Asymptotic Error Analysis

Transcription

1 Asymptotic Error Analysis Brian Wetton Mathematics Department, UBC wetton PIMS YRC, June 3, 2014

2 Outline Overview Some History Romberg Integration Cubic Splines - Periodic Case More History: Strang s Trick PDE Problem and Discretization Piecewise Regular Grids Example Scheme Analysis Summary

3 Overview of the Talk Errors from computational methods using regular grids to compute 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 be called Asymptotic Error Analysis. Needs smooth solutions and regular grids. Historical examples: Romberg Integration, Cubic Splines, Strang s Trick. New result: a numerical artifact from an idealized adaptive grid with hanging nodes.

4 Trapezoidal Rule Trapezoidal Rule T h approximation to b a f (x)dx is the sum of areas of red trapezoids. Widths h = (b a)/n where N is the number of sub-intervals. Error bound b f (x)dx T h a where K = max f Second order convergence. (b a) Kh 2 12

5 Proof of Error Bound-I Consider a subinterval x [0, h]. Let L(x) be linear interpolation on this subinterval and g(x) = f (x) L(x), so g(0) = g(h) = 0. The error E of trapezoidal rule on this subinterval is Integrate by parts twice E = = 1 2 h 0 h 0 E = h 0 (x h/2)g (x)dx g(x)dx (x 2 xh)g (x)dx = 1 2 h 0 (x 2 xh)f (x)dx

6 Proof of Error Bound-II Subinterval E = 1 2 h 0 (x 2 xh)f (x)dx E K 2 h 0 (xh x 2 )dx = Kh3 12. Summing over N = (b a)/h subintervals gives the result I T h (b a) Kh 2 12

7 Trapezoidal Rule Applied Trapezoidal Rule applied to the integral I = 1 0 h I T h 1/ / / / / Not only is but I T h I T h lim h 0 h 2 (b a) Kh 2 12 sin xdx exists. There is regularity in the error that can be exploited.

8 Error Analysis of Trapezoidal Rule-I We had E Kh3 12 I T (b a) h Kh 2 12 but with a bit more work it can be shown that E = f aveh /12 + O(h 5 ) I T h = (b a) Ch 2 + O(h 4 ) 12 where C is average value of f on the subinterval. with more work the error in Trapezoidal Rule can be written as a series of regular terms with even powers of h (Euler-McLaurin Formula).

9 Error Analysis of Trapezoidal Rule-II T h = I + (b a) Ch 2 + O(h 4 ) 12 This error regularity justifies Richardson extrapolation I = ( 4 3 T h/2 1 3 T h) + O(h 4 ) The O(h 4 ) error above is regular and so can also be eliminated by extrapolation. Repeated application of this idea is the Romberg method.

10 Interesting Facts Richardson extrapolation of the Trapezoidal Rule is Simpson s Rule Trapezoidal and Midpoint Rules are spectrally accurate for integrals of periodic functions over their period

11 Cubic Splines Given smooth f (x) on [0,1], spacing h = 1/N, and data a i = f (ih) for i = 0,... N the standard cubic spline fit is a C 1 piecewise cubic interpolation. Cubic interpolation on each sub-interval for given values and second derivative values c i at the end points is fourth order accurate. If the second derivative values are only accurate to second order, the cubic approximation is still fourth order accurate. For C 1 continuity, c i 1 + 4c i + c i+1 = 6 h 2 (a i+1 2a i + a i 1 )

12 Cubic Splines - Periodic Analysis c i 1 + 4c i + c i+1 = 6 h 2 (a i+1 2a i + a i 1 ) In this case, c has a regular asymptotic error expansion c = f + h 2 ( )f +... (the fact that c i 1 + c i+1 = 2c i + h 2 c +... is used). Since the c s are second order accurate, the cubic spline approximation is fourth order accurate. Notes: The earliest convergence proof for splines is in this equally spaced, periodic setting Ahlberg and Nilson, Convergence properties of the spline fit, J. SIAM, 1963 Lucas, Asymptotic expansions for interpolating periodic splines, SINUM, 1982.

13 Cubic Splines - Non-Periodic Case c i 1 + 4c i + c i+1 = 6 h 2 (a i+1 2a i + a i 1 ) In the non-periodic case, additional conditions are needed for the end values c 0 and c N : natural: c 0 = 0, O(1) derivative: 2c 0 + c 1 = 6 h 2 (a 1 a 0 ) 3 h f (0), O(h 2 ) not a knot: c 0 2c 1 + c 2 = 0, O(h 2 ) First convergence proof for derivative conditions Birkhoff and DeBoor, Error Bounds for Spline Interpolation, J Math and Mech, 1964.

14 Cubic Splines - Numerical Boundary Layer c i 1 + 4c i + c i+1 = 6 h 2 (a i+1 2a i + a i 1 ) No regular error can match the natural boundary condition c 0 = 0. However, note that 1 + 4κ + κ 2 = 0 has a root κ Error Expansion: c i = f (ih) h f (ih) f (0)κ i... The new term is a numerical boundary layer. In this case, the spline fit will be second order near the ends of the interval and fourth order in the interior. Reference?

15 Cubic Splines - Computation

16 PDE Problem and Discretization Strang, Accurate Partial Differential Methods II. Non-linear Problems, Numerische Mathematik, 1964 Suppose u(x, t), 0 t T, x [0, 1], periodic in x is smooth and solves u t = f (u x ). Divide [0, 1] into N sub-intervals of length h = 1/N. Let U i (t) u(ih, t), i = 1 N. Let D 1 be the second order centred finite approximation of the first derivative D 1 U i = U i+1 U i 1 2h = u x (ih, t) + h2 6 u xxx(ih, t) + O(h 4 ) Semi-discrete scheme (method of lines): du i dt = f (D 1 U)

17 Analysis of Scheme-I du i dt du i dt = f (D 1 U) Introduce the error E = U u = f (D 1 u) 1 6 f (u x )u xxx h 2 + O(h 4 ) de i dt = f (D 1 u)d 1 E + f (D 1 (u + θe))(d 1 E) 2 + O(h 2 ) Introduce the discrete l 2 norm E 2 = N h Note that i E 2 i E h 1/2 E 2

18 Analysis of the Scheme-II To handle the nonlinear error, a bootstrap argument is used. Assume E 2 h p for t T and show that the scheme converges with order greater than p to finish the argument. de i dt = f (D 1 u)d 1 E + f (D 1 (u + θe))(d 1 E) 2 + O(h 2 ) Multiplying the equation above by h and taking the inner product with E gives (after summation by parts) d E 2 dt Df (D 1 u) E 2 + f (D 1 (u+θe)) h p 5/2 E 2 +O(h 2 ) Note that the bootstrap argument can t be closed with this estimate.

19 Asymptotic Error Analysis-I The truncation error is smooth so it is reasonable to expect error regularity U = u + h 2 e + O(h 4 ) with e a smooth function of x and t, independent of h. I ll show how to construct this function e below, but assume it exists: u + h 2 e has truncation error O(h 4 ) in the discrete equations so by using E = U (u + h 2 e) in the analysis above, the bootstrap argument can be closed. The error expression above can be used to show full order convergence in maximum norm and difference approximations to derivatives (super-convergence) D 2 U = u xx + h 2 ( 1 12 u xxxx + e xx ) + O(h 4 ) Error regularity also leads to full order convergence of derivatives of solution functionals with respect to parameters.

20 Asymptotic Error Analysis-II Want u + h 2 e to satisfy the discrete equations to fourth order. du dt = f (D 1U) Plug in, expand, identify the O(h 2 ) term: e t = f (u x )e x f (u x )u xxx Realize that this problem (linearization about the exact solution forced by terms involving derivatives of the exact solution) has a smooth solution. You don t have to find the solution, just know that it exists.

21 More History Goodman, Hou and Lowengrub, Convergence of the Point Vortex Method for the 2-D Euler Equations, Comm. Pure Appl. Math, 1990 E and Liu, Projection Method I: Convergence and Numerical Boundary Layers, SINUM, 1995

22 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

23 Idealized Piecewise Regular Grid Consider the idealized setting of a coarse grid and fine grid with a straight interface:

24 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 is used. At the interface, ghost points are introduced, related to grid points by linear extrapolation.

25 Analysis of Piecewise Regular Grid-I I advise not to try to analyze the accuracy of the scheme using the 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.

26 Analysis of Piecewise Regular Grid-II The ghost point extrapolation is equivalent to 1 4 (U A + U A + U B + U B ) = 1 2 (U C + U C ) 1 h (U A U A + U B U B ) = 1 h (U C U C ) (U A U B U A + U B ) = 0 The first two conditions are second order approximations of the interface conditions [u] = 0 and [ u/ n] = 0. They contribute to the second order regular errors of the scheme (different on either side of the grid interface).

27 Analysis of Piecewise Regular Grid-III (U A U B U A + U B ) = 0 This is satisfied to second order by the exact solution, error h 2 u xy /4. Note that this only involves fine grid points. Expect a parity difference between fine grid solutions at the interface. This results in a numerical artifact of the form h 2 A(y)( 1) j κ i where (i, j) is the fine grid index and κ This is a numerical boundary layer on the fine grid side that alternates in sign between vertically adjacent points.

28 Piecewise Regular Grid - Analysis Summary Coarse grid has regular error U coarse = u + h 2 e coarse +... Fine grid has regular error and the artifact U fine = u + h 2 e fine + h 2 u xy(0, y) 8(1 κ) ( 1)j κ i + Second order convergence in U (surprising? the scheme violates the rule of thumb ) Artifact causes loss of convergence in D 2,y U and D 2,x on the fine grid side at the interface.

29 Additional Discussion h q A(y)( 1) j κ i This artifact is present in all schemes (FE, FD, FV) on the grid, although the q may vary. Determinant condition, satisfied for stable schemes. For variable coefficient elliptic problems, κ(y) smooth.

30 Summary Asymptotic error analysis can be used to describe regular errors and numerical artifacts in finite difference methods and other schemes on regular meshes. Historical examples of Romberg integration; finite difference approximations of nonlinear, first order equations; and spline interpolation were given. Asymptotic error analysis can be used to help understand the accuracy of different implementations of boundary and interface conditions. A new result describing the errors in methods for elliptic problems on piecewise regular grids with hanging nodes was given.