In this paper, a family of sub-cell finite volume schemes for solving the hyperbolic conservation laws is proposed and analyzed in one-dimensional cases. The basic idea of this method is to subdivide a control volume(...In this paper, a family of sub-cell finite volume schemes for solving the hyperbolic conservation laws is proposed and analyzed in one-dimensional cases. The basic idea of this method is to subdivide a control volume(main cell) into several sub-cells and the finite volume discretization is applied to each of the sub-cells. The averaged values on the sub-cells of current and face neighboring main cells are used to reconstruct the polynomial distributions of the dependent variables. This method can achieve arbitrarily high order of accuracy using a compact stencil. It is similar to the spectral volume method incorporating with PNPM technique but with fundamental differences. An elaborate utilization of these differences overcomes some shortcomings of the spectral volume method and results in a family of accurate and robust schemes for solving the hyperbolic conservation laws. In this paper, the basic formulation of the proposed method is presented. The Fourier analysis is performed to study the properties of the one-dimensional schemes. A WENO limiter based on the secondary reconstruction is constructed.展开更多
In this paper, the problem of determining the most general Lie point symmetries group and conservation laws of a well known nonlinear hyperbolic PDE in mathematical physics called the Hunter-Saxton equation （HSE） is...In this paper, the problem of determining the most general Lie point symmetries group and conservation laws of a well known nonlinear hyperbolic PDE in mathematical physics called the Hunter-Saxton equation （HSE） is anaiyzed. By applying the basic Lie symmetry method for the HSE, the classical Lie point symmetry operators are obtained. Also, the algebraic structure of the Lie algebra of symmetries is discussed and an optimal system of one- dimensional subalgebras of the HSE symmetry algebra which creates the preliminary classification of group invariant solutions is constructed. Particularly, the Lie invariants as well as similarity reduced equations corresponding to in- finitesimal symmetries are obtained. Mainly, the conservation laws of the HSE are computed via three different methods including Boyer＇s generalization of Noether＇s theorem, first homotopy method and second homotopy method.展开更多
基金supported by the National Natural Science Foundation of China(Grant Nos.U1430235,and 11672160)
文摘In this paper, a family of sub-cell finite volume schemes for solving the hyperbolic conservation laws is proposed and analyzed in one-dimensional cases. The basic idea of this method is to subdivide a control volume(main cell) into several sub-cells and the finite volume discretization is applied to each of the sub-cells. The averaged values on the sub-cells of current and face neighboring main cells are used to reconstruct the polynomial distributions of the dependent variables. This method can achieve arbitrarily high order of accuracy using a compact stencil. It is similar to the spectral volume method incorporating with PNPM technique but with fundamental differences. An elaborate utilization of these differences overcomes some shortcomings of the spectral volume method and results in a family of accurate and robust schemes for solving the hyperbolic conservation laws. In this paper, the basic formulation of the proposed method is presented. The Fourier analysis is performed to study the properties of the one-dimensional schemes. A WENO limiter based on the secondary reconstruction is constructed.
文摘In this paper, the problem of determining the most general Lie point symmetries group and conservation laws of a well known nonlinear hyperbolic PDE in mathematical physics called the Hunter-Saxton equation （HSE） is anaiyzed. By applying the basic Lie symmetry method for the HSE, the classical Lie point symmetry operators are obtained. Also, the algebraic structure of the Lie algebra of symmetries is discussed and an optimal system of one- dimensional subalgebras of the HSE symmetry algebra which creates the preliminary classification of group invariant solutions is constructed. Particularly, the Lie invariants as well as similarity reduced equations corresponding to in- finitesimal symmetries are obtained. Mainly, the conservation laws of the HSE are computed via three different methods including Boyer＇s generalization of Noether＇s theorem, first homotopy method and second homotopy method.
