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The second-order Nessyahu-Tadmor scheme

  In this section we overview the construction of high-resolution central schemes in one-space dimension. We begin with the reconstruction of the second-order, non-oscillatory Nessyahu and Tadmor (NT) scheme, [31]. To approximate solutions of (1.2.1), we introduce a piecewise-linear approximate solution at the discrete time levels, tex2html_wrap_inline5391, based on linear functions tex2html_wrap_inline5495 which are supported at the cells tex2html_wrap_inline5419 (see Figure 1.3.3),

Figure 1.3.3: The second-order reconstruction 

Second-order of accuracy is guaranteed if the discrete slopes approximate the corresponding derivatives, tex2html_wrap_inline5499. 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, tex2html_wrap_inline5501 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, tex2html_wrap_inline5507, then reads, (1.2.9)
The first integral is the staggered cell-average at time tex2html_wrap_inline5509, tex2html_wrap_inline5511, 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, tex2html_wrap_inline5515 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, tex2html_wrap_inline5419, is detected (by the limiter), it sets a zero slope, tex2html_wrap_inline5519, 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

next up previous contents
Next: The third-order central scheme Up: Central schemes in one-space Previous: Central schemes in one-space

Eitan Tadmor
Mon Dec 8 17:34:34 PST 1997