Abstract
In this work, we are concerned with the study and computing of stabilized radial basis function-generated finite difference (RBF-FD) approximations for shallow-water equations. In order to obtain both stable and highly accurate numerical approximations of convection-dominated shallow-water equations, we use stabilized flat Gaussians (RBFSGA-FD) and polyharmonic splines with supplementary polynomials (RBFPHS-FD) as basis functions, combined with modified method of characteristics. These techniques are combined with careful design for the spatial derivative operators in the momentum flux equation, according to a general criterion for the exact preservation of the “lake at rest” solution in general mesh-based and meshless numerical schemes for the strong form of the shallow-water equations with bottom topography. Both structured and unsructured point clouds are employed for evaluating the influence of cloud refinement, size of local supports and maximal permissible degree of the polynomials in RBFPHS-FD.
Highlights
Radial basis function-generated finite difference (RBFFD) meshless schemes have been applied in many research areas such as fluid dynamics, heat transfer and seismic modeling, but still, the state of art in this field is far more developed in elliptic problems [1] in timedependent convection-dominated problems
The results show that meshless radial basis function-generated finite difference (RBF-FD) methods produces very promissing results, satisfying both “lake at rest” and stability conditions and obtainning highly accurate approximattions for shallow-water equations’ solution
Recent researches [2][3] show that RBF-FD meshless schemes may have an important role on development of stabilized and highly accurate numerical methods for convection-dominated problems
Summary
Radial basis function-generated finite difference (RBFFD) meshless schemes have been applied in many research areas such as fluid dynamics, heat transfer and seismic modeling, but still, the state of art in this field is far more developed in elliptic problems [1] in timedependent convection-dominated problems. A stabilized well-balanced RBF-FD meshless method for shallow-water equations. We are concerned with the study and computing of stabilized radial basis function-generated finite difference (RBF-FD) approximations for shallow-water equations.
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