Single-Course English 5 ECTS

Scientific computing for 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 initial value problems (IVP) described by ordinary differential equation systems (ODEs), stochastic differential equation systems (SDEs), differential-algebraic equation systems (DAEs), and partial differential equation systems (PDEs). 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 initial value problems (IVPs) described by various categories of differential equation systems
  • Analyze and derive order, convergence, and stability properties of numerical methods
  • Discuss and analyze properties of differential equation systems and choose an appropriate numerical method for stiff as well as non-stiff systems
  • Analyze, implement and apply linear multi-step methods (explicit and implicit methods, Adam-Bashforth, Adams-Moulton, BDF)
  • Analyze, implement and apply Runge-Kutta methods (ERK, ESDIRK, SDIRK, DIRK, IRK)
  • Analyze, implement and apply numerical methods for automatic control of the local error
  • Describe, implement and apply numerical methods for systems of linear differential equations (the matrix exponential function)
  • Analyze, implement and apply numerical methods for solution of systems of ordinary differential equations (ODEs)
  • Analyze, implement and apply numerical methods for solution of systems of stochastic differential equations (SDEs)
  • Analyze, implement and apply numerical methods for systems of differential-algebraic equations (DAEs)
  • Analyze, implement and apply numerical methods for systems of partial differential equations (PDEs)
  • Analyze, implement and apply sensitivity computations for differential equations in connection with shooting methods

Course Content

Topics covered in the course include:
– Analysis of methods for numerical solution of systems of differential equations
– Properties of methods for numerical solution of systems of differential equations
(order, convergence, stability)
– Convergence and stability analysis
– Stiff and non-stiff systems of differential equations
– The initial value problem (IVP) and dynamical systems
– Newton’s method in connection to implicit methods for solution of differential equation
systems
– Basic numerical methods (explicit Euler, implicit Euler) for solution of systems of
differential equations (ODEs, SDEs, DAEs)
– Linear multi-step methods (Adams-Bashforth, Adams-Moulton, BDF)
– Runge-Kutta methods (ERK, ESDIRK, SDIRK, DIRK, IRK)
– The matrix exponential function for solution of systems of linear differential equations
– Numerical solution of ordinary differential equation systems (ODEs)
– Numerical solution of stochastic differential equation systems (SDEs)
– Numerical solution of differential-algebraic systems (DAEs)
– Sensitivity computations for differential equation systems in connection to shooting
methods

Recommended prerequisites

02601/02002/02631/02632/02633/02692, Basic course in numerical algorithms, eg 02601.
Basic knowledge in programming, eg 02631.
02687 can be followed in the same semester as 02686. Both courses can be taken independently.

Teaching Method

Lectures and computer exercises.

Faculty

See course in the course database.

Registration

Language

English

Duration

13 weeks

Institute

Compute

Place

DTU Lyngby Campus

Course code 02686
Course type Candidate
Semester start Week 5
Semester end Week 19
Price

7.500,00 DKK

Registration