Computational Methods for Ordinary and Partial Differential Equations

Aim

Many important problems from technology, science and economics are formulated in terms of differential equations. Thus it is important to be able to solve such equations accurately and efficiently. In the course we treat finite difference approximations of partial differential equations and numerical methods for solving ordinary differential equations. The theory is illustrated by using problems from relevant applications. After a completed course the student should be able to

• discuss important concepts
• derive difference approximations of derivatives with desired properties and explain how boundary conditions should be treated numerically.
• explain and use standard methods, in particular Runge-Kutta type methods, for solving time dependent problems.
• explain what stiffness is and use implicit time stepping methods for solving stiff problems.
• explain the requirements on the computational mesh that need to be fulfilled in order for a finite difference solution to give a good solution.
• write Matlab programs that solves different types of partial differential equations.
• judge the quality of a numerical solution

Content

Classification of differential equations, order of accuracy, consistency, convergence, wellposedness, stability, stability analysis using the Fourier ansatz. Ordinary differential equations: Runge-Kutta methods, explicit and implicit methods, stiff problems. Partial differential equations: finite difference methods, interpolation of boundary conditions, Crank-Nicholson method.