The Pierce Lab

ACM 210a (Fall 2000)

Numerical Methods for PDEs

Instructor

Niles A. Pierce
165 Broad
Open office hours

Linear convection

Well-posed initial value problems, consistency, Lax-Richtmyer stability, convergence, the Lax equivalence theorem, characteristics, the upwind scheme, truncation error, the CFL condition, Fourier stability analysis, Parseval's relation, the von Neumann stability condition, central difference schemes, a maximum principle, the Lax-Wendroff scheme, numerical dissipation, amplitude and phase errors, group velocity in finite difference schemes, modified equation analysis, semi-discrete schemes, stability regions, Fourier footprints, Fourier and eigenvalue stability analysis for discretizations of linear hyperbolic systems, stability analysis for discretizations of initial boundary value problems, spectra and pseudo-spectra of non-normal matrices, the Kreiss matrix theorem, the leapfrog scheme, direct stability analysis of boundary conditions, a group velocity approach, the Godunov-Ryabenkii condition, GKS normal mode analysis.

Conservation laws

Weak solutions, compression and breaking, jump conditions, Lax entropy conditions, Riemann problems, shocks, contacts and rarefactions, inviscid Burgers equation, the shock tube problem in 1D gas dynamics, non-conservative methods, conservative methods, discrete conservation, the Lax-Wendroff theorem, entropy violation, Godunov's method, Roe's linearization, flux difference splitting, an entropy fix, total variation stability, non-oscillatory TVD schemes, Godunov monotonicity theorem, high resolution switched and limited schemes, flux and slope limiters, TVD time-stepping schemes, generalization to systems, characteristic boundary conditions, finite volume discretizations in multiple dimensions, convergence acceleration, preconditioning, multigrid.

Adjoint sensitivity analysis

Lagrange multiplier and duality formulations, adjoint boundary conditions, optimal shape design, error correction for functionals.

Reserve Texts

  • B. Gustafsson, H.-O. Kreiss, J. Oliger, Time Dependent Problems and Difference Methods, Wiley, 1995.
  • C. Hirsch, Numerical Computation of Internal and External Flows, Vols 1 & 2, Wiley, 1988.
  • E. Isaacson and H.B. Keller, Analysis of Numerical Methods, Dover, 1994.
  • A. Iserles, A First Course in the Numerical Analysis of Differential Equations, Cambridge, 1996.
  • P.D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM, 1973.
  • R.J. LeVeque, Numerical Methods for Conservation Laws, Birkhauser, 1992.
  • K.W. Morton and D.F. Mayers, Numerical Solution of Partial Differential Equations, Cambridge,1994.
  • R.D. Richtmyer and K.W. Morton, Difference Methods for Initial-Value Problems, Wiley, 1967.
  • J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, 1983.
  • J.C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, Wadsworth & Brooks, 1989.
  • G.B. Whitham, Linear and Nonlinear Waves, Wiley, 1999.

Grading

Problem sets: 50%
Final project: 25%
Oral final exam: 25%