FSF3562 HT24 Numerical Methods for Partial Differential Equations (51139)

Numerical methods for partial differential equations

 

 

FSF3562 Numerical Methods for Partial Differential Equations 2024


Graduate course, starting Friday September 6th at  13.15-15.00  in room 3721 at the Department of Mathematics KTH, period 1-2, 7.5 ECTS.  The lectures will continue at 10.15-12.00 on September 10 and 20 in room 3418; and September 24, October 1, 8, 15,  in room 3721.

 

Goal: To understand and use basic methods and theory for numerical solution of partial differential equations.

Some topics: finite difference methods, finite element methods, multi grid methods, adaptive methods.

Some applications: elliptic problems (e.g. diffusion), parabolic problems (e.g. time-dependent diffusion), hyperbolic problems (e.g. convection), systems and nonlinear problems (conservation laws and optimal control).

Learning outcomes etc.

Prerequisites: undergraduate differential equations and numerics.

Literature  

The literature overlaps, so the list gives alternatives.

Plan (lectures by Anders period 1 (1-4 below + Lax equivalence theorem)  and by Jennifer period 2)

1. Introduction and overview: 1D elliptic finite element and finite difference methods, (LT chapters 2, 4.1, 5.1; CJ chapter 1, LN1)

2. Elliptic equations, minimization and variational methods, maximum principle (LT 3,4,5; CJ 2, 3)

3. A priori and a posteriori error estimates, adaptive methods (LT 5.4, 5.5; CJ 4; LN1)

4. Solution methods, multigrid (LT B, CJ 6,7)

5. Parabolic equations, Lax equivalence theorem, energy methods, finite difference and finite element methods (LT 8, 9,10; CJ 8; LN2)

6. hyperbolic and nonlinear problems

7. Hyperbolic conservation laws & introduction to discontinuous Galerkin Methods Download Hyperbolic conservation laws & introduction to discontinuous Galerkin Methods (1 and 12.0-12.1 in Hesthaven).  Suggested Exercises Download Suggested Exercises

8. Nonlinear hyperbolic equations Download Nonlinear hyperbolic equations, characteristics, weak solutions (2.0-2.1 in Hesthaven).

9. Weak solutions, entropy solutions Download Weak solutions, entropy solutions, Rankine-Hugoniot jump condition . (2.2-2.3 in Hesthaven).

10.  Entropy conditions and numerical flux functions Download numerical flux functions. (2.3, 4.1 in Hesthaven).

11. Stability, Dissipation and Dispersion (12.1.3-4 in Hesthaven, and this paper Links to an external site. by Wei Guo and Jing-Mei Qiu).

12. Nonlinear Stabilisation Techniques Download Nonlinear Stabilisation Techniques and monotonicity of schemes (10.1-2, 12.2.4 in Hesthaven).

13. Total Variation (Diminishing or not); extensions to multiple dimensions  (10.1-2, 12.2.4, 12.4 in Hesthaven).

14. Filtering and hidden accuracy (12.2 in Hesthaven)

 

 

Preliminary set of homework and computer lab:

  • Homework 1: Problem 1.2 (derive the wave equation), 2.1 (illustrate the maximum principle) in (LT) , due October 1.
  • Homework 2: do Problem 3.1 and 3.3 in (LT) (formulate variational form and prove existence of unique solution)  due October 7th, or do Computer lab 0, 
  • Homework 3: Problem 3.5, 3.9  in LT, due November 8th.
  • Hand in one of the two problems in Computer lab 1, due November 8th.



Here is a preliminary list of questions  for the exam.

Presentations (group of 1-2 presents a paper or chapter on numerics for PDE) for instance:

  • A. Brandt, Multi-level adaptive solutions of boundary value problems, Math. Comp. 31 (1977), 333-390.
  • Boundary element methods (CJ 10; LT 14.4)
  • Mixed finite elements ( CJ 11; LT 5.7)
  • Spectral methods (LT 14.2)
  • Stochastic elliptic PDE
  • Maxwell
  • MHD
  • Obstacle problems and variational inequalities
  • Your own choice 


To pass the course requires at least 16 credits and passed computer laborations. A good solution of the exam gives 15 credits. A good solution of a homework problem gives 1 credit.  Therefore one obtains 25 = 15 + 10x1  credits if everything is good.

 

Welcome!
Jennifer Ryan and Anders Szepessy,