Lagrangian numerical scheme

Lagrangian Numerical Scheme. The Lagrangian approach defines cells of whose corners, and hence boundaries, move with the local velocity. Cell corner movement is used to update densities, strains and stresses in the fluid. The velocity is then updated to complete one step in the time evolution of fluid acoustic response. 

The finite difference numerical simulation was carried out by solving the Euler equations by the Lagrangian approach. The DANE code was used for computation. The air bubble radius was 1.0 mm and the shock overpressure was 1 kbar. Computational grids were, in the axlsymmetric Cartesian coordinates, 150 x 300 and one grid size was 0.025 mm. Figure 5 shows the sequential isobars. It is clearly seen that the peak pressure appear on the side where the shock first impinged the bubble, and the bubble deformation starts which indicates the microjet initiation. However, the rebound shock is so weak that, if compared with the incident shock, the wave front could not be resolved in this numerical scheme. 

The SPH particles in the previous simulations had been spatially fixed, so that the method was used in a similar way as a mesh-based approach. However, the main advantages of a mesh-free method rely in its Lagrangian way of view with discretisation points moving according to the fluid motion. We will shortly explain the underlying algorithm and numerical schemes. For a deeper insight, we refer to... 

Ibid, pp 527-37 [A brief description of the following numerical methods for calculation a) "Finite Difference Scheme in Lagrangian Coordinates , previously described by Goad (Ref 5) b) Particle-in Cell Method, previously described by Evans Harlow (Ref 1)... 

The application of the chemical schemes to atmospheric phenomena requires a diffusion formulation that reflects time-dependence and spatial variability of meteorological conditions. An attempt has been made to keep the mathematical description near the level of detail and precision of the observational data. This has resulted in a Lagrangian air parcel formulation with finite-rate vertical diffusion. The approach avoids the artificial numerical diffusion because it uses natural (or intrinsic) coordinates that are aligned with fluid motion. This allows us simultaneously to include upward dispersion and chemical change. Figure 1 schematically illustrates the main features of the formulation. Highspeed memory requirements are limited by allowing sequential point-by-point output of the history of the air parcel. 

As mentioned above, Ryckaert et al. introduced the method of undetermined parameters, which is essentially the analytical method modified to ensure that the constraints are satisfied exactly at each time step. The method of undetermined parameters is discussed in detail, in its most general form, in a later section. Basically, the aforementioned modification requires that the highest derivatives with respect to time of the Lagrangian multipliers (i.e., of order be replaced by a set of undetermined parameters with values to be determined such that the constraints are satisfied exactly at each time step. As will be seen, the algorithm does not introduce into the trajectories additional numerical errors any worse than the error already present in the integration scheme itself. 

---