Lax-Wendroff scheme

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 ... 

The gas-dynamics of the studied problem is described by a system of two-dimensional non-stationaiy Euler equations [87, 88]. The system is solved by the Lax-Wendroff scheme together with a flow correction algorithm for shock capturing. The detailed solution of the chemical reactions uses CHEMKIN-II and is included in the calculations. The solution domain consists of a half of a rectangular tube, divided into 400 X 200 cells. The spatial resolution in both x and y directions is uniform and equal to 0.3 mm. 

Lax-Wendroff. This is a well known method to solve first-order hyperbolic partial differential equations in boundary value problems. The two step Richtmeyer implementation of the explicit Lax-Wendroff differential scheme is used (8). 

The Lax-Wendroff method is less accurate, but is faster than the method of lines. The scheme is also sensitive for discontinuities in Gn, although the oscillations are less severe. The first-order Lax scheme is not reliable. 

The solver used for this study is the same as in Chapter 9 a parallel fully compressible code for turbulent reacting two-phase flows, on both structured and unstructured grids. The fully explicit finite volume solver uses a cell-vertex discretization with a Lax-Wendroff centered numerical scheme [296] or a third order in space and time scheme named TTGC [268]. Characteristic boundary conditions NSCBC [339 329] are used for the gas phase. Boundary conditions are easier for the dispersed phase, except for solid walls where particles may bounce off. In the present study it is simply supposed that the particles stick to the wall, with either a slip or zero velocity. 

With the Lax-Wendroff two-step scheme, the difference term found in the RHS of Eq. 10.76 is a central difference term. The computational error made in this calculation scheme is of the order of O ( ). Thus, if we are interested in solving the... 

This equation is equivalent to Eq. 10.71 with Da = 0, which is expected because we are now writing the Lax-Wendroff equation for solving Eq. 10.72, which is equivalent to Eq. 10.61 with Da = 0. However, in this chapter we are interested in solving the equilibrium-dispersive model (Eq. 10.61), not the ideal model (Eq. 10.72). So, neither the Lax-Wendroff nor any similar scheme which gives a high-order truncation error is suitable for our purpose. The error made is too small to account for the dispersive effects in an actual column. 

A variety of explicit (Dufort-Frankel, Lax-Wendroff, Runge-Kutta) and implicit (approximate factorization, LU-SGS) or hybrid schemes have been employed for integration in time. Because of the complexity of the incompressible Navier-Stokes equations, stability analyses to determine critical time steps are difficult. As a general rule, the allowable time step for an explicit method is proportional to the ratio of the smallest grid size to the largest convective velocity (or the wave propagation speed for an artificial compressibility method). 

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. 

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