Single-Course English 5 ECTS

Advanced Numerical Methods for Differential Equations

DTU Compute

Overall Course Objectives

The use of model-based simulation tools on modern computers are increasingly being used in academia and industry for improving engineering designs and decision support.
The course goal is to give a solid grounding in the developments of theory and practice in the use of advanced numerical computational methods for efficient solution of differential equations and for prediction in science and engineering. This includes development, analysis and application of advanced numerical methods and algorithms for the solution of any type of differential equations (fx. ODEs/PDEs/SDEs). We will develop and generalize ideas from finite difference methods, Fourier methods and extend them to modern and flexible multi-domain methods such as Discontinuous Galerkin Finite Element Methods and Spectral Element Methods. The experience gained in the course is useful for the numerical solution and prediction and study of mathematical problems where no analytical solutions exist or are readily obtainable. The methods are suitable for scientific computing, high performance computing (HPC) and is relevant for topics in modern Uncertainty Quantification (UQ) and data-driven Scientific Machine Learning (SciML). The last project in the course can be defined in a scientific area of relevance to the participant.

Learning Objectives

  • Apply basic principles for numerical approximation/discretization of spectral methods.
  • Apply the Fourier Transforms.
  • Analyze convergence and stability properties of spectral methods.
  • Setup and solve boundary value problems for partial differential equations.
  • Implement methods for time integration of semi-discrete equation systems.
  • Implement and apply algorithms for periodic and non-periodic problems.
  • Construct spektral approximations for partial derivatives.
  • Implement spektral methods in Matlab.
  • Write reports and clearly communicate, discuss and draw conclusions based on main ideas and results.

Course Content

Subjects that will be covered in the course includes:
– Spektral approximation methods
– Fourier approximations methods (periodic)
– Polynomial approximation methods (non-periodic)
– Time-integration and stability analysis
– Solution of Linear and Nonlinear PDE problems
– Numerical solution of dynamical systems
– Consistency and convergence properties for spectral methods
– Numerical integration
– Derivation and analysis of advanced algorithms (fx. via projects)

Recommended prerequisites

02686/02687, A course in numerical methods for solving differential equations or similar scientific computing course for differential equations

Teaching Method

Lectures and assignment work in the Databar

Faculty

Allan Peter Engsig-Karup

Contact or read more about Allan Peter
Allan Peter Engsig-Karup

Limited number of seats

Minimum: 10.

Please be aware that this course will only be held if the required minimum number of participants is met. You will be informed 8 days before the start of the course, whether the course will be held.

See course in the course database.

Registration

Language

English
Duration

13 weeks
Institute

Compute
Place

DTU Lyngby Campus
Course code 02689
Course type Candidate
Semester start Week 35
Semester end Week 48
Days Thurs 8-12
Price

7.500,00 DKK

Please note that this course has participants limitation. Read more

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