Reading Assignments APMA 2550
Main textbooks: Gustafsson et al (1995 Edition) and Parallel Scientific Computing in C++ and MPI, in Karniadakis & Kirby,Cambridge University Press
Lecture 1: pp. 3 – 26. TIP — Parseval’s equality relates the continuous (functions) with the discrete worlds (array/vectors).
(2013 Book Edition: Appendix A, pp. 465-476, and also pp. 3-9)
Lecture 2: pp. 38-50. TIP — To gain stability in the difference equation we modify it by adding dissipation.
(2013 Book Edition: pp. 10- 20)
Lecture 3: pp. 47-58. TIP — In multistep schemes we need a one-step starter scheme.
(2013 Book Edition: pp. 20-26)
Lecture 4: pp. 59-73. TIP — The Lax equivalence theorem (consistency and stability imply convergence) is a very practical tool.
(2013 Book Edition: pp. 27-39)
Lecture 5-cont’d: pp. 74-79; also pp. 260-263 & 309-312 in Karniadakis & Kirby book (CUP). TIP — High-order accuracy is expensive but it pays-off!
(2013 Book Edition: pp. 39-46)
Lecture 6: pp. 106-115. TIP — Well-possedness depends on the norm.
(2013 Book Edition: pp. 65-72)
Lecture 7: pp. 116-133. TIP — Highest derivative controls well-posedness/stability.
(2013 Book Edition: pp. 73-81)
Lecture 8: pp. 142-149; pp. 157-162. TIP — Round-off error divided by time step should be less than truncation error!
(2013 Book Edition: pp. 93-97; pp. 119-122)
Lecture 9: pp. 163-170-149. TIP — The truncation error acts as forcing in the difference equation!
(2013 Book Edition: pp. 122-130)
Lecture 10: pp. 171-182; pp. 195-201. TIP — Splitting methods for multi-dimensional problems are fast but watch out for splitting errors!
(2013 Book Edition: pp. 131-141; 147-151)
Lecture 11: pp. 222-234; pp. 238-254. TIP — The stability region of Runge-Kutta methods increases as their order increases unlike the Adams family.
(2013 Book Edition: pp. 162-165; 167-171)
Lecture 12: Diffusion equation, pp. 270-282 (textbook 1995) and 347-355, 358-360, 368-373 in Karniadakis & Kirby (Cambridge University Press). TIP — The long-time stability requirement is equivalent to the requirement for zero overshoots for an isolated disturbance.
(2013 Book Edition: pp. 177-1187)