Numerical Methods for Time-Dependent PDEs

AMSC 715, Fall 2024

Course Information

Place Phys 4122 (unless otherwise stated)
Time Tuesdays & Thursdays, 14-15:15pm
InstructorProfessor Eitan Tadmor
Contacttel.: x5-0648   e-mail:
Office HoursBy appointment 4141 CSIC Bldg #406
Teaching AssistantMichael Rozowski (e-mail: )
Grading60% Homework + 40% Final
Prerequisite MATH673, AMSC666

Course Description

Initial and initial-boundary value problems associated with time-dependent PDEs of hyperbolic and parabolic type. Finite difference and spectral methods for such time-dependent problems. The linear theory: accuracy, stability and convergence. von-Neumann, normal mode analysis and the eneregy method for problems with constant and variable coefficeints.
Nonlinear problems: shock discontinuities, viscosity and entropy. Monotonicity, TVD and entropy stability. High-resolution upwind and central schemes. Finite Volume (FV) and Discontinuous Galerkin (DG) methods.

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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
Assignment #1 [ pdf file ] with selected answers [ pdf file ]
  • 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 -- A brief overview [ pdf file ] and more detailed notes [ 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
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 #2 [ pdf file ] with selected answers [ pdf file ]

Lecture notes: Finite Difference Methods [ pdf file ]

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

Assignment #3 [ 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 #4 [ pdf file ] with selected answers [ pdf file ]

Lecture notes: Convergence theory [ pdf file ]

Related material:  2D LxW scheme (1964) [Lax-Wendroff (1964)] its staggered versions [Lax-Richtmyer; MacCormack]

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
Related material:  Stability [Lax-Nirenberg (1966)] Numerical dissipation [Kreiss (1964)] [Strang (1966)]

Multi-Dimensional Problems

    • ADI and splitting methods
Assignment #5 [ pdf file ] with selected answers [ pdf file ]

Lecture notes: freezing coefficients, energy method, splitting... [ 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
Lecture notes: Stability --- power-boundedness vs. the resolvent condition [ notes ]

  • Difference approximations to initial-boundary value problems
    • Eigen-mode analysis: Godunov-Ryabenkii condition
    • UKC the resolvent condition and stability
    • Examples
Lecture Notes: A primer on [normal mode analysis]

Lecture notes(*): L2 vs. resolvant stability: the C-N example for system [ pdf file ] and [ futher details]

Related material: GKS (1972), Stability Theory of Difference Approximations for Mixed IBVP. II  [ pdf file ]

Assignment #6 [ pdf file ] with selected answers [ pdf file]

Related material: Scheme independent stability criteria: [ CalTech 1980] or [1981]; more can be found [here]

Spectral Methods

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

Related material: Why accurate methods [Kreiss & Oliger (1972)]
          From finite-difference to smoothing of pseudo-dpectral methods [Tadmor (1987)]
          On Gibbs phenomenon [Gottlieb & Tadmor (1985)] [On filters and mollifiers (2007)]

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

Assignments #7 [ pdf file ] and #8 [ 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 ]

  • LxF, upwind and Godunov schemes: 1st-order methpds
    • Conservative schemes: shock capturing
    • Lax-Friedrichs, upwind and Godunov schemes
    • Monotonicity and L1-contraction
Related material:
Monotonicity L1-contraction and entropy [notes]


Lecture notes: Numerical solution of Burgers & Buckley-Levertt equations [ notes ]

Assignment #9 [ pdf file ]

  • Finite-volume: higher order resolution
    • TVD and non-oscillatory reconstructions
    • Godunov and Godunov-type schemes: limiters and high-resolution
    • Entropy stability: entropy conservative schemes and numerical dissipation
    • Discontinuous Galerkin (DG) method
    Related material:
    An overview on [Approximate methods for nonlinear conservation laws] (Venice 1996)
    High-resolution schemes: [Harten 2nd-order TVD scheme ]; [2nd-order central scheme ] [vA limiter ] and [more]
    Limiters and high-resolution methods: reviews in [ LeVeque notes] and [Sjogreen notes]
    On entropy stability of difference schemes: [Acta Numetrica Survey (2003)]

    Assignment #10 [ pdf file ]



    • Additional reading

    ► Leonhard Euler [pdf file] (by W. Gautschi)

    ► John von-Neumann [Wikipedia page]

    ► P. Lax on [ John von Neumann] and on [Flowering of Applied Math]

    References

    B. Cockburn, C.-W. Shu, C. Johnson, E. Tadmor, Advanced Numerical Approximation of Nonlinear Hyperbolic Equations
    Lectures given at CIME, Cetraro, Italy, (1998)

    D. Gottlieb and S.Orszag, Numerical Analysis of Spectral Methods. Theory and Applications SIAM (1977)

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

    J. Hesthaven, Numerical Methods for Conservation Laws, SIAM 2018

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

    H.-O. Kreiss and O. Widlund,  Difference Approximations for initial value problems for PDEs, Uppsala (1967)

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

    P. D. Lax,  Hyperbolic Conservation Laws and the Mathematical Theory of Shock Waves CBMS 1973

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

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

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

    E. Tadmor, Approximate solution of nonlinear conservation laws and related equations,
    AMS Proc. Symposia in Applied Math. 54 (1998), 325-368.

    E. Tadmor, A review of numerical methods for nonlinear partial differential equations
    Bulletin of the AMS, 49(4) (2012) 507-554.

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

    Eitan Tadmor
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