8/7/2019 Computation of the L^2 norm in a mixed Fourier/B-spline collocation discretization

1/2

L2 norm within Suzerains discretization Page 1 of 2

We wish to compute the L2 norm of an instantaneous, real-valued field u(x,y,z) on the spatial domainLx

2,Lx2

[0,Ly]

Lz

2,Lz2

which has been discretized as

uh(x,y,z) =

Ny1l=0

Nx21

m=Nx

2

Nz21

n=Nz2

ulmnBl(y) ei2mLx xei 2nLz z =

l

m

n

ulmnBl(y) eikmxeiknz

where km = 2m/Lx, kn = 2n/Lz, and Bl(y) are a B-spline basis for some order and knot selection.

By direct computation we find

||u||2L2xyz

=

Ly0

Lx2

Lx2

Lz2

Lz2

uudz dx dy

Ly0

Lx2

Lx2

Lz2

Lz2

l

m

n

ulmnBl(y) eikmxeiknz

l

m

n

ulmnBl(y) eikmxeiknz

dz dx dy

=

l

l

Ly

0

Bl(y) Bl(y)m

m

Lx2

Lx2

eikmxeikmxn

n

Lz2

Lz2

eiknzeiknzulmnulmn

dz

dx dy

=

l

l

Ly0

Bl(y) Bl(y)m

m

Lx2

Lx2

eikmxeikmxLzn

ulmnulmn

dx dy

=

l

l

Ly0

Bl(y) Bl(y)LxLzm

n

ulmnulmn

dy

= LxLz

m

n

l

ulmn

l

ulmn

Ly0

Bl(y) Bl(y) dy

= LxLzm

n

umnHMumn where M=Ly

0

Bl(y) Bl(y) dy.

In the above expression the three-dimensional field of transform coefficients ulmn is treated as a two-dimensional

collection of length Ny vectors indexed by m and n. For a B-spline basis with piecewise polynomial order

k 1 giving rise to a real-valued, symmetric positive definite matrix Mwith bandwidth 2k 1, the compu-

tational cost to find ||u||2L2xyz

scales as OkNxN

2yNz

.

The y-varying norm over the x and z directions can be computed via

||u||2L2xz

(y) =

Lx2

Lx2

Lz2

Lz2

uudz dx

Lx2

Lx2

Lz2

Lz2

l

m

n

ulmnBl(y) eikmxeiknz

l

m

n

ulmnBl(y) eikmxeiknz

dz dx

=

l

Bl(y)l

Bl(y)m

m

Lx2

Lx2

eikmxeikmxn

n

Lz2

Lz2

eiknzeiknzulmnulmn

dz

dx= LxLz

l

Bl(y)l

Bl(y)m

n

ulmnulmn.

Using the limited, symmetric support of the productsBl (y) Bl (y) to reduce the required operations, one can

compute this fully functional representation of||u||2L2xz

(y) at a cost proportional to OkNxN

2yNz

.

8/7/2019 Computation of the L^2 norm in a mixed Fourier/B-spline collocation discretization

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L2 norm within Suzerains discretization Page 2 of 2

When ||u||2L2xz

(y) is only of interest at pointsyj, the previous expression yields

||u||2L2xz

(yj) LxLzm

n

l

Blyj ulmn

l

Blyj ulmn

= LxLz

m

n

l

Blyj

ulmn

2

= LxLz

m

n

(Mumn)y=yj2 where M= Blyj

where again the three-dimensional field is treated as as a two-dimensional field of vectors. The computa-

tional cost for this operation is much lower than the others at O(2k 1)NxNyNz

.

These three results include summations over all coefficients in the homogeneous x and z directions, namely

m {Nx/2, . . . , 0, . . . ,Nx/2 1} and n {Nz/2, . . . , 0, . . . ,Nz/2 1}. Because u(x,y,z) is real-valued, its

transform coeffi

cients ulmn exhibit conjugate symmetry in one of the homogeneous directions. Often, notall of these coefficients are stored. Consequently, care must be exercised when evaluating summations likem

n within such norm calculations.

More concretely, say one employs conjugate symmetry in thex direction when computing f(ulmn). Then

Nx21

m=Nx

2

Nz21

n=Nz2

f(ulmn) =

Nz21

n=Nz2

Nx

21

m=0

f(ulmn) +

1m=

Nx2

f(ulmn)

=

Nz21

n=Nz2

Nx21

m=0

f(ulmn) +

Nx2

m=1

fulmn

=

Nz21

n=Nz2

f(ulmn)|m=0 +Nx

21

m=1

f(ulmn) + f

ulmn

+ f

ulmn

m=

Nx2

.

When f(ulmn) = fulmn

holds, as it does for ||u||2

L2xyz, ||u||2

L2xz(yj), and ||u||

2

L2xz(yj), the summands further

simplify to yield

Nx21

m=Nx

2

Nz21

n=Nz2

f(ulmn) =

Nz21

n=Nz2

f(ulmn)|m=0 + 2Nx

21

m=1

f(ulmn) + f(ulmn)|m=Nx2

.