Spectral Vanishing Viscosity Method for Nonlinear
نویسندگان
چکیده
We propose a new spectral viscosity (SV) scheme for the accurate solution of nonlinear conservation laws. It is proved that the SV solution converges to the unique entropy solution under appropriate reasonable conditions. The proposed SV scheme is implemented directly on high modes of the computed solution. This should be compared with the original nonperiodic SV scheme introduced by Maday, Ould Kaber, and Tadmor in [SIAM J. Numer. Anal., 30 (1993), 321–342], where SV is activated on the derivative of the SV solution. The new proposed SV method could be viewed as a correction of the former, and it offers an improvement which is confirmed by our numerical experiments. A postprocessing method is implemented to greatly enhance the accuracy of the computed SV solution. The numerical results show the efficiency of the new method.
منابع مشابه
Computer Methods in Applied Mechanics and Engineering 80 (1990) 197-208 North-holland Shock Capturing by the Spectral Viscosity Method*
A main disadvantage of using spectral methods for nonlinear conservation laws lies in the formation of Gibbs phenomenon, once spontaneous shock discontinuities appear in the solution. The global nature of spectral methods then pollutes the unstable Gibbs oscillations over all the computational domain, and the lack of entropy dissipation prevents convergences in these cases. In this paper, we di...
متن کاملVanishing Moment Method and Moment Solutions for Fully Nonlinear Second Order Partial Differential Equations
This paper concerns with numerical approximations of solutions of fully nonlinear second order partial differential equations (PDEs). A new notion of weak solutions, called moment solutions, is introduced for fully nonlinear second order PDEs. Unlike viscosity solutions, moment solutions are defined by a constructive method, called the vanishing moment method, and hence, they can be readily com...
متن کاملA spectral vanishing viscosity for the LES of turbulent flows within rotating cavities
A spectral vanishing viscosity technique (SVV) is presented for the simulation of 3D turbulent incompressible flows within a rotor–stator cavity. One characteristic of this technique is that the SVV is active only for the short length scales, a feature which is reminiscent of Large Eddy Simulation models. The Spectral Vanishing Viscosity, first introduced by E. Tadmor for the inviscid Burgers e...
متن کاملFractional Burgers equation with nonlinear non-locality: Spectral vanishing viscosity and local discontinuous Galerkin methods
We consider the viscous Burgers equation with a fractional nonlinear term as a model involving non-local nonlinearities in conservation laws, which, surprisingly, has an analytical solution obtained by a fractional extension of the Hopf-Cole transformation. We use this model and its inviscid limit to develop stable spectral and discontinuous Galerkin spectral element methods by employing the co...
متن کاملOn the spectral vanishing viscosity method for periodic fractional conservation laws
Abstract. We introduce and analyze a spectral vanishing viscosity approximation of periodic fractional conservation laws. The fractional part of these equations can be a fractional Laplacian or other non-local operators that are generators of pure jump Lévy processes. To accommodate for shock solutions, we first extend to the periodic setting the Kružkov-Alibaud entropy formulation and prove we...
متن کامل