112
3. Beyond the One-Way Wave Equation
where the original system is approximated by finite differences that are centered
in space and time. For example, suppose the one-dimensional shallow-water system (3.1) and (3.2) is approximated using leapfrog-time centered-second-order
space differencing as
Ö2tU + U Ö2xU + gÖ2xh = 0,
Ö2th + UÖ2xh + H Ö2xU = O.
(3.9)
(3.10)
(The finite-difference operator ö nx is defined in the Appendix by (A.1).) As in
the case of the scalar advection equation discussed in Section 2.5.1, the construc -
tion of the discrete dispersion relation mimics the procedure used to obtain the
dispersion relation for the continuous system. Wave solutions to the discretized
shallow-water equations are sought in the form
uTI. = uoe i(kj6x- wn6t)
hTl. = hoe i(kj6 x -wn6t )
J
'
(3.11)
where Uo and ho are complex constants determining the wave amplitude, and the
physically relevant portion of the solution is the real part of uj and hj. Substitution of (3.11) into the finite-differenced governing equations (3.9) and (3.10)
yields
( -
sin
+ U
sin k
Uo + g
sin k
ho = 0 ,
_ U
_
-gH
,
( -
sin
+ U
sin k
ho + H
sin k
Uo = O.
Nontrivial values of Uo and ho will satisfy the preceding pair of homogeneous
equations when the determinant of the coefficients of Uo and ho is zero, which
requires
or defining c = JgH ,
.
= -(U ± )
c sm ' k Sx .
(3.12)
In the limit of
0, this discrete-dispersion relation approaches the dispersion relation for the continuous problem ltJ = (U ± c)k .
The discrete-dispersion relation for the linearized shallow-water system is identical to that for the scalar advection equation (2.92) except that it supports two
physical modes moving-at velocities U + c and U - c. The amplitude and phasespeed error in each wave may be analyzed in the same manner as that in the scalar
advection problem (see Section 2.5.1). The analysis of amplitude error, for exarnple, proceeds by examining the amplification factor e::l(w)6t by which the waves
(3.11) grow or decay during each time step. Since the horizontal wave number (k)
of periodic waves is real, the imaginary part of ltJ will be zero unless the right side
3. Beyond the One-Way Wave Equation
where the original system is approximated by finite differences that are centered
in space and time. For example, suppose the one-dimensional shallow-water system (3.1) and (3.2) is approximated using leapfrog-time centered-second-order
space differencing as
Ö2tU + U Ö2xU + gÖ2xh = 0,
Ö2th + UÖ2xh + H Ö2xU = O.
(3.9)
(3.10)
(The finite-difference operator ö nx is defined in the Appendix by (A.1).) As in
the case of the scalar advection equation discussed in Section 2.5.1, the construc -
tion of the discrete dispersion relation mimics the procedure used to obtain the
dispersion relation for the continuous system. Wave solutions to the discretized
shallow-water equations are sought in the form
uTI. = uoe i(kj6x- wn6t)
hTl. = hoe i(kj6 x -wn6t )
J
'
(3.11)
where Uo and ho are complex constants determining the wave amplitude, and the
physically relevant portion of the solution is the real part of uj and hj. Substitution of (3.11) into the finite-differenced governing equations (3.9) and (3.10)
yields
( -
sin
+ U
sin k
Uo + g
sin k
ho = 0 ,
_ U
_
-gH
,
( -
sin
+ U
sin k
ho + H
sin k
Uo = O.
Nontrivial values of Uo and ho will satisfy the preceding pair of homogeneous
equations when the determinant of the coefficients of Uo and ho is zero, which
requires
or defining c = JgH ,
.
= -(U ± )
c sm ' k Sx .
(3.12)
In the limit of
0, this discrete-dispersion relation approaches the dispersion relation for the continuous problem ltJ = (U ± c)k .
The discrete-dispersion relation for the linearized shallow-water system is identical to that for the scalar advection equation (2.92) except that it supports two
physical modes moving-at velocities U + c and U - c. The amplitude and phasespeed error in each wave may be analyzed in the same manner as that in the scalar
advection problem (see Section 2.5.1). The analysis of amplitude error, for exarnple, proceeds by examining the amplification factor e::l(w)6t by which the waves
(3.11) grow or decay during each time step. Since the horizontal wave number (k)
of periodic waves is real, the imaginary part of ltJ will be zero unless the right side