Hermitian Grid-Characteristic Scheme for Linear Transport Equation and Its Dissipative-Dispersion Properties

Abstract

AbstractThis chapter presents an interpolation-characteristic scheme for solving a non-stationary inhomogeneous transport equation based on Hermitian interpolation of the third order of approximation. Interpolation of the solution on the lower time layer is carried out within a single cell and uses not only the nodal values of the distribution function, but also the values of its spatial derivatives. Such schemes are not new and are used by several groups of researchers. To complete the procedure of finding a solution, this scheme needs an algorithm for calculation not only the nodal values of the function, but also the nodal values of its spatial derivatives at a new time layer. For this purpose, a continued transport equation is usually used. The continued equation has the form of transport equation with respect to the spatial derivative of the function. Some algorithm for closing the resulting system of two transport equations is required. In this chapter, a variant of the Hermitian interpolation-characteristic scheme that does not require continued transport equation is proposed. The Euler–Maclaurin formula of the fourth order of approximation is used for calculation of the nodal values of spatial derivatives by calculating the integral averages over edges. This scheme might be extended for it using at non-structured tetrahedron grids. Fourier analysis of the dissipative and dispersion errors is carried out. It should be mentioned that the integral averages are responsible for the truth of conservation laws.KeywordsAdvection equationTransport equationDissipative propertiesDispersionBicompact schemeCIP methodHermitian interpolationGrid-characteristic method