Entry requirements

120 credits with 60 credits mathematics (linear algebra, vector calculus, complex analysis, Fourier analysis must be covered), Computer Programming I and Scientific Computing III or the equivalent is included.

Learning outcomes

After finishing the course, the student should

• explain basic concepts in the numerical solution of partial differential equations like consistency, convergence, stability, efficiency;
• analyse consistency, convergence, stability, efficiency of finite difference methods for partial differential equations;
• analyse stability and efficiency of finite difference methods for linear hyperbolic and parabolic problems with periodic boundary conditions by means of the Fourier method;
• analyse stability of finite difference methods for simple initial-boundary value problems by means of the energy method and the normal mode analysis, i.e. GKS analysis;
• analyse properties of nonlinear partial differential equations and finite difference methods like hyperbolicity, Rankine-Hugoniot conditions, conservativity, total variation diminishing;
• analyse iterative methods for elliptic problems like two-grid multigrid method for model problems;
• program finite difference methods for simple one-dimensional hyperbolic, parabolic and elliptic problems;
• choose and implement suitable numerical methods for solving scientific and engineering problems described by partial differential equations;
• identify deficiencies and limitations of the considered method with regard to efficiency, accuracy and stability;

Content

Fundamental properties for numerical methods to solve partial differential equations: consistency, convergence, stability, efficiency. Fourier method to analyse stability and efficiency of finite difference methods for time-dependent partial differential equations. Energy method and normal mode analysis, i.e. GKS analysis, to analyse stability of finite difference methods for simple initial boundary value problems. Special properties of non-linear partial differential equations and finite difference methods: hyperbolicity, Rankine-Hugoniot conditions, conservativity, total variation diminishing. Multigrid methods for elliptic partial differential equations.

Instruction

Lectures, problem classes and compulsory assignments.

Assessment

Written exam at the end of the course. Passed laboratory course and approved assignments are also required.