Numerical solution of shallow-water equations （SWE） has been a challenging task because of its nonlinear hyperbolic nature, admitting discontinuous solution, and the need to satisfy the C-property. The presence of s...Numerical solution of shallow-water equations （SWE） has been a challenging task because of its nonlinear hyperbolic nature, admitting discontinuous solution, and the need to satisfy the C-property. The presence of source terms in momentum equations, such as the bottom slope and friction of bed, compounds the difficulties further. In this paper, a least-squares finite-element method for the space discretization and θ-method for the time integration is developed for the 2D non-conservative SWE including the source terms. Advantages of the method include： the source terms can be approximated easily with interpolation functions, no upwind scheme is needed, as well as the resulting system equations is symmetric and positive-definite, therefore, can be solved efficiently with the conjugate gradient method. The method is applied to steady and unsteady flows, subcritical and transcritical flow over a bump, 1D and 2D circular dam-break, wave past a circular cylinder, as well as wave past a hump. Computed results show good C-property, conservation property and compare well with exact solutions and other numerical results for flows with weak and mild gradient changes, but lead to inaccurate predictions for flows with strong gradient changes and discontinuities.展开更多
The treatment of advective fluxes in high-order finite volume models is well established, but this is not the case for diffusive fluxes, due to the conflict between the discontinuous representation of the solution and...The treatment of advective fluxes in high-order finite volume models is well established, but this is not the case for diffusive fluxes, due to the conflict between the discontinuous representation of the solution and the continuous structure of analytic solutions. In this paper, a derivative reconstruction approach is proposed in the context of spectral volume methods, for the approximation of diffusive fluxes, aiming at the reconciliation of this conflict. Two different reconstructions are used for advective and diffusive fluxes： the advective reconstruction makes use of the information contained in a spectral cell, and allows the formation of discontinuities at the spectral cells boundaries; the diffusive reconstruction makes use of the information contained in contiguous spectral cells, imposing the continuity of the reconstruction at the spectral cells boundaries. The method is demonstrated by a number of numerical experiments, including the solution of shallow-water equations, complemented with the advective-diffusive transport equation of a conservative substance, showing the promising abilities of the numerical scheme proposed.展开更多
基金the National Science Council of Taiwan for funding this research (NSC 96-2221-E-019-061).
文摘Numerical solution of shallow-water equations （SWE） has been a challenging task because of its nonlinear hyperbolic nature, admitting discontinuous solution, and the need to satisfy the C-property. The presence of source terms in momentum equations, such as the bottom slope and friction of bed, compounds the difficulties further. In this paper, a least-squares finite-element method for the space discretization and θ-method for the time integration is developed for the 2D non-conservative SWE including the source terms. Advantages of the method include： the source terms can be approximated easily with interpolation functions, no upwind scheme is needed, as well as the resulting system equations is symmetric and positive-definite, therefore, can be solved efficiently with the conjugate gradient method. The method is applied to steady and unsteady flows, subcritical and transcritical flow over a bump, 1D and 2D circular dam-break, wave past a circular cylinder, as well as wave past a hump. Computed results show good C-property, conservation property and compare well with exact solutions and other numerical results for flows with weak and mild gradient changes, but lead to inaccurate predictions for flows with strong gradient changes and discontinuities.
文摘The treatment of advective fluxes in high-order finite volume models is well established, but this is not the case for diffusive fluxes, due to the conflict between the discontinuous representation of the solution and the continuous structure of analytic solutions. In this paper, a derivative reconstruction approach is proposed in the context of spectral volume methods, for the approximation of diffusive fluxes, aiming at the reconciliation of this conflict. Two different reconstructions are used for advective and diffusive fluxes： the advective reconstruction makes use of the information contained in a spectral cell, and allows the formation of discontinuities at the spectral cells boundaries; the diffusive reconstruction makes use of the information contained in contiguous spectral cells, imposing the continuity of the reconstruction at the spectral cells boundaries. The method is demonstrated by a number of numerical experiments, including the solution of shallow-water equations, complemented with the advective-diffusive transport equation of a conservative substance, showing the promising abilities of the numerical scheme proposed.
