simulations

2027

High-performance coupling for monolithic microreactor analysis with preCICE

Elsevier BV2026

A new kinetic-MHD code based on the Case-Van Kampen approach for studying magnetic fusion relevant instabilities

EPFL2026

Related people (32)

Jean-François Molinari

Professor J.F. Molinari is the director of the Computational Solid Mechanics Laboratory (http://lsms.epfl.ch) at EPFL, Switzerland. He holds an appointment in the Civil Engineering institute, which he directed from 2013 to 2017, and a joint appointment in the Materials Science institute. He started his tenure at EPFL in 2007, and was promoted to Full Professor in 2012. He is currently an elected member of the Research Council of the Swiss National Science Foundation in Division 2 (Mathematics, Natural and Engineering Sciences), and co editor in chief of the journal Mechanics of Materials. J.F. Molinari graduated from Caltech, USA, in 2001, with a M.S. and Ph.D. in Aeronautics. He held professorships in several countries besides Switzerland, including the United States with a position in Mechanical Engineering at the Johns Hopkins University (2000-2006), and France at Ecole Normale Supérieure Cachan in Mechanics (2005-2007), as well as a Teaching Associate position at the Ecole Polytechnique de Paris (2006-2009). The work conducted by Prof. Molinari and his collaborators takes place at the frontier between traditional disciplines and covers several length scales from atomistic to macroscopic scales. Over the years, Professor Molinari and his group have been developing novel multiscale approaches for a seamless coupling across scales. The activities of the laboratory span the domains of damage mechanics of materials and structures, nano- and microstructural mechanical properties, and tribology.

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

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

Farhad Rachidi-Haeri

Farhad Rachidi (IEEE Fellow, EMP Fellow, Electromagnetics Academy Fellow) was born in Geneva in 1962. He received the M.S. degree in electrical engineering and the Ph.D. degree from the Swiss Federal Institute of Technology, Lausanne, in 1986 and 1991 respectively. He worked at the Power Systems Laboratory of the same institute until 1996 and had several short stays at the University of Florida and the NASA Kennedy Space Center. In 1997, he joined the Lightning Research Laboratory of the University of Toronto in Canada and from April 1998 until September 1999, he was with Montena EMC in Switzerland. He is currently a titular professor and the head of the EMC Laboratory at the Swiss Federal Institute of Technology, Lausanne, Switzerland. His research interests concern electromagnetic compatibility, lightning electromagnetics and electromagnetic field interactions with transmission lines. Dr. Rachidi is currently a member of the Advisory Board of the IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY and the President of the Swiss National Committee of the International Union of Radio Science. He has received numerous awards including the 2005 IEEE EMC Technical Achievement Award, the 2005 CIGRE Technical Committee Award, the 2006 Blondel Medal from the French Association of Electrical Engineering, Electronics, Information Technology and Communication (SEE), the 2016 Berger Award from the International Conference on Lightning Protection, the 2016 Best Paper Award of the IEEE Transactions on EMC, and the 2017 Motohisa Kanda Award for the most cited paper of the IEEE Transactions on EMC (2012-2016). In 2014, he was conferred the title of Honorary Professor of the Xi’an Jiaotong University in China. He served as the Vice-Chair of the European COST Action on the Physics of Lightning Flash and its Effects from 2005 to 2009, the Chairman of the 2008 European Electromagnetics International Symposium, the President of the International Conference on Lightning Protection from 2008 to 2014, the Editor-in-Chief of the Open Atmospheric Science Journal (2010-2012) and the Editor-in-Chief of the IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY from 2013 to 2015. He is a Fellow of the IEEE and of the SUMMA Foundation, and a member of the Swiss Academy of Sciences. He is the author or coauthor of over 200 scientific papers published in peer-reviewed journals and over 400 papers presented at international conferences.

Maude Girardin

Teaching & PhD Courses Analysis A (for MAN) PREPA-031(a) This course offers the basic tools in calculus and in solving of equations and inequations on real and complex numbers.The course explores the study of elementary fonctions and is a basis for the calculus course. Analysis II (for CMS) PREPA-005(a) This class focus on the studies of special functions : trigonometric, logarithmic and exponential functions. Linear Algebra (for CMS) PREPA-002 the goal is to

Marco Picasso

Expertise Computational Science and Engineering Mission I am teaching analysis and numerical analysis for engineers and mathematicians at bachelor and master levels.My research pertains to the numerical simulation of complex physical systems with partial differential equations.Here is some bibliometry: Google Scholar, Research Gate, Orcid, here is my most recent CV. Ongoing research projects: - Numerical simulation of aluminium electrolysis, collaboration with RioTinto LRF lab, Ycoorsystems SA and Jacques

Haomin Sun

Expertise I am currently working on turbulent momentum transport in tokamak using both theoretical and numerical methods.

Olivier Claude Martin Février

Teaching & PhD PhD Students Benjamin Yong Kwak Brown, Massimo Carpita, Garance Durr-Legoupil-Nicoud, Alysée Khan, Riccardo Ian Morgan Courses Nonlinear dynamics, chaos and complex systems PHYS-460 The course provides students with the tools to approach the study of nonlinear systems and chaotic dynamics. Emphasis is given to concrete examples and numerical applications are carried out during the exercise sessions.

Parmenion Mavrikakis

Expertise Computational electromagnetics, integral equations, finite element analysis, metasurfaces, photonics, plasmonics and microwaves. Current Work Developing a C++ code for the modeling of light scattering in layered media with the surface integral equation method. I am a PhD student in the Electrical Engineering Department (EDEE) at École Polytechnique Fédérale de Lausanne (EPFL), affiliated with the Nanophotonics and Metrology Laboratory (NAM), part of the Microengineering Department (IEM) at EPFL. M

Adrian Blumenthal

Teaching & PhD Courses Numerical integration of dynamical systems MATH-452 In this course we will introduce and study numerical integrators for multi-scale (or stiff) differential equations and dynamical systems with special geometric structures (symplecticity, reversibility, first integrals, etc.). These numerical methods are important for many applications.

Related units (32)

Chair of Computational Mathematics and Simulation Science

Picasso Group

Scientific Computing and Uncertainty Quantification - CADMOS Chair

Related concepts (5)

Additive Schwarz method

In mathematics, the additive Schwarz method, named after Hermann Schwarz, solves a boundary value problem for a partial differential equation approximately by splitting it into boundary value problems on smaller domains and adding the results. Partial differential equations (PDEs) are used in all sciences to model phenomena. For the purpose of exposition, we give an example physical problem and the accompanying boundary value problem (BVP). Even if the reader is unfamiliar with the notation, the purpose is merely to show what a BVP looks like when written down.

Finite difference method

In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time interval (if applicable) are discretized, or broken into a finite number of steps, and the value of the solution at these discrete points is approximated by solving algebraic equations containing finite differences and values from nearby points.

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.

Related exercises (10)

Car suspension problem

A car’s suspension is comfortable when it has a vertical oscillation period that our bodies are used to: the oscillation period of walking, Tu2004=u20040.8u2005s.

Find the spring constant k of the suspension in a car of mass $M = 1540\,{\rm kg}$ .
The car lowers when we place a trunk of mass $m = 70\,{\rm kg}$ in it. Find its vertical displacement.

To make the car more comfortable, we add a shock absorber/dampener that creates a friction force F⃗u2004=u2004u2005−u2005bv⃗_(z), where v⃗_(z)

Car in a bend

A sports car is driving around a circular track of radius R with a horizontal velocity of constant norm v. The track is on an incline with angle α from the the horizontal, such that the car is tilted inwards. The static friction coefficient between the tires and the track is μ.

Draw a diagram, considering the vertical plane perpendicular to the car’s velocity and represent all forces acting on the car.
What’s the condition on v for the car not to drift outwards ?

Numerical application: $g=10\

Conceptual question: Coriolis in your sink

Calculate the order of magnitude of the Coriolis acceleration of the water in your emptying sink. The Coriolis acceleration is expressed with the following equation:

\vec a_{\rm Coriolis}=2\vec{\Omega} \wedge \vec{V} \,

in which Ω⃗ is the Earth’s rotational velocity and V⃗ is the velocity of the water in the sink. We find the following expression for the Coriolis acceleration’s norm:

a_{\rm Coriolis}=2\Omega V \sin\theta \,

with

\Omega=\frac{2\pi }{86400\,\mathrm{s}}

the Earth’s rotational speed,

Related notebooks (1)

Exploring Chemical Potential for Fermions in 3D at Finite Temperature

This notebook gives a simple illustration to the variation of chemical potential for Fermions in 3D, following the Fermi-Dirac statistics in finite temperature. " In 3D, we have the density of states

\\rho \\propto \\sqrt{E}

. At zero temperature, the distribution of electrons just follows

g(E)=a \\sqrt{E}

and the Fermi energy is determined by the total occupied number

N_{occ}

E_f

is the energy such that

N_{occ}=\\int_{0}^{E_f} g(E) dE

\n", " \n", "In finite temperature, however, o