Numerical Methods for Time-Dependent PDEs

AMSC 612, Spring 2012

Course Information

Place 4122 CSIC Bldg. #406 (unless otherwise stated)
Time Tuesdays & Thursdays, 14-15:15pm
Thursday March 1 meeting, 14-15:15pm: class at CHM0124
Additional meetings on Wednesdays 10-12pm: Feb. 15, Mar. 14, Apr. 18 May 2 and May 9
InstructorProfessor Eitan Tadmor
Contacttel.: x5-0648   Email:
Office HoursBy appointment
Midterm Date TBA 4122 CSIC Bldg. #406
Final Wed. May 16, 10:30-12:30pm 4122 CSIC Bldg. #406
Grading50% Homework + midterm, 50% Final

Course Description

Time-dependent Partial Differential Equations (PDEs) of hyperbolic and parabolic type. Initial and initial-boundary value problems. Finite difference and spectral methods for time-dependent problems. The linear theory: accuracy, stability and convergence. Nonlinear problems: shock discontinuities, viscosity and entropy. Finite volume and discontinuous Galerkin numerical methods.

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Introduction

  • Examples of nonlinear conservation laws: Euler and Navier-Stokes equtions, the shallow-water eqs.
    PDEs from image processing, geometrical optics, incompressible Euler eqs, nonlinear models for traffic flow,...
  • Fron nonlinear to constant-coeffiecinets: linearization, method of characteristics
Assignment #1 [ pdf file ] with selected answers [ pdf file ]

Initial Value Problems

  • Initial value problems of hyperbolic type
    • The wave equation - the energy method and Fourier analysis
    • Weak and strong hyperbolicity -- systems with constant coefficients
    • Hyperbolic systems with variable coefficients
Lecture notes on the wave equation:
          The [ one-dimensional ] and [ simulation ] of the [ two-dimensional ] setup (J. Balbas)

  • Initial value problems of parabolic type
    • The heat equation -- Fourier analysis and the energy method
    • Parabolic systems
  • Well-posed problems
Lecture notes: Time dependent problems [ pdf file ]

Lecture notes: Matrices: Eigenvalues, Norms and Powers [ pdf file ]

Finite Difference Approximations for Initial Value Problems

  • Preliminaries
    • Discretization. grid functions, their Fourier representation, divided differences.
    • Upwind and centered Euler's schemes: accuracy, stability, CFL and convergence
Assignment #2 [ pdf file ] with selected answers [ pdf file ]

Related material:  [ CFL paper 1928 ]

  • Canonical examples of finite difference schemes
    • Lax-Friedrichs schemes: numerical dissipativity
    • Lax-Wendroff schemes: second- (and higher-) order accuracy
    • Leap-Frog scheme: the unitary case
    • Crank-Nicolson scheme: implicit schemes
    • Forward Euler for the heat equation
Assignment #3 [ pdf file ] with selected answers [ pdf file ]

Lecture notes: Fourier expansions, circulant matrices and Fourier differencing [ pdf file ]

Assignment #4 [ pdf file ] with selected answers [ pdf file ]

Convergence Theory

    • Accuracy
    • Stability: von Neumann analysis; power-bounded symbols
    • Convergence: stability implies convergence (Lax equivalence theorem)
    • Numerical dissipation; Kreiss matrix theorem


Assignment #5 [ pdf file ] with selected answers [ pdf file ]

Lecture notes: Stability --- power-boundedness vs. the resolvent condition [ notes ]

Related material:  From semi-discrete to fully-discrete: stability of Runge-Kutta schemes [ I ] [ II ]

Approximations of Problems with Variable Coefficients

    • Strong stability and freezing coefficients
    • Symmetrizers and the Lax-Nirenberg result
    • Dissipative schemes
    • The energy method: positive schemes, skew-symmetric differencing

Multi-Dimensional Problems

    • ADI and splitting methods
Assignment #6 [ pdf file ] with selected answers [ pdf file ]

Spectral Methods

  • Fourier and Chebyshev methods
  • Spectral accuracy, stability and convergence
Lecture notes: Chebyshev expansions [ pdf file ]

Lecture notes: Fourier & Chebyshev methods for time dependent problems [ pdf file ]

Assignment #7 (#8) [ pdf file ] with selected answers [ pdf file ]

Initial-Boundary Value Problems

  • One dimensional hyperbolic systems
    • Method of characteristics
    • The energy method: maximal dissipativity
  • Multi-dimensional hyperbolic systems
    • Eigen-mode analysis
    • Resolvent stability
  • Difference approximations to initial-boundary value problems
    • Eigen-mode analysis: Godunov-Ryabenkii condition
    • UKC the resolvent condition and stability
    • Examples
Lecture notes: Stability Theory of Difference Approximations for Mixed initial boundary value problems. II Gustafsson, Kreiss & Sundstrom, 1972. [ pdf file ]

Lecture notes: L2 vs. resolvant stability estimates for mixed systems [ pdf file ]

Assignment #8 (#7) [ pdf file ] with selected answers [ pdf file ]

Nonlinear Problems

  • Nonlinear Conservation laws
    • Smooth solutions: linearization. Strang's theorem
    • Nonlinear conservation laws: shock wand rarefaction waves
    • Viscosity solutions and L1 scalar theory
Lecture notes: Introduction to scalar conservation laws; traffic model [ notes ]

  • Finite-volume and discontinuous Galerkin methods
    • Conservative schemes
    • Godunov and Godunov-type schemes: limiters and high-resolution
    • Roe, Godunov and Lax-Friedrichs schemes: L1-contraction and TVD property
Lecture notes: Numerical solution of Burgers & Buckley-Levertt equations [ notes ]

Assignment #9 [ pdf file ]

    • TVD and non-oscillatory reconstructions
    • Godunov and Godunov-type schemes: limiters and high-resolution
    • Entropy stability
    • Discontinuous Galerkin (DG) method
Related material:
Limiters and high-resolution [ Randy LeVeque notes] and [Bjorn Sjogreen notes]
High-resolution schemes: [2nd-order TVD scheme ]; [2nd-order central scheme ] more can be found [here] Spectral viscosity [1989 paper]

Final Assignment #10 [pdf file ] and related material: [Numerical viscosity and entropy stability]


Epilogue - have you been paying attention in your numerical analysis course?


References

F. John, PDEs, Applied Mathematical Sciences 1, 4th ed., Springer-Verlag, 1982

R. Richtmyer and B. Morton,  Difference Methods for Initial-Value Problems, 2nd ed., Interscience, Wiley, 1967

H.-O. Kreiss and J. Oliger,  Discrete Methods for Time Dependent Problems

B. Gustafsson, H.-O. Kreiss, and J. Oliger, Time Dependent Problems and Difference Methods, 2nd edition, Wiley, 2013

J. Strikwerda, Finite Difference Schemes and Partial Differential Equations, Wadsworth & Brroks, 1989.

V. Thomee,  Stability Theory for PDEs, SIAM Rev. 11 (1969)

R. J. LeVeque, Numerical Methods for Conservation Laws, 2nd ed., Birkhäuser Verlag, 1992

Eitan Tadmor
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