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.

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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.