Development of multiquadric method for solving dam break problem

Introduction:
In order to avoid meshing and its difficulties and costs, Multiquadric Radial Basis Function (MQ-RBF) method has been developed (Kansa 1990) and has been examined for different types of physical phenomena. In this regard, the present study develops this meshless method for analysis of dam break problem. MQ is more convenient and accurate than other RBF methods for solving partial differential equations (Fallah et al. 2019). This meshless method have advantages such as; 1) creating a continuous response function all over the computational domain, 2) no need to discretize the entire domain with optimal usability in large-scale problems, 3) high capability in modelling irregular and complex geometries, 4) high ability to simulate discontinuities of responses, 5) easy generalization to 3D problems, and etc. Both the accuracy and the convergence rate of MQ depend strongly on its shape parameter (Koushki et al. 2019). So far, researchers have been working on many methods for determining the optimal shape parameter but a comprehensive method has not been developed yet (Babaee et al. 2019). In this study, the commonly previous methods have been investigated for determining the optimal shape parameter and a novel idea has been presented for analyzing the flood flow caused by dam break. The efficiency and accuracy of the present approach compared to other solutions have been shown through three examples.
Methodology:
The governing PDEs of dam break problem consist of the continuity equation and two momentum equations in two dimensions. MQ approximates solution of 2D equations system using an estimation function in which the unknown coefficients have to be determined for each unknown variable of the PDE, i.e. the velocities in two directions and the pressure. In one hand, for definition of the estimation function, the RBF methods need N center points inside the domain or on the boundaries which leads to N unknown coefficients. On the other hand, the governing PDEs and their boundary conditions again have to be satisfied on N collocation points which leads to N algebraic equations to be solved for the mentioned unknown coefficients. A critical parameter, namely, the shape parameter strongly affects the precision of the estimation function which may be considered constant or variable from point to point for each estimation function. Determining the optimal value of the shape parameter has always been a challenge in using MQ and other RBF methods. In this study it has been shown that the shape parameter in all time steps can be equal and a new high-speed idea is proposed to determine its optimal value. In this approach, the initial conditions of the problem will be estimated using MQ function and it has been shown that the optimal value of the shape parameter in the initial conditions is also the optimal value of the shape parameter for the next time steps and there is no need to be optimized for every next time steps. Therefore the computational cost will be considerably reduced. Also for discretizing the time dependent terms, the forward finite difference method is used and it was shown that for discretizing the local terms, the implicit method must be used by substituting MQ function. Consequently, the presented approach becomes unconditionally stable.
In order to verify and validate the proposed approach, three 2D numerical examples are presented. In two of the examples with 1D and 2D behaviors, discontinuities in initial conditions and run times are different. Sharp discontinuities highlight the capabilities of the approach while in long run time shows stability. Besides, results of the proposed approach have been compared with those of other numerical and analytical methods. Also, in this research, inefficiency of previously common methods for determining the optimal shape parameter in solving the dam-break problem was shown (Golbabai et al. 2015). In verification, the RMSE error criterion has been considered which results in errors less than 5 percent. In the third example capability of the numerical model has been demonstrated by a two dimensional dam break flow.
Conclusion:
Using the MQ-RBF, the disadvantages of mesh-based method including; high cost of meshing, need to fundamental solution dependence on the conditions of each problem, singularity, continuous discretization of domain and need to a regular mesh will be eliminated.
The benefits of the proposed idea for the optimal value of the shape parameter respect to existing methods include 1) independency of a secondary solution of the problem, 2) solving problem using just one set of computational centers, 3) independency of the geometry and physics of the problems, 4) needless to optimize the shape parameter at every time step, 5) low computational cost and 6) convenient to use. Finally result of several different examples compared to other numerical and analytical methods showed acceptable accuracy of the present approach.