**Figure 1.3.3:** *The second-order reconstruction*

Second-order of accuracy is guaranteed if the
discrete slopes approximate the corresponding derivatives,
.
Such a non-oscillatory approximation of the derivatives is possible, e.g., by using
built-in non-linear limiters of the form

Here and below, is a non-oscillatory limiter and *MM* denotes the Min-Mod
function

An *exact* evolution of *w*, based on integration of the conservation law over the
staggered cell, , then reads, (1.2.9)

The first integral is the staggered cell-average at time , ,
which can be computed directly from the above reconstruction,

The time integrals of the flux are computed by the second-order accurate
mid-point quadrature rule

Here, the Taylor expansion is being used to predict the required
mid-values of *w*

In summary, we end up with the central scheme,
[31], which consists of a
first-order *predictor step*,

followed by the second-order *corrector step*, (1.2.12),

The *scalar* non-oscillatory properties of (1.3.18)-(1.3.19)
were proved in [31],
[32], including the TVD property, cell entropy inequality,
error estimates, etc.
Moreover, the numerical experiments, reported in
[30], [31], [2],
[3], [45], [37],
[38], [39],
with one-dimensional *systems*
of conservation laws, show that such second-order central schemes enjoy the same
high-resolution as the corresponding second-order upwind schemes do.
Thus, the excessive smearing typical to the first-order LxF central scheme
is compensated here by the second-order accurate MUSCL reconstruction.

In figure 1.3.4 we compare, side by side, the upwind ULT scheme of Harten, [12], with our central scheme (1.3.18)-(1.3.19). The comparable high-resolution of this so called Lax's Riemann problem is evident.

At the same time, the central scheme (1.3.18)-(1.3.19) has the
advantage over the corresponding upwind schemes, in that no (approximate)
Riemann solvers, as in (1.2.7), are required. Hence, these Riemann-free
central schemes provide an efficient high-resolution alternative in
the one-dimensional case, and a particularly advantageous framework for
multidimensional computations, e.g., [3],
[2], [16].
This advantage in the multidimensional case will be explored in the next
section.
Also, *staggered* central differencing, along the lines
of the Riemann-free Nessyahu-Tadmor scheme (1.3.18)-(1.3.19),
admits simple efficient extensions in the presence of general source
terms, [8], and in particular, stiff source terms,
[4].
Indeed, it is a key ingredient behind the relaxation schemes studied
in [18].

It should be noted, however, that the component-wise version of these central schemes might result in deterioration of resolution at the computed extrema. The second-order computation presented in figure 1.3.2 below demonstrates this point. (this will be corrected by higher order central methods). Of course, this - so called extrema clipping, is typical to high-resolution upwind schemes as well; but it is more pronounced with our central schemes due to the built-in extrema-switching to the dissipative LxF scheme. Indeed, once an extrema cell, , is detected (by the limiter), it sets a zero slope, , in which case the second-order scheme (1.3.18)-(1.3.19) is reduced back to the first-order LxF, (1.2.14).

**Figure 1.3.4:** 2nd order: central (STG) vs.
upwind (ULT) -- Lax's Riemann problem

Mon Dec 8 17:34:34 PST 1997