Scientific computing for differential equations

• 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

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

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.

Lectures and computer exercises.