Scientific Computing for ordinary and partial differential equations

• Describe, analyze and apply fundamental principles for numerical solution of boundary value problems (BVPs) described by ordinary or partial differential equation systems
• Analyze and derive order, convergence, and stability properties of finite difference methods
• Analyze and derive finite difference schemes for numerical solution of boundary value problems (BVPs) with Dirichlet, Neumann and Robin boundary conditions
• Design, analyse, and implement numerical methods for the efficient solution of elliptic equations
• Derive and implement deferred correction methods for the solution of 2-point BVPs and elliptic equations
• Derive and analyze stationary iterative defect correction methods, including multigrid methods, to solve large linear systems of equations
• Describe and implement matrix-free conjugate gradient, preconditioned conjugate gradient and GMRES methods to solve linear systems of equations
• Describe, implement and analyze iterative multigrid methods
• Analyze the convergence and stability of numerical schemes for the solution of parabolic and hyperbolic equations
• Analyze, derive and implement numerical schemes for the solution of mixed and nonlinear systems of partial differential equations
• Gain experience with introduced techniques from scientific machine learning

Topics covered in the course include:
– Analysis of methods for numerical solution of 2-point boundary problems.
– Properties of methods for numerical solution of systems of ordinary and partial differential equations
(consistency, order of accuracy, stability, convergence).
– Newton’s method for the solution of nonlinear 2-point boundary problems.
– Numerical methods for 2-point boundary problems with varying conductivity.
– Deferred correction methods for boundary value problems.
– Analysis of methods for numerical solution of elliptic equations.
– Iterative methods for the solution of large linear systems of equations.
– Conjugate gradient, preconditioned conjugate gradient and GMRES methods for the solution of linear systems of equations.
– Newton-Krylov methods for nonlinear systems of equations.
– Multigrid methods for the efficient iterative solution of large linear system of equations.
– Analysis of methods for numerical solution of parabolic and hyperbolic equations.
– Methods for mixed equations and nonlinear PDEs.
– Methods for Scientific Machine Learning

02687 can be followed in the same semester as 02686. Both courses can be taken independently and in any order.

Lectures and computer exercises