Lax-Wendroff procedure

The time derivatives in (14) can be replaced by space derivatives by applying the Lax- Wendroff procedure in order to make the maximum use of the information given by the governing PDE in (13). 

Therefore, to replace the first order time derivative in (14) by space derivatives using the Lax-Wendroff procedure, we simply have to solve (16) for Ut leading to... 

Table 7 shows the CPU times in seconds needed to complete the different steps of each ADERm scheme, m = 1,. ..,4. Here, denotes the time in CPU seconds required to construct the stencils, tr is the time to compute the reconstruction polynomials, and to is the time required for all other computations, such as the flux evaluation and the update of cell averages. Note, that to also includes the Lax-Wendroff procedure described in Subsection 4.2 to replace the time derivatives by space derivatives. The total time ttot indicates the required CPU seconds in order to complete one time step. 

The error analysis of this calculation procedure can be done using the equations in the previous section. It shows that the error made in using this scheme is of the order of 0(h + t). Thus, the scheme introduces an error term equivalent to a second-order partial differential term, which would add up to the RHS of Eq. 10.61, t.e., would decrease the apparent column efficiency. This procedure should not be used, unless very small values of the time increment t are selected. This, in turn, would make the computation time very long. In order to overcome this type of problem. Lax and Wendroff have suggested the addition to the axial dispersion term of an extra term, equivalent to the numerical dispersion term but of opposite sign [51]. This term compensates the first-order error contribution. In linear chromatography, the new finite difference equation, or Lax-Wendroff scheme, can be written as follows ... 

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