Numerical methods for differential equations
- discretize ordinary differential equations with initial condition using a finite difference method of required precision
- discretize ordinary differential equations with boundary conditions using finite difference and finite element methods
- discretize elliptic partial differential equations using finite difference and finite element methods
- discretize transport equations by corresponding methods
- implement numerical schemes on computer using appropriate software and numerically solve them
- decide on appropriateness of a numerical scheme for certain types of problems
- carry out simulation tests and interpret observations
- differentiate between numerical artefacts of inappropriate numerical schemes and physical phenomena in systems under consideration
Forms of Teaching
Lectures will be held once per week in total duration of 3 hours. They will be focused on theoretical foundations of numerical methods for differential equations.Exercises
Auditory exercises will be held once per week in total duration of one hour. Based on specific exercises the understanding of theoretical concepts from lectures is adopted and deepened.Independent assignments
Students are given homework assignments. Homework points make 40% of the total number of points.Laboratory
Laboratory exercises are held over 10 weeks of semester (one hour each) and in those weeks together with auditory exercises. In laboratory exercises, the emphasis is on the implementation of numerical methods and they serve as preparation for homework assignments.
|Type||Threshold||Percent of Grade||Threshold||Percent of Grade|
|Homeworks||0 %||40 %||0 %||40 %|
|2. Mid Term Exam: Written||0 %||30 %||0 %|
|Final Exam: Written||0 %||30 %|
|Exam: Written||0 %||60 %|
Week by Week Schedule
- Introduction lecture: motivation examples. Finite difference approximations: derivation, truncation error.
- One-step methods for initial value problems: stability, error estimates, convergence.
- Multistep methods for initial value problems: stability, error estimates, convergence.
- Stiff systems.
- Finite difference methods for elliptic equations: Poisson equation. Iterative methods. Multigrid.
- Finite difference methods for parabolic equations: the heat equation. Crank-Nicolson method.
- Finite difference methods for transport equations: upwind scheme. CFL condition.
- Midterm exam
- Elements of functional analysis: Sobolev apaces and weak formulation.
- Galerkin finite element method for elliptic equations. Helmholtz equation.
- Finite element method for reaction - diffusion equations. Petrov-Galerkin method.
- Fluid flow simulations. Navier-Stokes equations.
- Traffic flow simulation. Discontinuous Galerkin finite element method.
- Maxwell equations. Boundary element method.
- Final exam