Instructor: Bojan Popov
Office: Blocker 507B
Email: popov"at"math.tamu.edu
Office Hours: TR 10:55-11:30, or by appointment
Textbooks, both
optional:
1. R.J. LeVeque, Numerical Methods for
Conservation Laws, 2nd ed., Birkauser 1992. (optional)
2. E.F.
Toro, Riemann solvers and numerical methods for fluid dynamics,
2nd edition, Springer Verlag, Berlin Heidelberg 1999.
Outline: This is a topic course for graduate students who are interested in nonlinear first order PDEs and their applications. I plan to cover: (i) The basic existence-uniqueness theory for scalar conservation laws; (ii) Consider special equations/systems of interest for various application and solve the Riemann problems for such systems; (iii) The stability for viscous perturbations and simple numerical methods. There will be no required book for the class but LeVeque's book above is a good reference. I will also use lecture notes, papers and the following references:
1. Lawrence C. Evans, Partial Differential Equations (Graduate Studies in Mathematics, V. 19) GSM/19. 2. H. Holden, and N.H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer Verlag, New York 2002. 3. C. Dafermos, Hyperbolic conservastion laws in continuum physics, Springer; 2nd edition (September 29, 2005). 4. E. Tadmor, Approximate solutions of nonlinear conservation laws, in "Advanced Numerical Approximation of Nonlinear Hyperbolic Equations", Lecture notes in Mathematics 1697, 1997 C.I.M.E. course in Cetraro, Italy, June 1997 (A.~Quarteroni ed.) Springer Verlag 1998, 1-149.
Grading system: Due to the wide range of applications, this course should be of interests to students in aerospace and atmospherics sciences, and also nuclear engineering, etc. Therefore, formal, in class theoretical exams are not appropriate for this class. I will give three assignments: two midterm and a final exam assignment.
Midterm and Final exam/projects: (1) projects link. (2) Link to the NT paper. (3) Exact solution for Riemann Problem for Buckley-Leveret flux: link
The basic scheme you should implement is given by (2.16a)-(2.16b) with the limiters given in (3.13a) and (3.16b).
Make-Up Policy: Make-ups for exams will only be given with documented University-approved excuses (see University Regulations). Consistent with University Student Rules, students are required to notify an instructor by the end of the next working day after missing an exam. Otherwise, they forfeit their rights to a make-up.
Scholastic Dishonesty: Students may work together and discuss the homework problems with each other. Copying work done by others is an act of scholastic dishonesty and will be prosecuted to the full extent allowed by University policy. For more information on university policies regarding scholastic dishonesty, see University Student Rules.
Copyright Policy: All printed hand-outs and web materials are protected by US Copyright Laws. No multiple copies can be made without written permission by the instructor.
Students with Disabilities: The Americans with Disabilities Act (ADA) is a federal anti-discrimination statute that provides comprehensive civil rights protection for persons with disabilities. Among other things, this legislation requires that all students with disabilities be guaranteed a learning environment that provides for reasonable accommodation of their disabilities. If you believe you have a disability requiring an accommodation, please contact Disability Services, currently located in the Disability Services building at the Student Services at White Creek complex on west campus or call 979-845-1637. For additional information, visit http://disability.tamu.edu
References for the class: Lucier CL notes, Bressan CL notes, Temple link
New schedule:
Lecture 1 (January 18, 2022):
Introduction to Conservation Laws: LeVeque (pages 1-35); For the
Euler system read Toro (pages 1-13).
Link to my notes here:
page1, page2
Lecture 2 (January 20, 2022):
Mathematical principles: method of characteristics, weak solutions,
scalar equations, Bounded
Variation, Rankine-Hugoniot condition. Link to my notes here:
page1, page2,
page3, page4,
page5
For functions
of bounded variation: Ziemer,
Weakly differentiable functions: Sobolev spaces and functions of
bounded variation
Lecture 3 (January 25, 2022):
Motivation for entropy solutions. Existence of a weak solution with
bounded variation and satisfying the entropy inequalities, elementary
waves
Link to my notes here: page1,
page2, page3,
page4, compactness
theorems - look at Frechet–Kolmogorov theorem and take p=1
Lecture 4 (January 27, 2022): Compactness theorems
(see lecture 3) and Exact solution to the 1D Riemann problem -
fundamental block of constructing numerical methods.
Link
to my notes here: page1,
page2, page3,
page4, page5
Lecture 5 (Tuesday, February 1, 2022
): Exact solution to the 1D Riemann problem - fundamental block of
constructing numerical methods.
For more information on convex
envelope and proofs of basic facts read Section 2 in the paper:
BIANCHINI_MODENA
Lecture 6 (February 8, 2022)
Summary of the exact solution construction. Derivation of the
Godunov's method and the Lax scheme (1953).
Link to my notes
here: page1,
page2, page3,
page4, page5,
page6
Lecture 7 (February 10, 2022):
Godunov, Lax, and Engquist-Osher Schemes. Read Lucier, Chaper 3
for the EO-scheme, and the NT
paper. for the Lax scheme.
Link to my notes here:
page1, page2,
page3, page4
Lecture 8 (February 17, 2020):
Godunov-type methods based on
cell averages. The Nessyahu-Tadmor scheme as an extension of the
Lax-Friedrichs scheme. Other second order methods. Here is the link
to the NT
paper. The basic scheme you should implement is given by
(2.16a)-(2.16b) with the limiters given in (3.13a) and (3.16b).
Link
to my notes here: page1
Lecture 9 (February 24, 2020):
Invariant domain preserving schemes for arbitrary hyperbolic systems.
Link to the paper here: invariant
domains
Link to my notes
here: page1, page2,
page3, page4,
page5
Lecture 10 (March 1, 2022):
Invariant domains, part 2. Entropy inequality. Recall what we have
done here: Invariant domain
notes
Link to my notes here: page1,
page2, page3,
page4, page5
Lecture 11 (March 3, 2022): Invariant domains examples: 1d linear wave equation, two component chromatography, the p-system. look at Bressan CL notes, pages 4-5 for linear systems, pages 16-19 and 26-27 for the p-system, and pages 25-26 for the two component chromatography.
Lecture 12 (March 8, 2022): Scalat case theorems. Global numerical solution, local trunkation error, consistency, conservation and stability.
Lax-Wendroff Theorem: P.D Lax; B. Wendroff (1960). "Systems
of conservation laws". Commun. Pure Appl Math. 13 (2):
217–237
Link to my notes here: page1,
page2, page3,
page4
General framework for error estimates in the scalar case, Lemma A3 from the paper (pages 22-27 in the linked pdf):
Lecture 13 (March 10, 2022) Systems
of conservation laws: elementary waves, read pages 16-22 here:
BressanCLnotes; Link to
my notes here: page1,
page2, page3,
page4
Every
system with a convex mathematical entropy is hyperbolic - read page 2
and 3 above Linear
algebra review here
Lecture 15 (March 22, 2022): The p-system: elementary waves, read pages 27-28 in Bressan CL notes
Link to my notes here: page1, page2, page3
Lecture 16 (March 24-29, 2022): The p-system: complete solution, read pages 27-28 in Bressan CL notes
Link to my notes here: page1, page2, read pages 5-6 in the invariant domain paper
Lecture 17-18 (March 29-31, 2022): Computation of the U* state in the Riemann problem. Godunov, Lax, Glimm and implementation for the p-system. Implementation of the invariant domain preserving method for for the p-system. Link to my notes here: page1, page2, page3
Lecture 19 (April 5, 2022): Isentropic gas: elementary waves, invariant domain, general solution of the Riemann problem. Link to my notes here: page1, page2
Lecture 20 (April 7): End of isentropic gas; Euler system of gas dynamics. Link to my notes here: page1, page2, page3, page4, page5
Lecture 21 (April 12): Euler system of gas
dynamics: hyperbolicity, genuine nonlinearity, elementary waves
Link
to my notes here: page1,
page2, page3,
page4
Lecture 22 (April 12): Euler system of gas dynamics: continued, the notion of specific entropy and Riemann invariants. Read the book Numerical Approximation of Hyperbolic Systems of Conservation Laws by Godlewski and Raviart: pages 40-97.
Lecture 23 (April 19): Derivation of the shock wave curves and maximum speed formula for ideal gas law.
Lecture 24 (April 21): Euler system of gas dynamics: elementary waves, complete solution of the Riemann problem, maximum speed of propagation. Link to my notes here: page1, page2, page3
Lecture 25 (April 26): Algorithm for computing p*: Fast estimation of the maximum wave speed in the Riemann problem for the Euler equations
Invariant domains
and first-order continuous finite element approximation for
hyperbolic systems
We are here in 2022 =====================================
