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| Assignments and schedule |
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- (1) 3/31: Introduction: exmaples and classification of PDEs.
- (2) 4/2: Elliptic PDEs of second order.
- HW 1 due 4/14:
452 and 552: do Chapter 3 exercises 3.1, 3.2.
452 do additionally one of 1,2,3 below.
552 do additionally two of 1,2,3 below.
In all exercises below, K is a 2x2 real constant coefficient matrix.
In 1,2 you can assume K is diagonal.
- In 3.1, modify the script and test to solve
$-\nabla \cdot (K \nabla u) = f$.
- In 3.2, derive a FD scheme and show its consistency for
$-\nabla \cdot (K \nabla u) = f$.
- Derive sufficient and necessary conditions on K for the PDE
$-\nabla \cdot (K \nabla u) = f$ to be of elliptic type.
- (3) 4/4: FD (5-pt stencil) for Poisson's equation. LTE and consistency.
- (4) 4/7: boundary conditions for second order elliptic PDEs and
their discretization.
- (5) 4/9: Method of deferred corrections and 9-pt stencil
for Poisson's equation. How to interpret $-\nabla \cdot (K \nabla u) = f$.
Recap the concept of stability.
- (6) 4/11: Stability of FD scheme for Poisson's equation. Grid norms.
- (7) 4/14: Heat equation, preliminaries. Basic scheme4s (FE,BE,CN)
and their stencils.
- (8) 4/16: Leapfrog and DuFort-Frankel schemes.
Method of Lines approach; revisit basic schemes. Define LTE.
- (9) 4/18: Stability of basic schemes with the MOL approach.
- (9-10-11) 4/18-21-23: Review of Fourier analysis: series, transform, DTFT, transform
for grid functions, introduction to von-Neumann analysis.
- HW 2 due May 5:
452: 9.1a and computational problem (1) with BE, FE, and CN.
552: 9.1a, computational problem (2), and show (conditional) consistency of DuFort-Frankel scheme
Computational problem (1): Using fd1d_heat.m , solve
the heat equation using BE. Confirm the desired order of convergence. Then modify the code
to include FE, BE, CN.
Computational problem (2): Same as (1) but let the true solution to be u(x,t)=sin(pi*x)*exp(t).
Modify the right hand side, boundary conditions etc. appropriately. Compare with the previous case.
If you wish, you can use codes from textbook website instead.
Extra:
You can do 9.1 b,c and modify the fd1d_heat.m code code to solve the same problem as in 9.3.
- (12) 4/28: Apply von-Neumann method to (BE). Recap how to choose
time step for consistency/etsability reasons and in order to show
optimal convergence.
- (13) 4/30: Implementation of algorithms for heat equation on example
of fd1d_heat.m .
- (14) 5/2: Review for exam.
- (15) 5/5: Midterm exam.
- (16) 5/7: Wrap up parabolic problems. Dissipation and diffusion.
Start Chapter 10: hyperbolic first order equations (transport, wave,
advection, convection).
- (17) 5/9: Start first order linear hyperbolic problems
(advection, convection, wave, transport). Exact solution,
characteristics, weak (integral) formulation.
- (18) 5/12: Basic numerical schemes for transport equation:
FT*S, Lax-Friedrichs, Lax-Wendroff.
- (19-20) 5/14-16: Stability of basic schemes, accuracy, and modified equations.
- HW 3 due May 23:
All: 10.8 a-b, (do c) extra)
All: Analyze stability and accuracy of the alpha-beta scheme (given in class)
or the scheme given in E10.4a (exercise section). (Do both for extra credit).
553 only: 10.5 or 10.7.
- (21-22) 5/19-21: Nonlinear scalar transport equations.
- (23) 5/28: Mixed transient equations (Chapter 11) with nonlinear
and linear differential/algebraic operators: overview of operator
splitting.
- (24-25) 5/30, 6/2: advection-diffusion equation: stability and discrete maximum principle.
- HW 4 due at the Final. .
- Office hours: Monday 6/9, 4:00pm-5:30pm.
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