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  • Numerical discretization of Hamiltonian PDEs

    Jason FrankCWI, Amsterdam

    6 May 2009Numerical Methods for Time-Dependent PDEs

  • Hamiltonian Systems

  • OutlineSymplectic integrators for Hamiltonian ODEs

    Hamiltonian systems

    Conservation laws

    Symplectic integrators

    Examples

    Symplectic integrators for Hamiltonian PDEs

    Infinite dimensional Hamiltonian systems

    Hamiltonian semi-discretizations

    Examples

  • Hamiltonian SystemsA Hamiltonian system of ordinary differential equations

    Compact notation

    More compact

    The J above is canonical. A non-canonical H.S. is defined equivalently, but with arbitrary skew-symmetric matrix J:

  • Examples of Hamiltonian ODEsNonlinear pendulum:

    Kepler problem

    N-body problem

  • Properties of Hamiltonian ODEsEnergy conservation. The Hamiltonian typically represents the total energy. Along a solution,

    The phase flow preserves volume. The divergence of a Hamiltonian vector field is zero.

  • Flow mapGiven an autonomous ordinary differential equation

    define the flow map to be the operator such that

    Properties:

    Semi-group property:

    Mapping of sets:

  • Symplectic mapsA symplectic map is one whose Jacobian satisfies

    The flow map of a Hamiltonian system is symplectic:

    The quantity is a quadratic first integral of the coupled system

  • Symplectic numerical methodA numerical method describes a discrete flow map. For example, Eulers method:

    The numerical map is symplectic if

    To prove method is symplectic, it is sufficient to show:

    That the derivative of the numerical flow with respect to the initial condition is equivalent to the method applied to the variational equation, and

    That the method conserves the invariant S, i.e.

  • Symplectic numerical methodsFor the implicit midpoint rule

    This is equivalent to implicit midpoint applied to the coupled system

    Implicit midpoint preserves arbitrary quadratic invariants. Suppose is a first integral. Hence,

  • Symplectic numerical methodsFor the canonical case, , the quadratic invariant S can

    be simplified. Partition , then

    It follows that methods for canonical problems which preserve invariants of the following form are symplectic:

    The Strmer-Verlet method is

    You can check that this method preserves arbitrary quadratic invariants of the above partitioned form.

  • Backward error analysisClassical error analysis for numerical integrators compares the numerical solution with the exact solution through the initial value.

    In backward error analysis we try to find a modified problem for which the numerical trajectory is exact.

    This modified differential equation is an expansion in the stepsize parameter:

    Forward Euler agrees with the first two terms on the left. But it is a more accurate approximation of the modified equation if we choose:

  • Backward error analysisFor Hamiltonian systems, it can be shown that the modified vector field associated to a symplectic method has the form

    That is, the modified equations are also Hamiltonian with a perturbed energy function.

    The asymptotic expansion generally diverges, but may be optimally truncated to a number of terms which grows exponentially as the stepsize is decreased.

    Due to conservation of the modified Hamiltonian, the original Hamiltonian is preserved approximately: with bounded variation over exponentially long intervals. For a method of order p:

  • Compare solutions of the pendulum equations:

    using Forward, Backward and Symplectic Eulers.

    PendulumFE

    BE SE

  • Statistical mechanicsA Lorenz model with Hamiltonian structure:

    This is a simplified, low-order model representing some typical behavior of the atmosphere. The solutions are chaotic. Sometimes it is desirable to solve such systems on long time intervals to produce a data set for statistics.

    The long time average of a function g of the first two variables is:

  • Statistical mechanicsSimulations using a standard (4th-order) time integrator show heavy dependence on the method parameters and integration time.

  • Statistical mechanicsSimulations using even a 1st-order symplectic integrator show give much better statistics.

    The combination of energy and volume conservation is crucial for statistics.

  • Hamiltonian PDEs arise as the infinite dimensional abstraction of Hamiltonian ODEs. Instead of we consider a domain and a Hilbert space with accompanying inner product:

    The Hamiltonian is a functional defined by integration over

    A Hamiltonian PDE is given by

    Where

    Examples:

    Hamiltonian PDEs

  • The variational derivative is also defined with respect to the inner product on

    For simplicity, we assume a periodic or unbounded domain. The variational derivative is

    Conservation of the Hamiltonian is seen by

    Variational derivative

    Fine print: integration by parts--we assume the boundary terms vanish. If not, there is energy flux across the boundary, so no energy conservation.

  • Sample calculation

    Variational derivative

    Only the terms survive

  • Nonlinear wave equation:

    Korteweg-de Vries equation:

    Examples of Hamiltonian PDEs

    The case V(u) = -cos(u) is the Sine-Gordon equation

  • The main idea of discretizing Hamiltonian PDEs is to preserve the Hamiltonian structure under spatial semi-discretization, so that we can take advantage of symplectic time integrators.

    To do this, it is enough to consider the discretization of the structure (mathematics) to preserve skew-symmetry, and the Hamiltonian (physics) using any convenient quadrature rule.

    Define a grid:

    Define a discrete inner product:

    Quadrature rule:

    Discrete structure:

    Numerical discretization

  • The variational derivative

    The semi-discretization defines a Hamiltonian ODE

    The discrete energy is a first integral of this ODE. But only that quadrature which was used to define H is exactly conserved. An issue of some confusion.

    Numerical discretization

  • The Kortweg-de Vries equation has Hamiltonian structure

    Choose Euclidean inner product on

    D = central difference operator

    Hamiltonian quadrature

    Example: KdV equation

  • Detailed calculation of the variational derivative

    Example: KdV equation

    Only the terms survive

    Rearranging the terms of the summation, using periodicity:

  • Example: Linear wave equation can be written as a Hamiltonian PDE:

    Choosing a collocated placement and central differences as used for KdV gives

    The even and odd grid points decouple!

  • A better approach is to define a staggered placement

    Define dual discrete function spaces

    Partition

    Define dual difference operators

    Discretization:

    Example: Linear wave equation

  • Experiments: KdV EquationApplying the two-step leapfrog method

    to the semi-discretization of the KdV equation, leads to a blow-up instability.

    The same spatial discretization is stable for a symplectic time stepping method.

  • KdV Equation

    In fact, with the symplectic method we can solve the KdV on an interval more than 10x as long.

  • Statistics of fluids

    The equations for an ideal fluid are Hamiltonian with a special structure.

    They conserve not only energy but an infinite class of vorticity functionals.

    In this experiment we compute the average vorticity and stream function fields on very long time intervals using (a) a finite difference method that conserves energy and one vorticity functional, and (b-d) a symplectic method that conserves energy in BEA sense and all vorticity integrals.

    Statistical studies, weather/climate.

    = 0

    0 2 4 60

    2

    4

    6 = 2

    0 2 4 60

    2

    4

    6

    = 4

    0 2 4 60

    2

    4

    6 = 6

    0 2 4 60

    2

    4

    6

    0.4

    0.2

    0

    0.2

    0.4

    0.6

    1 0.5 0 0.5 11

    0.5

    0

    0.5

    1

    = 2

    q

    1 0.5 0 0.5 11

    0.5

    0

    0.5

    1

    q

    = 0

    1 0.5 0 0.5 11

    0.5

    0

    0.5

    1

    = 4

    q

    1 0.5 0 0.5 11

    0.5

    0

    0.5

    1

    q

    = 6

  • Exercises1. Prove that the symplectic Euler method is symplectic

    2. Consider the nonlinear Schrdinger equation:

    a. Show that this is Hamiltonian with

    b. Show that there is an additional conserved quantity

    c. Derive a Hamiltonian semi-discretization for this equation

    d. Determine if your discretization preserves the second invariant

    e. Which time integrator would do the best job of preserving the invariants?

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