Single-Course English 5 ECTS

# Scientific Computing for ordinary and partial differential equations

## Overall Course Objectives

The course provides a solid knowledge of theory and practice of Scientific Computing for numerical solution of differential equation systems arising in science and engineering. The participants learn to develop, analyze, implement and apply various numerical methods and algorithms for solution of steady-state and time-dependent ordinary (ODEs) and partial differential equation systems (PDEs). The techniques are relevant for any type of differential equations, incl. stocastic differential equations (SDEs). Data-driven methods within the emerging area of Scientific Machine Learning (SciML) arre introduced. SciML combines techniques from mathematics-physics modelling, scientific computing and statistics. The experience gained is useful for the solution and study of mathematical problems arising in engineering and science applications.

Learning Objectives

• Describe, analyze and apply fundamental principles for numerical solution of boundary value problems (BVPs) described by ordinary or partial differential equation systems
• Analyze and derive order, convergence, and stability properties of finite difference methods
• Analyze and derive finite difference schemes for numerical solution of boundary value problems (BVPs) with Dirichlet, Neumann and Robin boundary conditions
• Design, analyse, and implement numerical methods for the efficient solution of elliptic equations
• Derive and implement deferred correction methods for the solution of 2-point BVPs and elliptic equations
• Derive and analyze stationary iterative defect correction methods, including multigrid methods, to solve large linear systems of equations
• Describe and implement matrix-free conjugate gradient, preconditioned conjugate gradient and GMRES methods to solve linear systems of equations
• Describe, implement and analyze iterative multigrid methods
• Analyze the convergence and stability of numerical schemes for the solution of parabolic and hyperbolic equations
• Analyze, derive and implement numerical schemes for the solution of mixed and nonlinear systems of partial differential equations
• Gain experience with introduced techniques from scientific machine learning

Course Content

Topics covered in the course include:
– Analysis of methods for numerical solution of 2-point boundary problems.
– Properties of methods for numerical solution of systems of ordinary and partial differential equations
(consistency, order of accuracy, stability, convergence).
– Newton’s method for the solution of nonlinear 2-point boundary problems.
– Numerical methods for 2-point boundary problems with varying conductivity.
– Deferred correction methods for boundary value problems.
– Analysis of methods for numerical solution of elliptic equations.
– Iterative methods for the solution of large linear systems of equations.
– Conjugate gradient, preconditioned conjugate gradient and GMRES methods for the solution of linear systems of equations.
– Newton-Krylov methods for nonlinear systems of equations.
– Multigrid methods for the efficient iterative solution of large linear system of equations.
– Analysis of methods for numerical solution of parabolic and hyperbolic equations.
– Methods for mixed equations and nonlinear PDEs.
– Methods for Scientific Machine Learning

Recommended prerequisites

02687 can be followed in the same semester as 02686. Both courses can be taken independently and in any order.

Teaching Method

Lectures and computer exercises

Faculty

See course in the course database.

## Registration

Language English 13 weeks Compute DTU Lyngby Campus 02687 Candidate Week 5 Week 19 Mon 8-12
Price 7.500,00 DKK