158
3. Beyond the One-Way Wave Equation
Both ofthe preceding schemes are in thc general form (3.109), and thereforc both
conserve the domain integral of 4J. The domain integral of 4J2 is, however, only
conserved by (3.110) , and is only conserved when the velocity field satisfies the
discret ized continuity equation (3.107).
It should be emphasized that desirable conservation properties are not limited to
finite-difference approximations to equations in flux form. In order for (3.110) to
conserve 114J 112, the flow field must satisfy the discrete continuity equation (3.107),
but in that case the finite-differcnce approximation to the advective form (3.106)
is algebraically equivalent to the flux form (3.110), and both schemes conserve (j)
and 114J1I2.
The Effect ofTime-Differencing on Conservation
Differential-difference equations that conserve 114J 112. such as (3.106) and (3.110) ,
generally cease to be conservative when the time derivative is approximated by finite differences . Nevertheless, one type of time-differencing that does preserve the
conservation properties of linear differential-difference equations is trapezoidal
differencing. The conservation properties of trapezoidal time differencing may be
demonstrated by writing the differential-difference equation in the general form
(3.111)
where L is a linear finite-difference operator including all the spatial differences.
As a preliminary step, note that in order for the differential-difference equation
(3.111) to conserve 114J 112, the linear operator L must have the algebraic property
Lf{JjL(f{Jj) = 0,
j
(3.1 12)
where tpj is any discrete function defined on the numerical mesh and the summation is taken over all the grid points.
Approximating (3.1 11) with trapezoidal time differences yields
-
+
J
J + J
J = o.
!i.t
2
Multiplying the preceding by (4J'J+1 +4J'J), using the linearity of L,summing over
the discrete mesh, and using (3.112), one obtains
[(4J'J+1)2 - (4J'J)2J =
[(4J'J + 1 + 4J'J) L (4J'J+1 + 4J'J) J = 0,
J
which implies that
J
3. Beyond the One-Way Wave Equation
Both ofthe preceding schemes are in thc general form (3.109), and thereforc both
conserve the domain integral of 4J. The domain integral of 4J2 is, however, only
conserved by (3.110) , and is only conserved when the velocity field satisfies the
discret ized continuity equation (3.107).
It should be emphasized that desirable conservation properties are not limited to
finite-difference approximations to equations in flux form. In order for (3.110) to
conserve 114J 112, the flow field must satisfy the discrete continuity equation (3.107),
but in that case the finite-differcnce approximation to the advective form (3.106)
is algebraically equivalent to the flux form (3.110), and both schemes conserve (j)
and 114J1I2.
The Effect ofTime-Differencing on Conservation
Differential-difference equations that conserve 114J 112. such as (3.106) and (3.110) ,
generally cease to be conservative when the time derivative is approximated by finite differences . Nevertheless, one type of time-differencing that does preserve the
conservation properties of linear differential-difference equations is trapezoidal
differencing. The conservation properties of trapezoidal time differencing may be
demonstrated by writing the differential-difference equation in the general form
(3.111)
where L is a linear finite-difference operator including all the spatial differences.
As a preliminary step, note that in order for the differential-difference equation
(3.111) to conserve 114J 112, the linear operator L must have the algebraic property
Lf{JjL(f{Jj) = 0,
j
(3.1 12)
where tpj is any discrete function defined on the numerical mesh and the summation is taken over all the grid points.
Approximating (3.1 11) with trapezoidal time differences yields
-
+
J
J + J
J = o.
!i.t
2
Multiplying the preceding by (4J'J+1 +4J'J), using the linearity of L,summing over
the discrete mesh, and using (3.112), one obtains
[(4J'J+1)2 - (4J'J)2J =
[(4J'J + 1 + 4J'J) L (4J'J+1 + 4J'J) J = 0,
J
which implies that
J