4.5 The Finite-Element Method
217
In the limit k S»
0
which matches the correct amplification factor, e-iwt:.1 = e-iJ.tkt:.x, through first
order except that the scheme is second-order accurate when v = J.L. Clearly one
should choose v = J.L since this allows the largest stable time step and gives
the best accuracy. When v = J.L , (4.91) is closely related to the standard LaxWendroff approximation (2.102); the only difference appears in the linear operator (i.e., the mass matrix) acting on the forward-time difference. As noted by
Morton and Parrott (1980), if v is set equal to J.L the truncation error in (4.91) is
always less than that for the standard Lax-Wendroff scheme ; the improvement is
particularly pronounced for small values of J.L . On the other hand, the standard
Lax-Wendroff method is stable for IJ.LI ::: I, whereas the Galerkin-Petrov method
(4.91) is a upstream method that requires J.L ::: 0 for stability. A formula analogous
to (4.91) can be derived for negative flow velocities using test functions in which
the sawtooth component has a negative slope.
A better scheme than that just derived via the Petrov-Galerkin approach can be
obtained using the Taylor-Galerkin method. The Taylor-Galerkin method does
not require the specification of a second set of test functions and yields centeredin-space methods . In the Taylor-Galerkin approach, the time derivative is discretized before invoking the finite-element formalism to approximate the spatial derivatives. Donea et al. (1987) present Taylor-Galerkin approximations to
several hyperbolic problems in which the time discretization is Lax-Wendroff,
leapfrog or trapezoidal. In the following we will focus on Lax-Wendroff-type
approximations to the constant-wind-speed advection equation.
If the spatial dependence of the solution is not discretized, an 0 [(dt)2]-accurate Lax-Wendroff approximation to the constant-wind-speed advection equation has the form
-'-----'-- + c-- = - - - - ,
rP
n
+
1 - rP
n
arP
n
c
2 dt a
2rPn
dt
ax
2 ax 2
(see Section 2.5.3). Suppose that the spatial structure in the preceding differentialdifference equation is approximated using the Galerkin finite-element method
with chapeau expansion functions, then the function value at each node satisfies
(4.92)
where once again 8 x = Sx li x and J.L = c dt / Sx , The stability condition for this
scheme is I/LI ::: 1/-J3, which is identical to that for the leapfrog approximation
to (4.89). The leading-order errors in the modified equations'" for this method
and the standard finite-difference Lax- Wendroff method (2.102) are shown in
Table 4.3. The leading-order errors in both schemes are second order and genIOThe "modified equation" is discussed in Section 2.5.2.
217
In the limit k S»
0
which matches the correct amplification factor, e-iwt:.1 = e-iJ.tkt:.x, through first
order except that the scheme is second-order accurate when v = J.L. Clearly one
should choose v = J.L since this allows the largest stable time step and gives
the best accuracy. When v = J.L , (4.91) is closely related to the standard LaxWendroff approximation (2.102); the only difference appears in the linear operator (i.e., the mass matrix) acting on the forward-time difference. As noted by
Morton and Parrott (1980), if v is set equal to J.L the truncation error in (4.91) is
always less than that for the standard Lax-Wendroff scheme ; the improvement is
particularly pronounced for small values of J.L . On the other hand, the standard
Lax-Wendroff method is stable for IJ.LI ::: I, whereas the Galerkin-Petrov method
(4.91) is a upstream method that requires J.L ::: 0 for stability. A formula analogous
to (4.91) can be derived for negative flow velocities using test functions in which
the sawtooth component has a negative slope.
A better scheme than that just derived via the Petrov-Galerkin approach can be
obtained using the Taylor-Galerkin method. The Taylor-Galerkin method does
not require the specification of a second set of test functions and yields centeredin-space methods . In the Taylor-Galerkin approach, the time derivative is discretized before invoking the finite-element formalism to approximate the spatial derivatives. Donea et al. (1987) present Taylor-Galerkin approximations to
several hyperbolic problems in which the time discretization is Lax-Wendroff,
leapfrog or trapezoidal. In the following we will focus on Lax-Wendroff-type
approximations to the constant-wind-speed advection equation.
If the spatial dependence of the solution is not discretized, an 0 [(dt)2]-accurate Lax-Wendroff approximation to the constant-wind-speed advection equation has the form
-'-----'-- + c-- = - - - - ,
rP
n
+
1 - rP
n
arP
n
c
2 dt a
2rPn
dt
ax
2 ax 2
(see Section 2.5.3). Suppose that the spatial structure in the preceding differentialdifference equation is approximated using the Galerkin finite-element method
with chapeau expansion functions, then the function value at each node satisfies
(4.92)
where once again 8 x = Sx li x and J.L = c dt / Sx , The stability condition for this
scheme is I/LI ::: 1/-J3, which is identical to that for the leapfrog approximation
to (4.89). The leading-order errors in the modified equations'" for this method
and the standard finite-difference Lax- Wendroff method (2.102) are shown in
Table 4.3. The leading-order errors in both schemes are second order and genIOThe "modified equation" is discussed in Section 2.5.2.