5.6 Approximation withLocal Polynomials
275
method suitable for problems with discontinuous solutions, it is necessary to modify the slope of the piecewise-linear interpolating functions near discontinuities
and poorly resolved gradients. This modification of the slope is accomplished
using "slope-limiting" algorithms that are closely related to the flux-limiting procedures discussed in Section 5.5.2.
Although the actual computations may be organized in a more efficient manner,
the procedure for advancing the numerical solution one time step is equivalent to
the following three-step process. In the first step the fully discrete solution is used
to define a piecewise-Iinear function within each grid cell of the form
for x J
' -2
I < X < x
-
-
J
'+ I •
2
In the second step the conservation law is integrated over a time !:!..t using ;P(x, t n )
as the initial condition. In the third and final step, tPj+1 is obtained by averaging
;P(x , t n +I) over each grid cell. The special considerations required to keep this
method TVD are entirely connected with the first step, since if the conservation
law is of the form (5.6), the solution obtained in the second step is TVD and the
averaging in the third step does not increase the total variation . The first step is
kept TVD by imposing limits on the slopes uJ.The most severe limitation would
be to set uJ = 0, in which case the seheme reduces to Godunov 's method .
In comparison with Godunov's method, in which exact solutions of the conservation law can be obtained relatively easily at each cell interface by solving a
series of Riemann problems, the problems to be solved at each interface in step
two of the piecewise-linear method are more difficult, because ;p is not constant
on each side of the initial discontinuity. General techniques for obtaining acceptable approximations to the solution required in step two are discussed in LeVeque (1992). In the following we will once again focus on the special case of the
constant-wind-speed advection equation (5.18), for which the solution required in
;P(x, t n + l ) = ;P(x - cSt , t"),
step two is simply
Assuming that e > 0 and averaging ;P(x, t n+ l) over (xj_!' xj+!)' steps two and
three yield
tPt
l = tPi - JL (tPi - tPi-l) -
- JL)!:!..x(Uj - Uj_)) ,
(5.48)
where JL = ctst / Sx, If the slopes of the piecewise-linear functions are defined in
step one such that
tPj+1 - tPj
Uj =
!:!..X
'
(5.49)
then (5.48) reduces to the Lax-Wendroff method, which is not TVD.
In order to make the preceding scheme TVD, the slope can be limited by a
multiplicative constant C j+! such that
(5.50)
275
method suitable for problems with discontinuous solutions, it is necessary to modify the slope of the piecewise-linear interpolating functions near discontinuities
and poorly resolved gradients. This modification of the slope is accomplished
using "slope-limiting" algorithms that are closely related to the flux-limiting procedures discussed in Section 5.5.2.
Although the actual computations may be organized in a more efficient manner,
the procedure for advancing the numerical solution one time step is equivalent to
the following three-step process. In the first step the fully discrete solution is used
to define a piecewise-Iinear function within each grid cell of the form
for x J
' -2
I < X < x
-
-
J
'+ I •
2
In the second step the conservation law is integrated over a time !:!..t using ;P(x, t n )
as the initial condition. In the third and final step, tPj+1 is obtained by averaging
;P(x , t n +I) over each grid cell. The special considerations required to keep this
method TVD are entirely connected with the first step, since if the conservation
law is of the form (5.6), the solution obtained in the second step is TVD and the
averaging in the third step does not increase the total variation . The first step is
kept TVD by imposing limits on the slopes uJ.The most severe limitation would
be to set uJ = 0, in which case the seheme reduces to Godunov 's method .
In comparison with Godunov's method, in which exact solutions of the conservation law can be obtained relatively easily at each cell interface by solving a
series of Riemann problems, the problems to be solved at each interface in step
two of the piecewise-linear method are more difficult, because ;p is not constant
on each side of the initial discontinuity. General techniques for obtaining acceptable approximations to the solution required in step two are discussed in LeVeque (1992). In the following we will once again focus on the special case of the
constant-wind-speed advection equation (5.18), for which the solution required in
;P(x, t n + l ) = ;P(x - cSt , t"),
step two is simply
Assuming that e > 0 and averaging ;P(x, t n+ l) over (xj_!' xj+!)' steps two and
three yield
tPt
l = tPi - JL (tPi - tPi-l) -
- JL)!:!..x(Uj - Uj_)) ,
(5.48)
where JL = ctst / Sx, If the slopes of the piecewise-linear functions are defined in
step one such that
tPj+1 - tPj
Uj =
!:!..X
'
(5.49)
then (5.48) reduces to the Lax-Wendroff method, which is not TVD.
In order to make the preceding scheme TVD, the slope can be limited by a
multiplicative constant C j+! such that
(5.50)