The procedure of Lax-Wendroff scheme is shown in eq.(65)-(66). This is a second order scheme and two computations are needed to compute a full time step. The result of the Lax-Wendroff scheme for shock tube problem is shown in Figure 7. From the figure, we can see that the simulation result fits the analytical result very well. Especially in those sharp changing places, it has excellent behavior. From the picture we can see two obvious "overshoots", this is because of numerical error. This error can be eliminated by adding artificial viscosity. The reason why we did not add it is because, first, it would make it more complex, second, we can see from this why further efforts are needed to make the simulation result much better. From the figure, we can also see that there are some wiggles. This is another characteristics of the Lax-Wendroff scheme(there is no such problem in Rusanov scheme). Each scheme has certain characteristics. Some schemes may have a strong point in some certain aspect, while it is weak in some other aspect. All schemes are like this. We must understand the characteristics of these schemes and to develop an ability to know when to use which one.
|Figure 7: Result of Lax-Wendroff scheme for one-dimensional shock-tube simulation.|
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