Introduction
In this research, the shallow water equations of dam-break flow have been rewritten in the semi-implicit form to create a system of linear equations. The existing methods used for solving the dam break problem have their pros and cons. For example, the finite volume method requires meshing of the entire problem domain; consequently, in large-scale domains such as the dam break problem, due to using large finite volumes, accuracy will be reduced. The finite difference method also requires points on a regular grid to define the geometry of the problem. Furthermore, such methods obtain the solutions in specific computational nodes or elements of the domain while other points require the interpolation procedure, in which, more errors may occur. Besides, most of the numerical methods for modeling discontinuities, which essentially occur in the dam-break problem, have a significant error and computational cost. In this research, the Multiquadric (MQ) method was employed to resolve some of the mentioned weaknesses in solving the problem. The development is carried out in solving systems of unsteady non-linear differential equations by making more convenient the implementation of Radial Basis Functions (RBFs). Since the accuracy of the MQ method drastically depends on its optimal shape parameter, an efficient optimization approach is implemented (Kahid et al., 2020).
Methodology
The Partial Differential Equations (PDE) of the dam break problem include the continuity and momentum equations in the x and y directions. In this research, the system of nonlinear equations is reproduced to solve the dam break problem due to the presence of slope and friction of the bed in the flow. Also, the forward finite difference method is used to discretize the time derivatives. Besides, a previously experienced optimization scheme (Kahid et al., 2020), is used for finding the optimal shape parameter as the critical factor in improving the accuracy of the method. In addition, in this approach, the initial conditions of the problem are estimated using the MQ function (for finding the optimized shape parameters), and it may be shown that the obtained optimal shape parameters are also the optimal values for the next time steps. Therefore, there is no need for the shape parameter to be optimized for all subsequent time steps. As a result, the computational cost was significantly reduced. Furthermore, to reduce the non-physical or numerical oscillations, viscous terms were added to the governing equations, which are coefficients of the second derivatives in the x and y directions. In this regard, the optimal values of the unknown coefficients were found as -0.25 and -0.25 in the x and y directions, respectively.
Results and Discussion
MQ-RBF resolves the disadvantages of mesh-based methods, such as the high cost of meshing, the need for interpolation between the grid nodes, etc. In this research, the MQ method has been successfully used to model the dam break problem by considering the friction and slope of the bed. An optimization scheme is used to find the optimal shape parameter as the main critical factor in increasing the accuracy of the MQ method. For instance, six numerical examples are presented to validate the reliability of the proposed approach, for 1D and 2D domains. The examples highlight the capabilities and demonstrate the long-term stability of the proposed approach. In addition, the results of the proposed approach were compared with other numerical and analytical methods (Stoker, 1992), which showed that the solution obtained by the present approach has an acceptable precision compared to the other solutions. Also, this approach may be developed to be used in solving partial and gradual dam break problems.
Conclusion
In this research, the dam break problem has been solved using the meshless MQ method in the presence of the bed and friction slopes. Besides, using viscous terms the numerical oscillations in the discontinuities were successfully captured. In this regard, the optimal values of the coefficients of the viscous terms were found. The capability and reliability of the numerical model were examined through some 1D and 2D examples. The results showed that the proposed method has acceptable accuracy in simulating the flood flow caused by the dam failure including the friction and slope of the bed, as well as its shock front, with less computational costs compared to other numerical methods.