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Overview Some History More History: Strang’s Trick Piecewise Regular Grids Summary
Asymptotic Error Analysis
Brian Wetton
Mathematics Department, UBCwww.math.ubc.ca/∼wetton
PIMS YRC, June 3, 2014
Overview Some History More History: Strang’s Trick Piecewise Regular Grids Summary
Outline
Overview
Some HistoryRomberg IntegrationCubic Splines - Periodic Case
More History: Strang’s TrickPDE Problem and Discretization
Piecewise Regular GridsExample SchemeAnalysis
Summary
Overview Some History More History: Strang’s Trick Piecewise Regular Grids Summary
Overview of the Talk
• Errors from computational methods using regular grids tocompute smooth solutions have additional structure.
• This structure can• allow Richardson Extrapolation• lead to super-convergence• help in the analysis of methods for non-linear problems
• Numerical artifacts (non-standard errors) can be present
• The process of finding the structure and order of errors can becalled Asymptotic Error Analysis. Needs smooth solutions andregular grids.
• Historical examples: Romberg Integration, Cubic Splines,Strang’s Trick.
• New result: a numerical artifact from an idealized adaptivegrid with hanging nodes.
Overview Some History More History: Strang’s Trick Piecewise Regular Grids Summary
Trapezoidal Rule
• Trapezoidal Rule Th approximation to∫ ba f (x)dx is the sum of areas of red
trapezoids.
• Widths h = (b − a)/N where N is thenumber of sub-intervals.
• Error bound∣∣∣∣∫ b
af (x)dx − Th
∣∣∣∣ ≤ (b − a)
12Kh2
where K = max |f ′′|• Second order convergence.
Overview Some History More History: Strang’s Trick Piecewise Regular Grids Summary
Proof of Error Bound-I
• Consider a subinterval x ∈ [0, h].
• Let L(x) be linear interpolation on this subinterval andg(x) = f (x)− L(x), so g(0) = g(h) = 0.
• The error E of trapezoidal rule on this subinterval is
E =
∫ h
0g(x)dx
• Integrate by parts twice
E = −∫ h
0(x − h/2)g ′(x)dx
=1
2
∫ h
0(x2 − xh)g ′′(x)dx =
1
2
∫ h
0(x2 − xh)f ′′(x)dx
Overview Some History More History: Strang’s Trick Piecewise Regular Grids Summary
Proof of Error Bound-II
Subinterval E = 12
∫ h0 (x2 − xh)f ′′(x)dx
|E | ≤ K
2
∫ h
0(xh − x2)dx =
Kh3
12.
Summing over N = (b − a)/h subintervals gives the result
|I − Th| ≤(b − a)
12Kh2
Overview Some History More History: Strang’s Trick Piecewise Regular Grids Summary
Trapezoidal Rule Applied
Trapezoidal Rule applied to the integral I =∫ 10 sin xdx
h I − Th
1/2 0.00961/4 0.00241/8 0.000601/16 0.000151/32 0.00004
Not only is
|I − Th| ≤(b − a)
12Kh2
but
limh→0
I − Th
h2
exists. There is regularity in the error that can be exploited.
Overview Some History More History: Strang’s Trick Piecewise Regular Grids Summary
Error Analysis of Trapezoidal Rule-I
• We had
|E | ≤ Kh3
12⇒ |I − Th| ≤
(b − a)
12Kh2
• but with a bit more work it can be shown that
E = −f ′′aveh3/12 + O(h5) ⇒
I − Th = −(b − a)
12Ch2 + O(h4)
where C is average value of f ′′ on the subinterval.
• with more work the error in Trapezoidal Rule can be writtenas a series of regular terms with even powers of h(Euler-McLaurin Formula).
Overview Some History More History: Strang’s Trick Piecewise Regular Grids Summary
Error Analysis of Trapezoidal Rule-II
Th = I +(b − a)
12Ch2 + O(h4)
• This error regularity justifies Richardson extrapolation
I = (4
3Th/2 −
1
3Th) + O(h4)
• The O(h4) error above is regular and so can also beeliminated by extrapolation. Repeated application of this ideais the Romberg method.
Overview Some History More History: Strang’s Trick Piecewise Regular Grids Summary
Interesting Facts
• Richardson extrapolation of the Trapezoidal Rule is Simpson’sRule
• Trapezoidal and Midpoint Rules are spectrally accurate forintegrals of periodic functions over their period
Overview Some History More History: Strang’s Trick Piecewise Regular Grids Summary
Cubic Splines
• Given smooth f (x) on [0,1], spacing h = 1/N, and dataai = f (ih) for i = 0, . . . N the standard cubic spline fit is a C1
piecewise cubic interpolation.• Cubic interpolation on each sub-interval for given values and
second derivative values ci at the end points is fourth orderaccurate.
• If the second derivative values are only accurate to secondorder, the cubic approximation is still fourth order accurate.
• For C1 continuity,
ci−1 + 4ci + ci+1 =6
h2(ai+1 − 2ai + ai−1)
Overview Some History More History: Strang’s Trick Piecewise Regular Grids Summary
Cubic Splines - Periodic Analysis
ci−1 + 4ci + ci+1 =6
h2(ai+1 − 2ai + ai−1)
In this case, c has a regular asymptotic error expansion
c = f ′′ + h2(1
12− 1
6)f ′′′′ + . . .
(the fact that ci−1 + ci+1 = 2ci + h2c ′′ + . . . is used). Since thec ’s are second order accurate, the cubic spline approximation isfourth order accurate.
Notes:
• The earliest convergence proof for splines is in this equallyspaced, periodic setting Ahlberg and Nilson, “Convergenceproperties of the spline fit”, J. SIAM, 1963
• Lucas, “Asymptotic expansions for interpolating periodicsplines,” SINUM, 1982.
Overview Some History More History: Strang’s Trick Piecewise Regular Grids Summary
Cubic Splines - Non-Periodic Case
ci−1 + 4ci + ci+1 =6
h2(ai+1 − 2ai + ai−1)
In the non-periodic case, additional conditions are needed for theend values c0 and cN :
natural: c0 = 0, O(1)
derivative: 2c0 + c1 = 6h2 (a1 − a0)− 3
h f ′(0), O(h2)
not a knot: c0 − 2c1 + c2 = 0, O(h2)
First convergence proof for “derivative” conditions Birkhoff andDeBoor, “Error Bounds for Spline Interpolation”, J Math andMech, 1964.
Overview Some History More History: Strang’s Trick Piecewise Regular Grids Summary
Cubic Splines - Numerical Boundary Layer
ci−1 + 4ci + ci+1 =6
h2(ai+1 − 2ai + ai−1)
No regular error can match the natural boundary condition c0 = 0.However, note that
1 + 4κ + κ2 = 0
has a root κ ≈ −0.268.
Error Expansion:
ci = f ′′(ih)− h2 1
6f ′′′′(ih)− f ′′(0)κi . . .
The new term is a numerical boundary layer. In this case, thespline fit will be second order near the ends of the interval andfourth order in the interior. Reference?
Overview Some History More History: Strang’s Trick Piecewise Regular Grids Summary
Cubic Splines - Computation
Overview Some History More History: Strang’s Trick Piecewise Regular Grids Summary
PDE Problem and DiscretizationStrang, “Accurate Partial Differential Methods II. Non-linearProblems,” Numerische Mathematik, 1964
• Suppose u(x , t), 0 ≤ t ≤ T , x ∈ [0, 1], periodic in x issmooth and solves ut = f (ux).
• Divide [0, 1] into N sub-intervals of length h = 1/N. LetUi (t) ≈ u(ih, t), i = 1 · · ·N.
• Let D1 be the second order centred finite approximation ofthe first derivative
D1Ui =Ui+1 − Ui−1
2h
= ux(ih, t) +h2
6uxxx(ih, t) + O(h4)
• Semi-discrete scheme (method of lines):
dUi
dt= f (D1U)
Overview Some History More History: Strang’s Trick Piecewise Regular Grids Summary
Analysis of Scheme-I
dUi
dt= f (D1U)
dui
dt= f (D1u)− 1
6f ′(ux)uxxxh
2 + O(h4)
Introduce the error E = U − u
dEi
dt= f ′(D1u)D1E + f ′′(D1(u + θE ))(D1E )2 + O(h2)
Introduce the discrete l2 norm
‖E‖2 =
√√√√hN∑i
E 2i
Note that‖E‖∞ ≤ h−1/2‖E‖2
Overview Some History More History: Strang’s Trick Piecewise Regular Grids Summary
Analysis of the Scheme-II
To handle the nonlinear error, a bootstrap argument is used.Assume
‖E‖2 ≤ hp for t ≤ T
and show that the scheme converges with order greater than p tofinish the argument.
dEi
dt= f ′(D1u)D1E + f ′′(D1(u + θE ))(D1E )2 + O(h2)
Multiplying the equation above by h and taking the inner productwith E gives (after summation by parts)
d‖E‖2
dt≤ ‖Df ′(D1u)‖∞‖E‖2+‖f ′′(D1(u+θE ))‖∞hp−5/2‖E‖2+O(h2)
Note that the bootstrap argument can’t be closed with thisestimate.
Overview Some History More History: Strang’s Trick Piecewise Regular Grids Summary
Asymptotic Error Analysis-IThe truncation error is smooth so it is reasonable to expect errorregularity
U = u + h2e + O(h4)
with e a smooth function of x and t, independent of h. I’ll showhow to construct this function e below, but assume it exists:
• u + h2e has truncation error O(h4) in the discrete equationsso by using E = U − (u + h2e) in the analysis above, thebootstrap argument can be closed.
• The error expression above can be used to show full orderconvergence in maximum norm and difference approximationsto derivatives (super-convergence)
D2U = uxx + h2(1
12uxxxx + exx) + O(h4)
• Error regularity also leads to full order convergence ofderivatives of solution functionals with respect to parameters.
Overview Some History More History: Strang’s Trick Piecewise Regular Grids Summary
Asymptotic Error Analysis-II
• Want u + h2e to satisfy the discrete equations
dU
dt= f (D1U)
to fourth order.
• Plug in, expand, identify the O(h2) term:
et = f ′(ux)ex +1
6f ′(ux)uxxx
• Realize that this problem (linearization about the exactsolution forced by terms involving derivatives of the exactsolution) has a smooth solution. You don’t have to find thesolution, just know that it exists.
Overview Some History More History: Strang’s Trick Piecewise Regular Grids Summary
More History
• Goodman, Hou and Lowengrub, “Convergence of the PointVortex Method for the 2-D Euler Equations,” Comm. PureAppl. Math, 1990
• E and Liu, “Projection Method I: Convergence and NumericalBoundary Layers,” SINUM, 1995
Overview Some History More History: Strang’s Trick Piecewise Regular Grids Summary
Piecewise Regular Grids
• Computations on regular grids have many advantages.
• To retain some of the advantages but allow adaptivity,refinement in regular blocks is often done.
Clinton Groth, University of Toronto
Overview Some History More History: Strang’s Trick Piecewise Regular Grids Summary
Idealized Piecewise Regular Grid
Consider the idealized setting of a coarse grid and fine grid with astraight interface:
Overview Some History More History: Strang’s Trick Piecewise Regular Grids Summary
Problem and Discretization
• Consider the problem ∆u = f .
• The grid spacing is h (coarse) and h/2 (fine).
• The discrete approximation is cell-centred, denoted by U.
• Away from the interface, a five point stencil approximation isused.
• At the interface, ghost points are introduced, related to gridpoints by linear extrapolation.
Overview Some History More History: Strang’s Trick Piecewise Regular Grids Summary
Analysis of Piecewise Regular Grid-I
• I advise not to try to analyze the accuracy of the scheme usingthe discrete approximation of the PDE near the interface
• Instead, determine the accuracy at which the “interface”conditions [u] = 0 and [∂u/∂n] = 0 are approximated.
Overview Some History More History: Strang’s Trick Piecewise Regular Grids Summary
Analysis of Piecewise Regular Grid-II
• The ghost point extrapolation is equivalent to
1
4(UA + UA∗ + UB + UB∗) =
1
2(UC + UC∗)
1
h(UA − UA∗ + UB − UB∗) =
1
h(UC∗ − UC∗)
(UA − UB − UA∗ + UB∗) = 0
• The first two conditions are second order approximations ofthe “interface” conditions [u] = 0 and [∂u/∂n] = 0.
• They contribute to the second order regular errors of thescheme (different on either side of the grid interface).
Overview Some History More History: Strang’s Trick Piecewise Regular Grids Summary
Analysis of Piecewise Regular Grid-III
(UA − UB − UA∗ + UB∗) = 0
• This is satisfied to second order by the exact solution, errorh2uxy/4.
• Note that this only involves fine grid points.
• Expect a parity difference between fine grid solutions at theinterface.
• This results in a numerical artifact of the form
h2A(y)(−1)jκi
where (i , j) is the fine grid index and κ ≈ 0.172.
• This is a numerical boundary layer on the fine grid side thatalternates in sign between vertically adjacent points.
Overview Some History More History: Strang’s Trick Piecewise Regular Grids Summary
Piecewise Regular Grid - Analysis Summary
• Coarse grid has regular error Ucoarse = u + h2ecoarse + . . .
• Fine grid has regular error and the artifact
Ufine = u + h2efine + h2 uxy (0, y)
8(1− κ)(−1)jκi + · · ·
• Second order convergence in U (surprising? the schemeviolates the “rule of thumb”)
• Artifact causes loss of convergence in D2,yU and D2,x on thefine grid side at the interface.
Overview Some History More History: Strang’s Trick Piecewise Regular Grids Summary
Additional Discussion
hqA(y)(−1)jκi
• This artifact is present in all schemes (FE, FD, FV) on thegrid, although the q may vary.
• Determinant condition, satisfied for stable schemes.
• For variable coefficient elliptic problems, κ(y) smooth.
Overview Some History More History: Strang’s Trick Piecewise Regular Grids Summary
Summary
• Asymptotic error analysis can be used to describe regularerrors and numerical artifacts in finite difference methods andother schemes on regular meshes.
• Historical examples of Romberg integration; finite differenceapproximations of nonlinear, first order equations; and splineinterpolation were given.
• Asymptotic error analysis can be used to help understand theaccuracy of different implementations of boundary andinterface conditions.
• A new result describing the errors in methods for ellipticproblems on piecewise regular grids with hanging nodes wasgiven.