We study the sonic point glitch of the Burgers' equation, which is formed in a sonic rarefaction fan. The reason of a sonic point glitch and the relation between the sonic point glitch and the entropy condition are discussed. According to these relations, several well-known schemes are classified into two and analyzed. In fact, the sonic point glitch appears only in the case of a transonic rarefaction wave. If the problem to be solved does not include transonic rarefaction waves, the difficulty in computation vanishes. Based on this idea, a new two-step splitting method eliminating the sonic point glitch is proposed. Numerical tests of applying the method to different schemes show that it is good in eliminating the sonic point glitch.

In multidimensional fluid dynamics, methods based on mesh meet difficulties frequently, especially in problems with multimaterial media and large deformation grids.In this paper, a new meshless Lagrangian finite point method to compute unsteady compressible flows is presented. In this method discrete points are distributed in the physical domain, and are regarded as Lagrangian points with mass, velocity and energy. At a given point, a "cloud" of points in the vicinity are chosen and the relations between them are set. The Lagrangian fluid equations other than the SPH ones are discreted with the Godunov method in which the interface is in a position of connect line between the given point and its neighbors. To enhance robustness and accuracy of the algorithm, more neighbor cloud points are introduced and the least square approximation is facilitated in the simulation. Computed results are good for classical examples.

A series of adaptive corrdinate transformation methods are proposed in which the grid angles are preserved approximately,and the material interface is kept to be Lagrangian description and a minimum difference (in the least-squares sense) between the mesh velocity and the fluid velocity is achieved.The new coordinate system is adapted to important features of flow fields.

It is well known that lattice Boltzmann methods(LBM) make great success in many computational physics fields,expecially in fluid mechanics.A lattice Boltzmann method with BGK model is developed to solve Burgers equation.Detailed analysis shows that the calculating scheme is a three level nonlinear finite difference one.The maximum value principle has been proved and the existence,uniqueness and stability are also discussed.The computational results agree with second order finite difference solutions very well.