Unevenly Spaced Grid

Ae232a. © Tim Colonius 1

Finite difference schemes

• Alternative to function approximations for (spatial) discretization of PDE

• Collocation method is the limit of FD • Related to Lagrange Poly

– Similar to Lagrange Polynomial, but applied locally

• Measure of errors – points per wavelength


FD discretization of PDE Method of lines

• Semi-discrete approach • Approximate each term in the equation locally based on

function values at nearby points • Not always unambiguous because continuous terms can

be written in different forms (e.g. apply chain rule), and discrete approximations do not necessarily obey these continuous identities

• Different approximations usually needed for “interior” points than boundary or near-boundary points

• Leads to a system of ODE


Finite-difference Formulae:

• Simplest to derive using Taylor-series expansion

Stencil Size: l+r+1


Simply done in maple


Classroom Note:Calculation of Weights in Finite Difference Formulas Bengt Fornberg, SIAM Rev. 40, 685 (1998)


Notes

• Higher order derivatives similar • On evenly spaced grid, symmetry/antisymmetry for even/

odd derivatives automatically zeros the odd/even equations

• “Biased” (no symmetry) schemes are useful for representing points near the ‘edge’ of a domain.

• Construction of stencils larger than ~3 points need to be constructed with care due to instability.

• Geometry handled by overset grids.


Unevenly spaced grid

A better approach is to solve on a curvilinear grid

Geometry is handled by overset grids



Can we do better using same stencil?

• Suppose we use derivatives at adjacent points in our formula


Notes

• Terminology: – “Compact Finite Difference” – “Padé Finite Difference” – “Implicit Finite Difference”

• Better accuracy and smaller stencil for same order • Can play same games to derive forward/backward

biased schemes • But….we have coupled terms on the LHS • In matrix form:


Solution

• If we take at most L=R=1 tridiagonal matrix • If we take at most L=R=2 penta-diagonal matrix

• Still O(N) computations to compute the derivative • More computations • Bad for parallel computations • Obvious question…is the better accuracy for, say, the 4th order

compact scheme (compared to explicit scheme) cost effective?

– 4 times smaller leading error – E4 requires 3A + 2M per N – C4 requires 3A + 3M per N

• The answer is generally yes, but we will discuss more later


Optimized Finite Difference Schemes

• Note that we have used all of the coefficients to minimize the error in the limit of h 0.

• Perhaps we should examine errors at finite h and see if we can choose (some or all of) the coefficients more wisely?

• How do we examine error at finite h?

• Need to pick a function (or class of functions)

• How about cosine and sine?


Fourier Analysis of Differencing Errors

• Consider FD schemes applied to periodic problems • Compare FD derivative to spectral derivative for each

wavenumber • Later this analysis will be crucial in understanding the

stability of FD schemes


Fourier analysis of differencing errors

• For local approximations (Finite-difference) it is useful to compare results to Spectrally accurate method.

• Analyze how accurate local differencing is for complex exponential at a given wavelength

• Gives nonlocal information about error (i.e. information at all scales)

• Rigorously restricted to periodic domains, but also useful as an approximate analysis for the interior of large domains


The modified wavenumber


Explicit formula for modified wavenumber


Centered 1st Derivative Examples

3pt

5pt 7pt


2 ppw

3pt

5pt

7pt

3 ppw 4 ppw 10 ppw

Points per wavelength


Biased schemes

• Have complex modified wavenumbers


2nd Derivative Approximations (symmetric)

3 pt (2nd order)

5 pt (4th order)

7 pt (6th order)


6th order compact (r=l=2, R=L=1),

6th order explicit (r=l=3,R=L=0)



Compact schemes (1st deriv)


The modified wavenumber (another derivation)


The modified wavenumber


Centered 1st Derivative Examples

3pt

5pt 7pt





Explicit schemes

Implicit (compact, Padé) schemes


Optimized FD schemes

• Choose some coefficients to give a certain order of accuracy

• Choose some coefficients to give a good modified wavenumber relation:

– e.g. Make k’=k at some particular points – Minimize the error in k’-k

• Choose some coefficients to give other desirable properties – Efficiency? – Low storage? – Parallalizability? – And so on


Efficiency

• Different schemes have different operation counts

• Real cost is CPU time to compute a derivative, can be estimated with operation count

• Weight PPW with operation count to get normalized cost – Aside: scheme can also impact maximum stable time step for a

given PDE. The plot on the next page accounts for this for the model advection equation, ut + ux = 0


Efficiency

Ref: Colonius & Lele, Prog. Aerosp. Science, 2005

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Unevenly Spaced Grid