Numerical Methods for PDEs - kjerland/nm-seminar/intro.pdfNumerical Methods for PDEs Outline 1 Numerical Methods for PDEs 2 Finite Di erence method 3 Finite Volume method 4 Spectral methods 5 Finite Element method

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  • Numerical Methods for PDEsAn introduction

    Marc Kjerland

    University of Illinois at Chicago

    Marc Kjerland (UIC) Numerical Methods for PDEs January 24, 2011 1 / 39

  • Numerical Methods for PDEs

    Outline

    1 Numerical Methods for PDEs

    2 Finite Difference method

    3 Finite Volume method

    4 Spectral methods

    5 Finite Element method

    6 Other considerations

    Marc Kjerland (UIC) Numerical Methods for PDEs January 24, 2011 2 / 39

  • Numerical Methods for PDEs

    Preliminaries

    We seek to solve the partial differential equation

    Pu = f

    where u is an unknown function on a domain RN , P is a differentialoperator, and f is a given function on . Typically u also satisfies someinitial and/or boundary conditions. It is seldom possible to find exactsolutions analytically.A numerical method will typically find an approximation to u by making adiscretization of the domain or by seeking solutions in a reduced functionspace.

    Marc Kjerland (UIC) Numerical Methods for PDEs January 24, 2011 3 / 39

  • Finite Difference method

    Outline

    1 Numerical Methods for PDEs

    2 Finite Difference method

    3 Finite Volume method

    4 Spectral methods

    5 Finite Element method

    6 Other considerations

    Marc Kjerland (UIC) Numerical Methods for PDEs January 24, 2011 4 / 39

  • Finite Difference method

    Finite differences

    The basic idea for the finite difference method is to replace derivativeswith finite differences. Consider u = u(x , t) and let h, k > 0. Then wecould use the following approximations:

    u

    x u(x + h, t) u(x , t)

    hu

    t u(x , t + k) u(x , t)

    k

    Marc Kjerland (UIC) Numerical Methods for PDEs January 24, 2011 5 / 39

  • Finite Difference method

    Domain discretization

    Let us define a regular grid of points (xm, tn) = (mh, nk) for some integersm and n. In general these points need not be equally spaced. The finitedifference algorithm will generate approximations to u at each grid point.

    Marc Kjerland (UIC) Numerical Methods for PDEs January 24, 2011 6 / 39

  • Finite Difference method

    Notation

    We introduce the notation unm = u(xm, tn). Then we can write:

    xunm

    unm+1 unmh

    tunm

    un+1m unmk

    .

    These are called forward differences; there are many other possible choices.

    Marc Kjerland (UIC) Numerical Methods for PDEs January 24, 2011 7 / 39

  • Finite Difference method

    Example

    Consider the one-way wave equation:

    ut + aux = 0

    with initial conditionu(x , 0) = u0(x).

    Here is the forward-time forward-space approximation:

    un+1m unmk

    + aunm+1 unm

    h= 0,

    from which we can derive the following explicit scheme:

    un+1m = (1 +akh )u

    nm akh u

    nm+1.

    Marc Kjerland (UIC) Numerical Methods for PDEs January 24, 2011 8 / 39

  • Finite Difference method

    Consistency

    Will our finite difference scheme actually generate a solution of the PDE?Consider the one-way wave equation with a = 1:{

    ut + ux = 0,

    u(x , 0) = u0(x).

    The exact solution is u(x , t) = u0(x t), found using the method ofcharacteristics. As time increases, initial data is propagated to the rightwith speed 1. The forward-time forward-space scheme cannot reproducethis behavior, so the scheme is inconsistent with the PDE.

    A scheme which is consistent with the one-way wave equation for all a isthe forward-time center-space scheme:

    un+1m unmk

    + aunm+1 unm1

    2h= 0.

    Marc Kjerland (UIC) Numerical Methods for PDEs January 24, 2011 9 / 39

  • Finite Difference method

    Consistency

    Will our finite difference scheme actually generate a solution of the PDE?Consider the one-way wave equation with a = 1:{

    ut + ux = 0,

    u(x , 0) = u0(x).

    The exact solution is u(x , t) = u0(x t), found using the method ofcharacteristics. As time increases, initial data is propagated to the rightwith speed 1. The forward-time forward-space scheme cannot reproducethis behavior, so the scheme is inconsistent with the PDE.A scheme which is consistent with the one-way wave equation for all a isthe forward-time center-space scheme:

    un+1m unmk

    + aunm+1 unm1

    2h= 0.

    Marc Kjerland (UIC) Numerical Methods for PDEs January 24, 2011 9 / 39

  • Finite Difference method

    Stability

    A poorly-chosen numerical scheme can sometimes result in uncontrolled(and incorrect) growth of the solution. We say that a finite differencescheme for a first-order equation is stable if there is an integer J such thatfor any positive time T , there is a constant CT such that

    un2h CTJ

    j=0

    uj2h.

    This is typically shown using Von Neumann analyis in Fourier space; thereis often a strong dependence on the relation between h and k .The forward-time center-space scheme for the one-way wave equation isunstable.

    Marc Kjerland (UIC) Numerical Methods for PDEs January 24, 2011 10 / 39

  • Finite Difference method

    Implicit schemes

    The backward-time center-space scheme is both consistent with theone-way wave equation and unconditionally stable:

    un+1m unmk

    + aun+1m+1 u

    n+1m1

    2h= 0

    This is an implicit scheme, since unknown values appear multiple times inthe equation. Implicit schemes often allow for much larger grid spacingbut require significant additional calculations at each step.

    Marc Kjerland (UIC) Numerical Methods for PDEs January 24, 2011 11 / 39

  • Finite Difference method

    Order of accuracy

    Consider Poissons equation in 2D:

    uxx + uyy = f .

    The discrete five-point Laplacian approximation is given by:

    um+1,n + um1,n + um,n+1 + um,n1 4um,n = h2fm,n

    Marc Kjerland (UIC) Numerical Methods for PDEs January 24, 2011 12 / 39

  • Finite Difference method

    Order of accuracy

    The discrete nine-point Laplacian is of higher accuracy: o(h4) vs. o(h2).However, it requires more information at each step:

    16 (um+1,n+1 + um+1,n1 + um1,n+1 + um1,n1)+

    + 23 (um+1,n + um1,n + um,n+1 + um,n1)103 um,n =

    = h2

    12 (fm+1,n + fm1,n + fm,n+1 + fm,n1)

    Marc Kjerland (UIC) Numerical Methods for PDEs January 24, 2011 13 / 39

  • Finite Difference method

    Pros and cons of finite difference methods

    Advantages:

    Fast Easy to code

    Disadvantages:

    Hard to generalize in complex geometries

    Marc Kjerland (UIC) Numerical Methods for PDEs January 24, 2011 14 / 39

  • Finite Volume method

    Outline

    1 Numerical Methods for PDEs

    2 Finite Difference method

    3 Finite Volume method

    4 Spectral methods

    5 Finite Element method

    6 Other considerations

    Marc Kjerland (UIC) Numerical Methods for PDEs January 24, 2011 15 / 39

  • Finite Volume method

    Conservation laws

    The finite volume method is used to find numerical solutions toconservation laws. A conservation law is a PDE written generally in theform:

    tq(x , t) + div F(x , t) = f (x , t),

    where q is a conserved quantity (i.e. mass, energy), F is a fluxrepresenting a transport mechanism for q, and f is a forcing term. Thereis an implicit dependence on an unknown u(x , t)

    Marc Kjerland (UIC) Numerical Methods for PDEs January 24, 2011 16 / 39

  • Finite Volume method

    Examples of conservation laws

    Linear transport equation:{tu(x , t) + div(vu)(x , t) = 0, t > 0,

    u(x , 0) = u0(x).

    [q = u(x , t),F = vu(x , t), f = 0]

    Stationary diffusion equation:{u = f , on = (0, 1) (0, 1)

    u = 0, on

    [q = u(x),F = u(x)]

    Marc Kjerland (UIC) Numerical Methods for PDEs January 24, 2011 17 / 39

  • Finite Volume method

    Examples of conservation laws

    1D Euler equations:

    t

    uE

    + x uu2 + p

    u(E + p)

    = 0.

    Marc Kjerland (UIC) Numerical Methods for PDEs January 24, 2011 18 / 39

  • Finite Volume method

    Intro to the Finite Volume method

    Let Rd be our spatial domain, and let T be a polygonal mesh on .Consider a control volume K T . Integrating over K , we have by thedivergence theorem:

    Ktq(x , t)dx +

    K

    F(x , t) nK (x)d =K

    f (x , t)dx .

    Let NK T be the set of neighbors of K . ThenKtq(x , t)dx +

    LNK

    KL

    F(x , t) nK (x)d =K

    f (x , t)dx .

    Marc Kjerland (UIC) Numerical Methods for PDEs January 24, 2011 19 / 39

  • Finite Volume method

    Discretization and approximation

    Using an explicit forward-time scheme, we can writeK

    qn+1(x) qn(x)k

    dx +LNK

    nK ,L =

    K

    f (x , tn)dx ,

    where nK ,L is an approximation to the flux from K to L.

    Marc Kjerland (UIC) Numerical Methods for PDEs January 24, 2011 20 / 39

  • Finite Volume method

    The numerical flux

    In a finite volume scheme, we seek an appropriate time discretization aswell as numerical flux terms F nK ,L which are

    Conservative, i.e. F nK ,L = F nL,K . Consistent with the PDE.

    Marc Kjerland (UIC) Numerical Methods for PDEs January 24, 2011 21 / 39

  • Finite Volume method

    Pros and cons of finite volume methods

    Advantages:

    Robust and cheap for conservation laws Can handle complex geometries

    Disadvantages:

    High precision difficult (see: Discontinuous Galerkin)

    Marc Kjerland (UIC) Numerical Methods for PDEs January 24, 2011 22 / 39

  • Spectral methods

    Outline

    1 Numerical Methods for PDEs

    2 Finite Difference method

    3 Finite Volume method

    4 Spectral methods

    5 Finite Element method

    6 Other considerati

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